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%I M4736 N2028 #70 Dec 19 2023 12:52:22
%S 1,10,105,1260,17325,270270,4729725,91891800,1964187225,45831035250,
%T 1159525191825,31623414322500,924984868933125,28887988983603750,
%U 959493919812553125,33774185977401870000,1255977541034632040625
%N Exponential generating function: (1+3*x)/(1-2*x)^(7/2).
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
%D F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 296.
%D C. Jordan, Calculus of Finite Differences. Eggenberger, Budapest and Röttig-Romwalter, Sopron 1939; Chelsea, NY, 1965, p. 172.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A000457/b000457.txt">Table of n, a(n) for n = 0..200</a>
%H 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 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 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 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 J. Riordan, <a href="/A001820/a001820.pdf">Notes to N. J. A. Sloane, Jul. 1968</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html">Stirling Number of the First Kind.</a>
%F a(n) = (2n+3)!/( 3!*n!*2^n ).
%F a(n) = (n+1)*(2*n+3)!!/3, n>=0, with (2*n+3)!! = A001147(n+2).
%F a(n) = Sum_{j=0..n} (j + 1) * Eulerian2(n + 2, n - j). - _Peter Luschny_, Feb 13 2023
%e G.f. = 1 + 10*x + 105*x^2 + 1260*x^3 + 17325*x^4 + 270270*x^5 + ... - _Michael Somos_, Dec 15 2023
%t Table[(2n+3)!/(3!*n!*2^n), {n,0,30}] (* _G. C. Greubel_, May 15 2018 *)
%o (PARI) for(n=0, 30, print1((2*n+3)!/(3!*n!*2^n), ", ")) \\ _G. C. Greubel_, May 15 2018
%o (Magma) [Factorial(2*n+3)/(6*Factorial(n)*2^n): n in [0..30]]; // _G. C. Greubel_, May 15 2018
%Y Equals (1/2)*A000906.
%Y Third column of triangle A001497.
%Y Second column (m=1) of unsigned Laguerre-Sonin a=1/2 triangle |A130757|.
%Y Diagonal k=n-1 of triangle A134991.
%Y Cf. A160473, A163939.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Sascha Kurz_, Aug 15 2002
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