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%I M2124 N0841 #50 Sep 08 2022 08:44:28
%S 2,20,210,2520,34650,540540,9459450,183783600,3928374450,91662070500,
%T 2319050383650,63246828645000,1849969737866250,57775977967207500,
%U 1918987839625106250,67548371954803740000,2511955082069264081250
%N Exponential generating function: 2*(1+3*x)/(1-2*x)^(7/2).
%C Ramanujan polynomials -psi_{n+2}(n+2,x) evaluated at 1.
%C With offset 2, second Eulerian transform of 0,1,2,3,4... - _Ross La Haye_, Mar 05 2005
%C With offset 1, a strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - _Michael Somos_, Dec 30 2016
%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. Budapest, 1939, p. 152.
%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 Vincenzo Librandi, <a href="/A000906/b000906.txt">Table of n, a(n) for n = 0..200</a>
%H H. W. Gould, Harris Kwong, 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.
%F a(n) = (2n+5)!!/3 - (2n+3)!!.
%F a(n) -2*(n+4)*a(n-1) +3*(2*n+1)*a(n-2) = 0. - _R. J. Mathar_, Feb 20 2013
%F a(n) ~ 2^(n+7/2)*n^(n+3)/(3*exp(n)). - _Ilya Gutkovskiy_, Aug 17 2016
%F a(n) = (2n+3)!/( 3!*n!*2^(n-1) ). - _G. C. Greubel_, May 15 2018
%e G.f. = 2 + 20*x + 210*x^2 + 2520*x^3 + 34650*x^4 + 540540*x^5 + ...
%t Table[(2 n + 5)!!/3 - (2 n + 3)!!, {n, 0, 20}] (* _Vincenzo Librandi_, Apr 11 2012 *)
%o (PARI) a(n)=(2*n+6)!/(n+3)!/2^(n+3)/3-(2*n+4)!/(n+2)!/2^(n+2)
%o (Magma) [Factorial(2*n+3)/(6*Factorial(n)*2^(n-1)): n in [0..30]]; // _G. C. Greubel_, May 15 2018
%Y a(n) = 2*A000457(n) = A051577(n+1) - A001147(n+2).
%Y Negative coefficient of x of polynomials in A098503.
%K nonn
%O 0,1
%A _N. J. A. Sloane_
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