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%I A000457 M4736 N2028 #70 Dec 19 2023 12:52:22
%S A000457 1,10,105,1260,17325,270270,4729725,91891800,1964187225,45831035250,
%T A000457 1159525191825,31623414322500,924984868933125,28887988983603750,
%U A000457 959493919812553125,33774185977401870000,1255977541034632040625
%N A000457 Exponential generating function: (1+3*x)/(1-2*x)^(7/2).
%D A000457 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
%D A000457 F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
%D A000457 C. Jordan, Calculus of Finite Differences. Eggenberger, Budapest and Röttig-Romwalter, Sopron 1939; Chelsea, NY, 1965, p. 172.
%D A000457 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000457 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000457 G. C. Greubel, <a href="/A000457/b000457.txt">Table of n, a(n) for n = 0..200</a>
%H A000457 Selden Crary, Richard Diehl Martinez, and Michael Saunders, <a href="https://arxiv.org/abs/1707.00705">The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters</a>, arXiv:1707.00705 [stat.ME], 2017, Table 1.
%H A000457 H. W. Gould, Harris Kwong, and Jocelyn Quaintance, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Kwong/kwong9.html">On Certain Sums of Stirling Numbers with Binomial Coefficients</a>, J. Integer Sequences, 18 (2015), #15.9.6.
%H A000457 C. Jordan, <a href="https://www.jstage.jst.go.jp/article/tmj1911/37/0/37_0_254/_pdf">On Stirling's Numbers</a>, Tohoku Math. J., 37 (1933), 254-278.
%H A000457 Alexander Kreinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Kreinin/kreinin4.html">Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity</a>, Journal of Integer Sequences, 19 (2016), #16.6.2.
%H A000457 J. Riordan, <a href="/A001820/a001820.pdf">Notes to N. J. A. Sloane, Jul. 1968</a>
%H A000457 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html">Stirling Number of the First Kind.</a>
%F A000457 a(n) = (2n+3)!/( 3!*n!*2^n ).
%F A000457 a(n) = (n+1)*(2*n+3)!!/3, n>=0, with (2*n+3)!! = A001147(n+2).
%F A000457 a(n) = Sum_{j=0..n} (j + 1) * Eulerian2(n + 2, n - j). - _Peter Luschny_, Feb 13 2023
%e A000457 G.f. = 1 + 10*x + 105*x^2 + 1260*x^3 + 17325*x^4 + 270270*x^5 + ... - _Michael Somos_, Dec 15 2023
%t A000457 Table[(2n+3)!/(3!*n!*2^n), {n,0,30}] (* _G. C. Greubel_, May 15 2018 *)
%o A000457 (PARI) for(n=0, 30, print1((2*n+3)!/(3!*n!*2^n), ", ")) \\ _G. C. Greubel_, May 15 2018
%o A000457 (Magma) [Factorial(2*n+3)/(6*Factorial(n)*2^n): n in [0..30]]; // _G. C. Greubel_, May 15 2018
%Y A000457 Equals (1/2)*A000906.
%Y A000457 Third column of triangle A001497.
%Y A000457 Second column (m=1) of unsigned Laguerre-Sonin a=1/2 triangle |A130757|.
%Y A000457 Diagonal k=n-1 of triangle A134991.
%Y A000457 Cf. A160473, A163939.
%K A000457 nonn,easy
%O A000457 0,2
%A A000457 _N. J. A. Sloane_
%E A000457 More terms from _Sascha Kurz_, Aug 15 2002

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