A381415 -id:A381415 - cp's OEIS frontend

A385440 E.g.f. A(x) satisfies A(x) = exp( arcsinh(x * A(x)^2) ).

1, 1, 5, 48, 693, 13440, 328185, 9676800, 334639305, 13284311040, 595505854125, 29756856729600, 1640160546688125, 98860780014796800, 6469121228247302625, 456736803668361216000, 34607895888408878660625, 2801319062499282124800000, 241247999301688986945463125

Offset: 0

Seiichi Manyama, Jun 29 2025

nonn

Cf. A001147, A385369, A385441, A385442.

Cf. A381415, A385343.

a(n) = 2^n*n!*binomial((3*n+1)/2, n)/(3*n+1);

E.g.f. A(x) satisfies A(x) = (1 + 2*x*A(x)^3)^(1/2).
a(n) = 2^n * n! * binomial((3*n+1)/2,n)/(3*n+1).
a(n) = Sum_{k=0..n} (2*n+1)^(k-1) * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ 3^(3*n/2) * n^(n-1) / exp(n). - Vaclav Kotesovec, Jul 04 2025

A381416 E.g.f. A(x) satisfies A(x) = exp( 2 * sinh(x * A(x)) ).

Original entry on oeis.org

1, 2, 12, 130, 2080, 44354, 1185856, 38188546, 1439993088, 62261776002, 3037542875136, 165090563653250, 9892965209886720, 648064548551770562, 46075919968420085760, 3533725068594022938626, 290804441398399410503680, 25561250854199444302177538, 2390133356713125694150017024

Offset: 0

Seiichi Manyama, Feb 23 2025

nonn

Cf. A136630, A381415.

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = 2*sum(k=0, n, (2*n+2)^(k-1)*a136630(n, k));

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381415.

a(n) = 2 * Sum_{k=0..n} (2*n+2)^(k-1) * A136630(n,k).

cp's OEIS Frontend

A385440 E.g.f. A(x) satisfies A(x) = exp( arcsinh(x * A(x)^2) ).

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