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Coefficient of x^4 in expansion of (x+1)*(x+3)*...*(x+2*n-1).
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%I #21 Nov 25 2016 13:33:14

%S 1,25,505,10045,208054,4574934,107494190,2702025590,72578259391,

%T 2078757113719,63324503917311,2046225352864875,69953125893139644,

%U 2523698606200763196,95853765344939263692,3824294822931302783964,159940198124792648875341,6998152417792503243516261

%N Coefficient of x^4 in expansion of (x+1)*(x+3)*...*(x+2*n-1).

%C Equals fifth left hand column of A161198 triangle divided by 16. - _Johannes W. Meijer_, Jun 08 2009

%H Robert Israel, <a href="/A028341/b028341.txt">Table of n, a(n) for n = 4..369</a>

%F a(n) = Sum_{i=k+1,..,n} (-1)^(k+1-i)*2^(n-1)*binomial(i-1, k)*s1(n, i) with k = 4, where s1(n, i) are unsigned Stirling numbers of the first kind. - Victor Adamchik (adamchik(AT)ux10.sp.cs.cmu.edu), Jan 23 2001

%F E.g.f.: (log(1-2*x))^4/( 384*sqrt(1-2*x) ). - _Vladeta Jovovic_, Feb 19 2003

%e G.f. = x^4 + 25*x^5 + 505*x^6 + 10045*x^7 + 208054*x^8 + 4574934*x^9 + ...

%p N:= 50: # to get a(4) to a(N)

%p P[0]:= 1;

%p for n from 1 to N do

%p P[n]:= rem(P[n-1]*(x + 2*n-1), x^5,x)

%p od:

%p seq(coeff(P[n],x,4),n=4..N); # _Robert Israel_, Nov 13 2014

%t Table[Coefficient[Product[x + 2*k - 1, {k, 1, n}], x, 4], {n,4,50}] (* _G. C. Greubel_, Nov 24 2016 *)

%o (PARI) a(n) = polcoeff(prod(k=1, n, x+2*k-1), 4); \\ _Michel Marcus_, Nov 12 2014

%Y Cf. A028338, A004041, A028339, A028340, A161198.

%K nonn

%O 4,2

%A _Bill Gosper_

%E More terms from _Michel Marcus_, Nov 12 2014