A322188 G.f.: exp( Sum_{n>=1} A322187(n)*x^n/n ), where A322187(n) is the coefficient of x^n*y^n/n in log( Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)) ).

1, 1, 2, 6, 15, 45, 140, 448, 1483, 5027, 17311, 60469, 213678, 762284, 2741864, 9932346, 36202666, 132677658, 488605698, 1807176452, 6710206574, 25003642942, 93468147306, 350425771854, 1317330452697, 4964398631867, 18751217069083, 70975750129731, 269180061675328, 1022750160098864, 3892577330120307, 14838784128136803, 56651259287153670, 216586672901518164, 829142137823283601, 3178107527615273349

Offset: 0

Author

Paul D. Hanna, Dec 07 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 15*x^4 + 45*x^5 + 140*x^6 + 448*x^7 + 1483*x^8 + 5027*x^9 + 17311*x^10 + 60469*x^11 + 213678*x^12 + ...
such that
log(A(x)) = x + 3*x^2/2 + 13*x^3/3 + 35*x^4/4 + 131*x^5/5 + 471*x^6/6 + 1723*x^7/7 + 6435*x^8/8 + 24349*x^9/9 + 92393*x^10/10 + 352727*x^11/11 + 1352183*x^12/12 + ... + A322187(n)*x^n/n + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 46*x^4 + 144*x^5 + 466*x^6 + 1536*x^7 + 5187*x^8 + 17842*x^9 + 62209*x^10 + 219504*x^11 + 782272*x^12 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..512

Crossrefs

Cf. A322187, A322198, A322204.

Programs

PARI

N=35;
{L = sum(n=1, N+1, -log(1 - x^(2*n-1) - y^(2*n-1) +x*O(x^N) +y*O(y^N)) ); }
{A322187(n) = polcoeff( n*polcoeff( L, n, x), n, y)}
{a(n) = polcoeff( exp( sum(m=1, n, A322187(m)*x^m/m ) +x*O(x^n) ), n) }
for(n=0, N, print1( a(n), ", ") )

Formula

a(n) ~ c * 4^n / n^(3/2), where c = 0.57389010009720382786456367148681469430628117317... - Vaclav Kotesovec, Jun 18 2019