0,3

Permutations avoiding 12-3 that contain the pattern 31-2 exactly once.

Kekule numbers for certain benzenoids. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 18 2005

If Y is a 3-subset of an n-set X then, for n>=6, a(n-5) is the number of 6-subsets of X having at least two elements in common with Y. - Milan R. Janjic (agnus(AT)blic.net), Nov 23 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 195.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 227, #16.

S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 166, Table 10.4/I/3).

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.

L. Hogben, Choice and Chance by Cardpack and Chessboard. Vol. 1, Chanticleer Press, NY, 1950, p. 36.

T. D. Noe, Table of n, a(n) for n=0..1000

Index entries for sequences related to linear recurrences with constant coefficients

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

A. F. Labossiere, Sobalian Coefficients.

A. F. Labossiere, Miscellaneous.

T. Mansour, Restricted permutations by patterns of type 2-1.

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Eric Weisstein's World of Mathematics, Stirling numbers of the 2nd kind.

G.f.: x(1+2x)/(1-x)^5 - Paul Barry (pbarry(AT)wit.ie), Jul 23 2003

Sum(j=1, n, j*triangle(j)) - Jon Perry (perry(AT)globalnet.co.uk), Jul 28 2003

E.g.f. with offset -1: exp(x)*(1*(x^2)/2! + 4*(x^3)/3! + 3*(x^4)/4!). For the coefficients [1, 4, 3] see triangle A112493.

E.g.f. x*exp(x)*(24 + 60*x + 28*x^2 + 3*x^3)/24 (above e.g.f. differentiated).

Partial sums of A002411, pentagonal pyramidal numbers: n^2 (n+1)/2. - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 16 2006

a(n) = n(1+n)(2+n)(1+3n)/24. - T. D. Noe, Jan 21 2008

a(n)=4a(n-1)-6a(n-2)+4a(n-3)-a(n-4)+3. [From Kieren MacMillan (kieren(AT)alumni.rice.edu), Sep 29 2008]

a(n)=5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+a(n-5) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Nov 23 2008]

A001296:=-(1+2*z)/(z-1)**5; [Conjectured by S. Plouffe in his 1992 dissertation.]

lst={}; Do[f=StirlingS2[n, n-2]; AppendTo[lst, f], {n, 2, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 27 2008]

(PARI) t(n)=n*(n+1)/2 for(i=1, 30, print1(", "sum(j=1, i, j*t(j))))

(Other) SAGE:[stirling_number2(n+2, n)for n in xrange(0, 38)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 14 2009]

a(n)=f(n, 2) where f is given in A034261.

Cf. A008277, A094262, A001297, A001298.

a(n)= A093560(n+3, 4), (3, 1)-Pascal column.

Cf. A002411.

Sequence in context: A155305 A155290 A056685 this_sequence A000970 A155245 A155291

Adjacent sequences: A001293 A001294 A001295 this_sequence A001297 A001298 A001299

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).