A322188 - OEIS

%I #12 Jun 18 2019 05:56:26

%S 1,1,2,6,15,45,140,448,1483,5027,17311,60469,213678,762284,2741864,

%T 9932346,36202666,132677658,488605698,1807176452,6710206574,

%U 25003642942,93468147306,350425771854,1317330452697,4964398631867,18751217069083,70975750129731,269180061675328,1022750160098864,3892577330120307,14838784128136803,56651259287153670,216586672901518164,829142137823283601,3178107527615273349

%N 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)) ).

%H Paul D. Hanna, <a href="/A322188/b322188.txt">Table of n, a(n) for n = 0..512</a>

%F a(n) ~ c * 4^n / n^(3/2), where c = 0.57389010009720382786456367148681469430628117317... - _Vaclav Kotesovec_, Jun 18 2019

%e 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 + ...

%e such that

%e 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 + ...

%e RELATED SERIES.

%e 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 + ...

%o (PARI) N=35;

%o {L = sum(n=1, N+1, -log(1 - x^(2*n-1) - y^(2*n-1) +x*O(x^N) +y*O(y^N)) ); }

%o {A322187(n) = polcoeff( n*polcoeff( L, n, x), n, y)}

%o {a(n) = polcoeff( exp( sum(m=1, n, A322187(m)*x^m/m ) +x*O(x^n) ), n) }

%o for(n=0, N, print1( a(n), ", ") )

%Y Cf. A322187, A322198, A322204.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 07 2018