A386505 a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (1 + k) * k^2 * binomial(n-1,k) * a(k) * a(n-1-k).

1, 1, 3, 43, 1717, 146261, 22851301, 5923208845, 2370243182889, 1386889039102537, 1137386506152214441, 1263728857603292729441, 1850186029852575829090909, 3487711314718246830637945549, 8300937715895750334611432889933, 24529666348754849148034163067487381

Offset: 0

Author

Seiichi Manyama, Jul 24 2025

Keywords

• nonn

Crossrefs

Cf. A143925, A386506, A386507, A386508, A386509.

Cf. A385939, A386443.

Programs

Mathematica

A386505[0] = 1;
A386505[n_] := A386505[n] = If[n==0,
            1,
            A386505[n-1]+ Sum[(1+k)*k^2*Binomial[n-1,k]*A386505[k]*A386505[n-1-k] ,{k,0,n-1} ]
        ] ;
Do [ Print[A386505[n]],{n,0,20}] (* R. J. Mathar, Aug 02 2025 *)

PARI

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

Formula

E.g.f. A(x) satisfies A(x) = exp( x + x^2 * (d/dx A(x)) + x^3 * (d^2/dx^2 A(x)) ).