A221972 G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (2*k-1) / (1 + k*(2*k-1)*x).

1, 1, 5, 49, 797, 19417, 661829, 30067105, 1755847661, 128153307433, 11430887275733, 1223433282301681, 154741998546660605, 22833118232808363769, 3887374029443206242917, 756359660427618330221377, 166781979021653656537782029, 41372815623877107580771950025

Offset: 0

Author

Paul D. Hanna, Feb 01 2013

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 49*x^3 + 797*x^4 + 19417*x^5 + 661829*x^6 +...
where
A(x) = 1 + x/(1+x) + 2!*1*3*x^2/((1+x)*(1+2*3*x)) + 3!*1*3*5*x^3/((1+x)*(1+2*3*x)*(1+3*5*x)) + 4!*1*3*5*7*x^4/((1+x)*(1+2*3*x)*(1+3*5*x)*(1+4*7*x)) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..160

Crossrefs

Cf. A110501, A024283.

Programs

PARI

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

Formula

a(n) = Sum_{k, 0<=k<=n} A211183(n,k)*4^(n-k). - Philippe Deléham, Feb 03 2013

G.f.: G(0) where G(k) = 1 + x*(2*k+1)*(4*k+1)/( 1 + x + 6*x*k + 8*x*k^2 - 2*x*(k+1)*(4*k+3)*(1 + x + 6*x*k + 8*x*k^2)/(2*x*(k+1)*(4*k+3) + (1 + 6*x + 14*x*k + 8*x*k^2)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 11 2013

a(n) ~ 2^(3*n+9/2) * n^(2*n+2) / (exp(2*n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, Nov 02 2014