0,11

B_{2n} = (-1)^(m-1)/2^(2m+1) * Integral{-inf..inf, [d^(m-1)/dx^(m-1) sech(x)^2 ]^2 dx} (see Grosset/Veselov).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810.

H. Bergmann, Eine explizite Darstellung der Bernoullischen Zahlen, Math. Nach. 34 (1967), 377-378. Math Rev 36#4030.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 230.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.1.

H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.6.

L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.

H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.

K. Dilcher, A Bibliography of Bernoulli Numbers (Alphabetically Indexed Authorwise)

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

Niels Nielsen, Traite Elementaire des Nombres de Bernoulli, Gauthier-Villars, 1923, pp. 398.

S. Plouffe, The First 498 Bernoulli numbers [Project Gutenberg Etext]

E. W. Weisstein, More information.

Index entries for sequences related to Bernoulli numbers.

Index entries for "core" sequences

M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons

E.g.f: x/(e^x - 1). Recurrence: B^n = (1+B)^n, n >= 2 (interpreting B^j as B_j).

B_{2n}/(2n)! = 2*(-1)^(n-1)*(2*Pi)^(-2n) Sum_{k=1..inf} 1/k^(2n) (gives asymptotics) - Rademacher, p. 16, Eq. (9.1). In particular, B_{2*n} ~ (-1)^(n-1)*2*(2*n)!/(2*Pi)^(2*n).

Sum_{i=1..n-1} i^k = ((n+B)^(k+1)-B^(k+1))/(k+1) (interpreting B^j as B_j).

B_{n-1} = - Sum_{r=1..n} (-1)^r binomial(n,r) r^(-1) Sum_{k=1..r} k^(n-1). More concisely, B_n = 1 - (1-C)^(n+1), where C^r is replaced by the arithmetic mean of the first r n-th powers of natural numbers in the expansion of the right-hand side. [Bergmann]

Sum_{i=1..inf} 1/i^(2k) = zeta(2k) = (2*Pi)^(2k)*|B_{2k}|/(2*(2k)!).

B_n sequence begins 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, ...

B := proc(n) sum( (-1)^'m'*'m'!*combinat[stirling2](n, 'm')/('m'+1), 'm'=0..n); end;

B := proc(n) numtheory[bernoulli](n); end;

Table[ Numerator[ BernoulliB[ n]], {n, 0, 40}] (from Robert G. Wilson v Oct 11 2004)

Numerator[ Range[0, 40]! CoefficientList[ Series[x/(E^x - 1), {x, 0, 40}], x]]

(PARI) a(n)=if(n<0, 0, numerator(bernfrac(n)))

This is the main entry for the Bernoulli numbers, and has all the references, links and formulae. Sequences A027642 (the denominators of B_n), and A000367/A002445 = B_{2n} are also important!

Cf. A027642, A000146, A000367, A002445.

Sequence in context: A103709 A073911 A036946 this_sequence A098173 A058177 A079508

Adjacent sequences: A027638 A027639 A027640 this_sequence A027642 A027643 A027644

sign,frac,nice

njas

0,2

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810.

.....................................................

0,5

C(n+k-1,n-1) is the number of ways of placing k indistinguishable balls into n boxes (the "bars and stars" argument - see Feller).

Row n gives coefficients in expansion of (1+x)^n.

C(n-1,m-1) is the number of compositions of n with m summands.

If thought of as an infinite lower triangular matrix, inverse begins:

+1

-1 +1

+1 -2 +1

-1 +3 -3 +1

+1 -4 +6 -4 +1

The string of 2^n palindromic binomial coefficients starting after the A006516(n)-th entry are all odd. - Lekraj Beedassy (boodhiman(AT)yahoo.com), May 20 2003

C(n+k-1,n-1) is the number of standard tableaux of shape (n,1^k). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 13 2004

Can be viewed as an array, read by antidiagonals, where the entries in the first row and column are all 1s, and A(i,j) = A(i-1,j) + A(i,j-1) for all other entries. The determinants of all its n X n subarrays starting at (0,0) are all 1. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 17 2004

Equals differences between consecutive terms of A102363 - David G. Williams (davidwilliams(AT)Paxway.com), Jan 23 2006

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 63ff.

B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 4.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 306.

W. Feller, An Introduction to Probability Theory and Its Application, Vol. 1, 2nd ed. New York: Wiley, p. 36, 1968.

D. Fowler, The binomial coefficient function, Amer. Math. Monthly, 103 (1996), 1-17.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, p. 155.

D. E. Knuth, The Art of Computer Programming, Vol. 1, 2nd ed., p. 52.

S. K. Lando, Lecture on Generating Functions, Amer. Math. Soc., Providence, R.I., 2003, pp. 60-61.

D. Merlini, F. Uncini and M. C. Verri, A unified approach to the study of general and palindromic compositions, Integers 4 (2004), A23, 26 pp.

Y. Moshe, The density of 0's in recurrence double sequences, J. Number Theory, 103 (2003), 109-121.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 6.

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 2.

R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996, p. 143.

L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, The Riordan group, Discrete Applied Math., 34 (1991), 229-239.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp 115-8, Penguin Books 1987.

Anonymous, Pascal's Triangle and Its Patterns

V. Asundi, Generate a Yanghui Triangle

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps

L. Euler, On the expansion of the power of any polynomial (1+x+x^2+x^3+x^4+etc)^n

L. Euler, De evolutione potestatis polynomialis cuiuscunque (1+x+x^2+x^3+x^4+etc)^n E709

Matthew Hubbard and Tom Roby, Pascal's Triangle From Top to Bottom

S. Kak, The Golden Mean and the Physics of Aesthetics

W. Knight, Short Table of Binomial Coefficients

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Mathforum, Pascal's Triangle

Mathforum, Links for Pascal's triangle

C. McDermottroe, n-th row generator of Pascal's triangle

D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees

A. Necer, Series formelles et produit de Hadamard

G. Sivek et al., ThinkQuest, Pascal's Triangle Row Generator

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

H. Verrill, Pascal's Triangle and related triangles

G. Villemin's Almanach of Numbers, Triangle de Pascal

E. W. Weisstein, More information.

Thomas Wieder, Home Page.

Thomas Wieder, (Old) Home Page.

Wikipedia, Pascal's triangle

H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, pp. 12ff.

K. Williams, Mathforum, Interactive Pascal's Triangle

K. Williams, MathForum, Pascal's Triangle to Row 19

D. Zeilberger, [math/9809136] The Combinatorial Astrology of Rabbi Abraham Ibn Ezra

Index entries for triangles and arrays related to Pascal's triangle

a(n,m)=binomial(n,m); a(n+1,m) = a(n,m)+a(n,m-1), a(n,-1) := 0, a(n,m) := 0, n<m; a(0,0)=1.

C(n,k)=n!/(k!(n-k)!) if 0<=k<=n, otherwise 0.

G.f.: 1/(1-y-xy)=Sum(C(n,k)x^k*y^n, n,k>=0); also g.f.: 1/(1-x-y)=Sum(C(n+k,k)x^k*y^n, n,k>=0). G.f. for row n: (1+x)^n = sum(k=0..n,C(n,k)x^k). G.f. for column n: x^n/(1-x)^n.

E.g.f.: A(x,y)=exp(x+xy). E.g.f. for column n: x^n*exp(x)/n!.

In general the m-th power of A007318 is given by: T(0,0) = 1, T(n,k) = T(n-1,k-1) + m*T(n-1,k), where n is the row-index and k is the column; also T(n,k) = m^(n-k) C(n,k).

Triangle T(n, k) read by rows; given by A000007 DELTA A000007, where DELTA is Deleham's operator defined in A084938.

With P(n+1) = the number of integer partitions of (n+1), p(i) = the number of parts of the i-th partition of (n+1), d(i) = the number of different parts of the i-th partition of (n+1), m(i,j) = multiplicity of the j-th part of the i-th partition of (n+1), sum_[p(i)=k]_{i=1}^{P(n+1)} = sum running from i=1 to i=P(n+1) but taking only partitions with p(i)=(k+1) parts into account, prod_{j=1}^{d(i)} = product running from j=1 to j=d(i) one has B(n,k) = sum_[p(i)=(k+1)]_{i=1}^{P(n+1)} 1/prod_{j=1}^{d(i)} m(i,j)! E.g. B(5,3) = 10, because n=6 has the following partitions with m=3 parts: (114), (123), (222). For their multiplicities one has: (114): 3!/(2!*1!) = 3, (123): 3!/(1!*1!*1!) = 6, (222): 3!/3! = 1. The sum is 3+6+1=10=B(5,3). - Thomas Wieder (wieder.thomas(AT)t-online.de), Jun 03 2005

C(n,k) = Sum_{j, 0<=j<=k} = (-1)^j*C(n+1+j,k-j)*A000108(j) . - Philippe DELEHAM (kolotoko(AT)lagoon.nc), Oct 10 2005

Triangle begins:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

A007318 := (n, k)->binomial(n, k);

Flatten[Table[Binomial[n, k], {n, 0, 11}, {k, 0, n}]] (from Robert G. Wilson v Jan 19 2004)

(PARI) C(n, k)=if(k<0|k>n, 0, n!/k!/(n-k)!)

(PARI) C(n, k)=if(n<0, 0, polcoeff((1+x)^n, k))

(PARI) C(n, k)=if(k<0|k>n, 0, if(k==0&n==0, 1, C(n-1, k)+C(n-1, k-1)))

Cf. A047999, A026729, A052553. Row sums give A000079 (powers of 2).

Cf. A083093 (triangle read mod 3).

Partial sums of rows give triangle A008949.

Infinite matrix squared: A038207, cubed: A027465

Cf. A101164. If rows are sorted we get A061554 or A107430.

Another version: A108044.

Cf. A008277.

Sequence in context: A095144 A034932 A094495 this_sequence A108086 A108363 A076831

Adjacent sequences: A007315 A007316 A007317 this_sequence A007319 A007320 A007321

nonn,tabl,nice,easy,new

njas, Mira Bernstein (mira(AT)math.berkeley.edu)

1,2

Sum_{m = 1..inf } 1/m^2.

"In 1736 he [Leonard Euler, 1707-1783] discovered the limit to the infinite series, Sum 1/n^2. He did it by doing some rather ingenious mathematics using trigonometric functions that proved the series summed to exactly Pi^2/6. How can this be? ... This demonstrates one of the most startling characteristics of mathematics - the interconnectedness of, seemingly, unrelated ideas.", Clawson

Also dilogarithm(1). - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Jul 21 2004

Also Integral_{x=0..inf} x/(exp(x)-1).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

R. Calinger, "Leonard Euler: The First St. Petersburg Years (1727-1741)," Historia Mathematica, Vol. 23, 1996, pp. 121-166.

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Perseus Books, 1996, p. 97.

W. Dunham, Euler: The Master of Us All, The Mathematical Association of America, Washington, D.C., 1999, p. xxii.

A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhaeuser, Boston, 1984; see p. 261.

David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 23.

R. Chapman, Evaluating Zeta(2):14 Proofs to Zeta(2)= (pi)^2/6

R. W. Clickery, Probability of two numbers being coprime

Math. Reference Project, The Zeta Function, Zeta(2)

Math. Reference Project, The Zeta Function, Odds That Two Numbers Are Coprime"

J. Perry, Prime Product Paradox

S. Plouffe, Plouffe's Inverter, Zeta(2) or Pi**2/6 to 100000 digits

S. Plouffe, Zeta(2) or Pi**2/6 to 10000 places

C. Tooth, Pi squared over six

E. W. Weisstein, Link to a section of The World of Mathematics

H. Wilf, Accelerated series for universal constants, by the WZ method

E. W. Weisstein, Dilogarithm MathWorld page

D. H. Bailey, J. M. Borwein and D. M. Bradley, Experimental determination of Ap'ery-like identities for zeta(4n+2)

L. Euler, On the sums of series of reciprocals

L. Euler, De summis serierum reciprocarum, E41.

A. L. Robledo, PlanetMath.org, value of the Riemann zeta function at s=2

Limit(n-->+oo) of (1/n)*(sum(k=1,n,frac((n/k)^(1/2)))) = zeta(2) and in general have limit(n-->+oo) of (1/n)*(sum(k=1,n,frac((n/k)^(1/m)))) = zeta(m), m >= 2. - Aktar Yalcin (aktaryalcin(AT)msn.com), Jul 14 2005

1.6449340668482264364724151666460251892189499012067984377355582293700074704032...

RealDigits[N[Pi^2/6, 100]][[1]]

(PARI) \p 200; Pi^2/6

(PARI) a(n)=if(n<1, 0, default(realprecision, n+2); floor(Pi^2/6*10^(n-1))%10)

(PARI) \p 200 dilog(1) \p 200 zeta(2)

Cf. A013679, A013631, A013680, 1/A059956.

Sequence in context: A029680 A021612 A110756 this_sequence A019174 A019166 A058158

Adjacent sequences: A013658 A013659 A013660 this_sequence A013662 A013663 A013664

cons,nonn,nice

njas

1,5

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

David Wells, "The Penguin Dictionary of Curious and Interesting Numbers," Revised Edition, Penguin Books, London, England, 1997, page 23.

G. Xiao, Contfrac

Index entries for continued fractions for constants

1.6449340668482264364724151666460251892189...

ContinuedFraction[ Pi^2/6, 100]

Cf. A013661.

Sequence in context: A021706 A007005 A066978 this_sequence A096428 A091007 A110638

Adjacent sequences: A013676 A013677 A013678 this_sequence A013680 A013681 A013682

nonn,cofr,nice,easy

njas