A028339 Coefficient of x^2 in expansion of (x+1)*(x+3)*...*(x+2*n-1).

1, 9, 86, 950, 12139, 177331, 2924172, 53809164, 1094071221, 24372200061, 590546123298, 15467069396610, 435512515705695, 13121113142970855, 421214220916438680, 14354510691610713240, 517596339235489288425, 19688993487602867898225, 787995759739909824183150

Offset: 2

Author

Bill Gosper

Keywords

• nonn

Comments

Equals third left hand column of A161198 triangle divided by 4. - Johannes W. Meijer, Jun 08 2009

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 2..400

Crossrefs

Cf. A028338, A161198.

Programs

Mathematica

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

PARI

a(n) = polcoeff(prod(k=1, n, x+2*k-1), 2); \\ Michel Marcus, Nov 12 2014

Formula

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

E.g.f.: (log(1-2*x))^2/(8*sqrt(1-2*x)). - Vladeta Jovovic, Feb 19 2003

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

Extensions

More terms from Michel Marcus, Nov 12 2014