Search: id:A008275 Results 1-1 of 1 results found. %I A008275 %S A008275 1,1,1,2,3,1,6,11,6,1,24,50,35,10,1,120,274,225,85,15,1,720, %T A008275 1764,1624,735,175,21,1,5040,13068,13132,6769,1960,322,28, %U A008275 1,40320,109584,118124,67284,22449,4536,546,36,1,362880,1026576 %V A008275 1,-1,1,2,-3,1,-6,11,-6,1,24,-50,35,-10,1,-120,274,-225,85,-15,1,720, %W A008275 -1764,1624,-735,175,-21,1,-5040,13068,-13132,6769,-1960,322,-28, %X A008275 1,40320,-109584,118124,-67284,22449,-4536,546,-36,1,-362880,1026576 %N A008275 Triangle read by rows of Stirling numbers of first kind, s(n,k), n >= 1, 1<=k<=n. %C A008275 The unsigned numbers are also called Stirling cycle numbers: |s(n,k)| = number of permutations of n objects with exactly k cycles. %C A008275 With P(n) = the number of integer partitions of n, T(i,n) = the number of parts of the i-th partition of n, D(i,n) = the number of different parts of the i-th partition of n, p(j,i,n) = the j-th part of the i-th partition of n, m(j,i,n) = multiplicity of the j-th part of the i-th partition of n, sum_[T(i,n)=k]_{i=1}^{P(n)} = sum running from i=1 to i=p(n) but taking only partitions with T(i,n)=k parts into account, prod_{j=1}^{T(i,n)} = product running from j=1 to j=T(i, n), prod_{j=1}^{D(i,n)} = product running from j=1 to j=D(i,n) one has S1(n,k) = sum_[T(i,n)=k]_{i=1}^{P(n)} (n!/prod_{j=1}^{T(i,n)} p(j,i,n))* (1/prod_{j=1}^{D(i,n)} m(j,i,n)!). For example, S1(6,3) = 225 because n=6 has the following partitions with k=3 parts: (114), (123), (222). Their complexions are: (114): (6!/1*1*4)*(1/2!*1!) = 90, (123): (6!/1*2*3)*(1/1!*1!*1!) = 120, (222): (6!/2*2*2)*(1/ 3!) = 15. The sum of the complexions is 90+120+15=225=S1(6,3). - Thomas Wieder (wieder.thomas(AT)t-online.de), Aug 04 2005 %C A008275 Row sums equal 0 - Jon Perry (perry(AT)globalnet.co.uk), Nov 14 2005 %C A008275 |s(n,k)| enumerates unordered n-vertex forests composed of k increasing non-plane (unordered) trees. Proof from the e.g.f. of the first column and the F. Bergeron et al. reference, especially Table 1, last row (non plane ``recursive"), given in A049029. W. Lang Oct 12 2007. %C A008275 |s(n,k)| enumerates unordered increasing n-vertex k-forests composed of k unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=0 come in j+1 colors (j=0 for the k roots). W. Lang, Oct 12 2007, Feb 22 2008 %D A008275 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833. %D A008275 E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42. %D A008275 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 93ff. %D A008275 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. 32. %D A008275 L. Comtet, Advanced Combinatorics, Reidel, 1974; Chapter V, also p. 310. %D A008275 J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 93. %D A008275 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226. %D A008275 H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.7. %D A008275 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 245. %D A008275 J. Hines, A generalization of the S-Stirling numbers, Math. Mag., 29 (1956), 200-203. %D A008275 Yoshinari Inaba, Hyper-Sums of Powers of Integers and the Akiyama-Tanigawa Matrix, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.7. %D A008275 Knessl, Charles; Keller, Joseph B. Stirling number asymptotics from recursion equations using the ray method. Stud. Appl. Math. 84 (1991), no. 1, 43-56. %D A008275 B. H. Margolius, Transient and periodic solution to the time-inhomogeneous quasi-birth death process, Queueing Systems, Volume 56, Numbers 3-4 / August, 2007. [From N. J. A. Sloane, Jul 09 2009] %D A008275 J. Riordan, An Introduction to Combinatorial Analysis, p. 48. %D A008275 R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996. %D A008275 J. Stirling, The Differential Method, London, 1749; see p. 10. %D A008275 N. M. Temme, Asymptotic estimates of Stirling numbers, Stud. Appl. Math. 89 (1993), no. 3, 233-243. %D A008275 Timashev, A. N. On asymptotic expansions of Stirling numbers of the first and second kinds. (Russian) Diskret. Mat. 10 (1998), no. 3,148-159 translation in Discrete Math. Appl. 8 (1998), no. 5, 533-544. %H A008275 T. D. Noe, Rows 1 to 100 of triangle, flattened. %H A008275 K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. vol. 50, 083512 (2009) %H A008275 Joerg Arndt, Fxtbook %H A008275 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]. %H A008275 R. M. Dickau, Stirling numbers of the first kind %H A008275 D. B. Gruenberg, On asymptotics, Stirling numbers, Gamma function and polylogs %H A008275 A. F. Labossiere, Sobalian Coefficients. %H A008275 A. F. Labossiere, Miscellaneous. %H A008275 W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. %H A008275 D. E. Loeb, [math/9502217] A generalization of Stirling numbers %H A008275 R. P. Stanley, Ordering events in Minkowski space %H A008275 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A008275 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A008275 Thomas Wieder, Comments on A008275 %F A008275 s(n, k)=s(n-1, k-1)-(n-1)*s(n-1, k), n, k >= 1; s(n, 0)=s(0, k)=0; s(0, 0)=1. %F A008275 The unsigned numbers a(n, k)=|s(n, k)| satisfy a(n, k)=a(n-1, k-1)+(n-1)*a(n-1, k), n, k >= 1; a(n, 0)=a(0, k)=0; a(0, 0)=1. %F A008275 E.g.f. for m-th column (unsigned): ((-ln(1-x))^m)/m!. %F A008275 s(n, k) = T(n-1, k-1), n>1 and k>1, where T(n, k) is the triangle [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, -6, -6, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, ...]and DELTA is Deleham's operator defined in A084938. The unsigned numbers are also |s(n, k)| = T(n-1, k-1), for n>0 and k>0, where T(n, k) = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]. %F A008275 Sum[(-1)^(n-i) StirlingS1[n, i] binomial[i, k], {i,0,n}] == (-1)^(n-k) StirlingS1[n+1, k+1]. - Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007 %F A008275 G.f.: S(n) = product[j=1, n, (x-j)] (i.e. (x-1)(x-2)(x-3) = x^3 - 6x^2 + 11x - 6) - Jon Perry (perry(AT)globalnet.co.uk), Nov 14 2005 %F A008275 a(n,k) = s(k,n) = (-1)^(k-n) * S1(k,n) = ( (-1)^(k-n) ) * ( k!/{(n-1)!*2^(k-n)} ) * [ { 1/(k-n)! }*k^(k-n-1) - { (1/6)*(1/(k-n-2)!) }*k^(k-n-2) + { (1/72)*(1/(k-n-4)!) }*k^(k-n-3) - { (1/6480)*(5/(k-n-6)! -36/(k-n-4)!) }*k^(k-n-4) + { (1/155520)*(5/(k-n-8)!-144/(k-n-6)!) }*k^(k-n-5) - { (1/6531840)*(7/(k-n-10)! -504/(k-n-8)!+2304/(k-n-6)!) }*k^(k-n-6) + { (1/1175731200)*(35/(k-n-12)!-5040/(k-n-10)!+87264/(k-n-8)!) }*k^(k-n-7) - { (1/7054387200)*(5/(k-n-14)!-1260/(k-n-12)!+52704/(k-n-10)!-186624/ (k-n-8)!) }*k^(k-n-8) + { (1/338610585600)*(5/(k-n-16)!-2016/(k-n-14)!+164736/ (k-n-12)!-2156544/(k-n-10)!) }*k^(k-n-9) - ..... ]. - Andre F. Labossiere (boronali(AT)laposte.net), Mar 27 2006 %e A008275 |s(3,2)| = 3 for the three unordered 2-forest with 3 vertices and two increasing (non plane) trees: ((1),(2,3)), ((2),(1,3)), ((3),(1,2)). %e A008275 Triangle begins: %e A008275 ..................................1 %e A008275 ................................-1, 1 %e A008275 ...............................2, -3, 1 %e A008275 ............................-6, 11, -6, 1 %e A008275 .........................24, -50, 35, -10, 1 %e A008275 .....................-120, 274, -225, 85, -15, 1 %e A008275 .................720, -1764, 1624, -735, 175, -21, 1 %e A008275 ............-5040, 13068, -13132, 6769, -1960, 322, -28, 1 %e A008275 ......40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1 %e A008275 Another version of the same triangle, from Joerg Arndt, Oct 05 2009: %e A008275 s(n,k) := number of permutations of n elements with exactly k cycles ("Stirling cycle numbers") %e A008275 ..n:...total...m=..1......2......3.....4.....5.....6....7...8...9 %e A008275 ..1:.......1.......1 %e A008275 ..2:.......2.......1......1 %e A008275 ..3:.......6.......2......3......1 %e A008275 ..4:......24.......6.....11......6.....1 %e A008275 ..5:.....120......24.....50.....35....10.....1 %e A008275 ..6:.....720.....120....274....225....85....15.....1 %e A008275 ..7:....5040.....720...1764...1624...735...175....21....1 %e A008275 ..8:...40320....5040..13068..13132..6769..1960...322...28...1 %e A008275 ..9:..362880...40320.109584.118124.67284.22449..4536..546..36...1 %e A008275 |s(4,2)| = 11 for the eleven unordered 2-forest with 4 vertices, composed of two increasing (non plane) trees: ((1),((23)(24))), ((2),((13)(14)), ((3),((12)(14)), ((4),((12)(13)); ((1),(2,3,4)),((2),(1,2,3)), ((3), (1,2,4)), ((4),(1,2,3)); ((1,2),(3,4)), ((1,3),(2,4)), ((1,4),(2, 3)). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 22 2008 %p A008275 with (combinat):seq(seq(stirling1(n, k), k=1..n), n=1..10); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 03 2007 %p A008275 for i from 0 to 9 do seq(stirling1(i, j), j = 1 .. i) od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 29 2007 %o A008275 (PARI) T(n,k)=if(n<1,0,n!*polcoeff(binomial(x,n),k)) %o A008275 (PARI) T(n,k)=if(n<1,0,n!*polcoeff(polcoeff((1+x+x*O(x^n))^y,n),k)) %o A008275 (PARI) vecstirling(n)=Vec(factorback(vector(n-1,i,1-i*'x))) (A function that returns all the s(n,k) as a vector) - Bill Allombert (Bill.Allombert(AT)math.u-bordeaux1.fr), Mar 16 2009 %Y A008275 Cf. A048994, A008277 (Stirling numbers of second kind), A039814-A039817, A048993. %Y A008275 Cf. A084938, A094216, A008276, A094262, A008277, A008278, A121632. %Y A008275 A130534 is a signed version. %Y A008275 A000142(n) = sum(|s(n, k)|) k=0..n, n >= 0, %Y A008275 Sequence in context: A165675 A138771 A121748 this_sequence A130534 A107416 A105613 %Y A008275 Adjacent sequences: A008272 A008273 A008274 this_sequence A008276 A008277 A008278 %K A008275 sign,tabl,nice,core %O A008275 1,4 %A A008275 N. J. A. Sloane (njas(AT)research.att.com). Search completed in 0.003 seconds