A221094 O.g.f.: Sum_{n>=0} n!^2 * x^n / Product_{k=1..n} (1 + k*(n-k+1)*x).

1, 1, 3, 21, 263, 5165, 146335, 5649397, 285069735, 18214525629, 1437313035887, 137272113393413, 15605422414146487, 2082375903282194893, 322303158660868063359, 57271523430269553109269, 11579903781095519639058119, 2643368434346324374530280157, 676521525314300179793917303951

Offset: 0

Author

Paul D. Hanna, Jan 03 2013

Keywords

• nonn

Comments

Compare to: Sum_{n>=0} n!^2 * x^n / Product_{k=1..n} (1 + k^2*x) = Sum_{n>=0} A110501(n)*x^n, where A110501 is unsigned Genocchi numbers of even index.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 263*x^4 + 5165*x^5 + 146335*x^6 +...
where
A(x) = 1 + x/(1+x) + 2!^2*x^2/((1+1*2*x)*(1+2*1*x)) + 3!^2*x^3/((1+1*3*x)*(1+2*2*x)*(1+3*1*x)) + 4!^2*x^4/((1+1*4*x)*(1+2*3*x)*(1+3*2*x)*(1+4*1*x)) + 5!^2*x^5/((1+1*5*x)*(1+2*4*x)*(1+3*3*x)*(1+4*2*x)*(1+5*1*x)) +...
		

Programs

Mathematica

CoefficientList[Series[Sum[(n!)^2 x^n/Product[1+k(n-k+1)x,{k,n}],{n,0,20}],{x,0,20}],x] (* Harvey P. Dale, Aug 03 2020 *)

PARI

{a(n)=polcoeff(sum(m=0, n, m!^2*x^m/prod(k=1, m, 1+(m-k+1)*k*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))