A385619 E.g.f. A(x) satisfies A(x) = exp( x*(A(x) + A(2*x)) ).

1, 2, 16, 320, 14176, 1363872, 288285760, 135499302976, 142083696478720, 331241746024775168, 1705949708332396248064, 19272264281263882812337152, 474329882865823082358501265408, 25275628582523724268037232839274496, 2899873213836728319564120809900380069888

Offset: 0

Author

Seiichi Manyama, Jul 05 2025

Keywords

• nonn

Crossrefs

Cf. A385617.

Programs

Mathematica

terms = 15; A[] = 1; Do[A[x] =Exp[x*(A[x] + A[2*x])]+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jul 05 2025 *)

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+1)*(2^j+1)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * (2^k+1) * binomial(n-1,k) * a(k) * a(n-1-k).

a(n) ~ c * n! * 2^(n*(n-1)/2), where c = 13.440025845363170742648943305743503903268661246000630477... - Vaclav Kotesovec, Jul 05 2025