login
Coefficient of x^2 in expansion of (x+1)*(x+3)*...*(x+2*n-1).
6

%I #19 Oct 18 2017 05:00:50

%S 1,9,86,950,12139,177331,2924172,53809164,1094071221,24372200061,

%T 590546123298,15467069396610,435512515705695,13121113142970855,

%U 421214220916438680,14354510691610713240,517596339235489288425,19688993487602867898225,787995759739909824183150

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

%C Equals third left hand column of A161198 triangle divided by 4. - _Johannes W. Meijer_, Jun 08 2009

%H G. C. Greubel, <a href="/A028339/b028339.txt">Table of n, a(n) for n = 2..400</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 = 2, 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))^2/(8*sqrt(1-2*x)). - _Vladeta Jovovic_, Feb 19 2003

%F a(n) ~ n! * log(n)^2 * 2^(n-3) / sqrt(Pi*n) * (1 + (2*gamma + 4*log(2))/log(n)), where gamma is the Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 18 2017

%e G.f. = x^2 + 9*x^3 + 86*x^4 + 950*x^5 + 12139*x^6 + 177331*x^7 + ...

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

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

%Y Cf. A028338, A161198.

%K nonn

%O 2,2

%A _Bill Gosper_

%E More terms from _Michel Marcus_, Nov 12 2014