A363405 - OEIS

%I #20 Jun 01 2023 11:11:42

%S 1,1,1,1,1,2,2,2,2,4,5,5,5,10,12,13,13,26,34,36,37,74,97,105,107,215,

%T 293,320,328,658,905,998,1025,2058,2878,3194,3292,6611,9316,10412,

%U 10748,21594,30697,34470,35663,71668,102446,115575,119761,240740,345940,391726,406571,817453,1179322,1339851

%N G.f. satisfies A(x) = exp( Sum_{k>=1} (A(x^k) + A(i*x^k) + A(-x^k) + A(i^3*x^k))/4 * x^k/k ), where i = sqrt(-1).

%H Seiichi Manyama, <a href="/A363405/b363405.txt">Table of n, a(n) for n = 0..1000</a>

%F A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-x^(4*k+1))^a(4*k).

%F A(x) * A(i*x) * A(-x) * A(i^3*x) = A(x^4).

%F a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k and d==1 mod 4} d * a(d-1) ) * a(n-k).

%o (PARI) seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, sum(m=0, 3, subst(A, x, I^m*x^k))/4*x^k/k)+x*O(x^n))); Vec(A);

%Y Cf. A195865, A363404.

%Y Cf. A363337.

%K nonn

%O 0,6

%A _Seiichi Manyama_, May 31 2023