A351798 a(0) = 1; a(n) = (1/2) * Sum_{k=0..n-1} (k+1) * (k+2) * a(k) * a(n-k-1).

1, 1, 4, 31, 377, 6531, 152452, 4619130, 176631345, 8334329638, 476245005316, 32437793281489, 2597918907028430, 241796318654003869, 25886976434072903664, 3159556047500264255868, 436160347706069120482893, 67621917400663695356651589, 11700923494462411106797164208

Offset: 0

Author

Ilya Gutkovskiy, Feb 19 2022

Keywords

• nonn

Crossrefs

Cf. A000108, A088716.

Programs

Maple

A351798 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        add((1+k)*(2+k)*procname(k)*procname(n-k-1),k=0..n-1) ;
        %/2 ;
    end if;
end proc:
seq(A351798(n),n=0..30) ; # R. J. Mathar, Aug 19 2022

Mathematica

a[0] = 1; a[n_] := a[n] = (1/2) Sum[(k + 1) (k + 2) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; A[] = 0; Do[A[x] = 1 + x A[x]^2 + 2 x^2 A[x] A'[x] + x^3 A[x] A''[x]/2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + x * A(x)^2 + 2 * x^2 * A(x) * A'(x) + x^3 * A(x) * A''(x) / 2.