A386532 -id:A386532 - cp's OEIS frontend

A385322 E.g.f. A(x) satisfies A(x) = exp(x + x^2/2 * A'(x)).

1, 1, 2, 10, 94, 1386, 29146, 822928, 29927612, 1359897724, 75429391276, 5013317213136, 393252908602720, 35947138247529952, 3787896504780614864, 455830291523748797776, 62132218311858089750416, 9523376536749853754046864, 1630878867943272798422352400, 310238352508716702644975872192

Offset: 0

Seiichi Manyama, Jul 24 2025

nonn

Cf. A386531, A386532.

Cf. A143925, A156325.

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

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

A386531 E.g.f. A(x) satisfies A(x) = exp(x + x^3/6 * A''(x)).

Original entry on oeis.org

1, 1, 1, 2, 13, 181, 4551, 188021, 11924753, 1103029649, 142906232381, 25095114042461, 5813156139567261, 1736262706526700925, 655797361805578202939, 308047913827328021014851, 177358895717746915172030241, 123578165227603044619210348321

Offset: 0

Seiichi Manyama, Jul 24 2025

nonn

Cf. A385322, A386532.

Cf. A362381, A385066.

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)*sum(k=1, 2, stirling(2, k, 1)*j^k)*binomial(i-1, j)*v[j+1]*v[i-j])/6); v;

a(0) = 1; a(n) = a(n-1) + (1/6) * Sum_{k=0..n-1} (-k + k^3) * binomial(n-1,k) * a(k) * a(n-1-k).

cp's OEIS Frontend

A385322 E.g.f. A(x) satisfies A(x) = exp(x + x^2/2 * A'(x)).

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