A385322 -id:A385322 - cp's OEIS frontend

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

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).

A386532 E.g.f. A(x) satisfies A(x) = exp(x + x^4/24 * A'''(x)).

Original entry on oeis.org

1, 1, 1, 1, 2, 16, 316, 12986, 1004886, 135218098, 29704320346, 10121902661146, 5129873988803026, 3733767262251907596, 3789310639383117177330, 5228858971600792496346106, 9599732445822506117447745156, 23009850124770465122843408990196, 70822854191943114816789796493816596

Offset: 0

Seiichi Manyama, Jul 24 2025

nonn

Cf. A385322, A386531.

Cf. A386502.

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, 3, stirling(3, k, 1)*j^k)*binomial(i-1, j)*v[j+1]*v[i-j])/24); v;

a(0) = 1; a(n) = a(n-1) + (1/24) * Sum_{k=0..n-1} (1 + k) * (2*k - 3*k^2 + k^3) * binomial(n-1,k) * a(k) * a(n-1-k).

cp's OEIS Frontend

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

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