A392372 - OEIS

%I #21 Apr 06 2026 09:46:18

%S 1,1,7,106,2693,101136,5208427,350079136,29655632169,3084806690560,

%T 386157558367151,57239257050526464,9913670289956500333,

%U 1983894690594669669376,454331789822164009288275,118077198411780842498154496,34569903816081852421748189393,11327237659616427357082928676864

%N E.g.f. A(x) satisfies A(x) = exp( -x * d/dx log(1 - x*A(x)) ).

%H Seiichi Manyama, <a href="/A392372/b392372.txt">Table of n, a(n) for n = 0..252</a>

%F E.g.f. A(x) satisfies A(x) = exp( x * (A(x) + x*d/dx A(x))/(1 - x*A(x)) ).

%F a(0) = 1; a(n) = Sum_{i=0..n-1} (i^2+2*n-1) * binomial(n-1,i) * a(i)*a(n-1-i) + Sum_{i, j, k>=0 and i+j+k=n-2} (i*j+i-j^2-2*j) * (n-1)!/(i!*j!*k!) * a(i)*a(j)*a(k).

%F a(n) ~ c * n^(2*n + 3) / exp(2*n), where c = 2.034311414558208... - _Vaclav Kotesovec_, Apr 06 2026

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0,i-1, (j^2+2*i-1)*binomial(i-1,j)*v[j+1]*v[i-j])+sum(j=0, i-2, sum(k=0, i-2-j, (j*k+j-k^2-2*k)*(i-1)!/(j!*k!*(i-2-j-k)!)*v[j+1]*v[k+1]*v[i-1-j-k]))); v;

%Y Cf. A000262, A052873.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Apr 06 2026