A322192 - OEIS

%I #6 Jun 18 2019 07:37:22

%S 1,1,2,7,16,49,158,480,1565,5372,18168,63018,223069,790675,2837099,

%T 10275237,37365238,136780746,503454552,1860283966,6903032032,

%U 25710869751,96062102703,360005362169,1352895525992,5096746479429,19245661967963,72829157526334,276144309118166,1048989168151209,3991676310364631,15213832997014866,58073559070913632,221990591912157497,849708949683300960

%N G.f.: exp( Sum_{n>=1} A322191(n)*x^n/n ), where A322191(n) is the coefficient of x^n*y^n/n in log( Product_{n>=1} 1/(1 - (x^(2*n) - y^(2*n))/(x - y)) ).

%H Paul D. Hanna, <a href="/A322192/b322192.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) ~ c * 4^n / n^(3/2), where c = 0.585811817455537... - _Vaclav Kotesovec_, Jun 18 2019

%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 16*x^4 + 49*x^5 + 158*x^6 + 480*x^7 + 1565*x^8 + 5372*x^9 + 18168*x^10 + 63018*x^11 + 223069*x^12 + ...

%e such that

%e log( A(x) ) = x + 3*x^2/2 + 16*x^3/3 + 35*x^4/4 + 141*x^5/5 + 528*x^6/6 + 1744*x^7/7 + 6435*x^8/8 + 25225*x^9/9 + 92743*x^10/10 + 352782*x^11/11 + 1364216*x^12/12 + ... + A322191(n)*x^n/n + ...

%o (PARI) N=35;

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

%o {A322191(n) = polcoeff( n*polcoeff( L, n, x), n, y)}

%o {a(n) = polcoeff( exp( sum(m=1, n, A322191(m)*x^m/m ) +x*O(x^n) ), n) }

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

%Y Cf. A322191.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 10 2018