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 A259456 Triangle read by rows, giving coefficients in an expansion of absolute values of Stirling numbers of the first kind in terms of binomial coefficients. 4
 1, 2, 3, 6, 20, 15, 24, 130, 210, 105, 120, 924, 2380, 2520, 945, 720, 7308, 26432, 44100, 34650, 10395, 5040, 64224, 303660, 705320, 866250, 540540, 135135, 40320, 623376, 3678840, 11098780, 18858840, 18288270, 9459450, 2027025, 362880, 6636960, 47324376, 177331440, 389449060, 520059540, 416215800 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES L. Comtet, Advanced Combinatorics (1974), Chapter VI, page 256. DJ Jeffrey, GA Kalugin, N Murdoch, Lagrange inversion and Lambert W, Preprint 2015; http://www.apmaths.uwo.ca/~djeffrey/Offprints/JeffreySYNASC2015paper17.pdf Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 152. Table C_{m, nu}. LINKS Table of n, a(n) for n=0..42. L. Berg, On polynomials related with generalized Bernoulli numbers, Rostock Math. Kolloq. (2002). S. Butler, P. Karasik, A note on nested sums, J. Int. Seq. 13 (2010) #10.4.4 page 4. T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms D. E. Knuth, Convolution polynomials, The Mathematica J., 2 (1992), 67-78. D. E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA], 1992. R. B. Paris, An asymptotic approximation for incomplete Gaussian sums. II., J. Comp. Appl. Math 212 (2008) 16-30, Table 1. G. Rzadkowski, On some expansions for the Euler Gamma function and the Riemann Zeta function, arxiv:1007.1955 [math.CA], Table 1. J. Comp. Appl. Math. 236 (15) (2012), 3710-3719. L. Takacs, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581. Table 3 gives a version of the triangle. FORMULA T(n,k) = (n-k-1)*( T(n-1,k-1)+T(n-1,k) ), n>=1, 1<=k<=n. [Berg, Eq. 6] The general results on the convolution of the refined partition polynomials of A133932, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these unsigned polynomials. - Tom Copeland, Sep 20 2016 EXAMPLE Triangle begins: 1, 2,3, 6,20,15, 24,130,210,105, 120,924,2380,2520,945, ... For k=4 and j=2 in Knuth's equation, |S1(4,4-2)| = |S1(4,2)| = |A008275(4,2)| = 11 = p_{2,1}*C(4,3) +p_{2,2}*C(4,4) = 2*4+3*1. - R. J. Mathar, Jul 16 2015 MAPLE A259456 := proc(n, k) option remember; if k < 1 or k > n then 0 ; elif n = 1 then 1; else procname(n-1, k-1)+procname(n-1, k); %*(n+k-1) ; end if; end proc: seq(seq(A259456(n, k), k=1..n), n=1..10) ; # R. J. Mathar, Jul 18 2015 MATHEMATICA T[n_, k_] := T[n, k] = If[k < 1 || k > n, 0, If[n == 1, 1, (T[n-1, k-1] + T[n-1, k])(n+k-1)]]; Table[T[n, k], {n, 1, 10}, { k, 1, n}] // Flatten (* Jean-François Alcover, Sep 26 2019, from Maple *) CROSSREFS Cf. This is a row reversed and unsigned version of A111999. Cf. A008275, A000276 (2nd column), A000483 (3rd column), A000142 (1st column). Cf. A133932. Sequence in context: A319204 A093447 A321203 * A334724 A328218 A254441 Adjacent sequences: A259453 A259454 A259455 * A259457 A259458 A259459 KEYWORD nonn,tabl,easy AUTHOR N. J. A. Sloane, Jun 30 2015 STATUS approved

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Last modified August 3 05:44 EDT 2024. Contains 374875 sequences. (Running on oeis4.)