A380727 -id:A380727 - cp's OEIS frontend

A380726 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2 / (1 - xA(x)^2)) / (1 - xA(x)^2).

1, 2, 23, 526, 18345, 865426, 51606511, 3725086590, 315869177777, 30781410753250, 3390102419068071, 416446509483046318, 56455962861401232025, 8372599773137199223794, 1348414830158700569758655, 234364024637335981658563486, 43725325359127416298442233569

Offset: 0

Seiichi Manyama, Jan 31 2025

nonn

Cf. A380663, A380727.

Cf. A380723.

a(n) = n!*sum(k=0, n, (2*n+1)^(k-1)*binomial(3*n, n-k)/k!);

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

A380915 E.g.f. A(x) satisfies A(x) = exp(x / (1 - xA(x)^3)) / (1 - xA(x)^3).

Original entry on oeis.org

1, 2, 19, 421, 14453, 676741, 40225525, 2901397997, 246222420841, 24038780973913, 2654362957336481, 327087730518759937, 44498835149618922253, 6624743172003104909957, 1071295799491745519081629, 186999332904147675923216341, 35044146207707289182759039825

Offset: 0

Seiichi Manyama, Feb 08 2025

nonn

Cf. A380769, A380914.

Cf. A380727.

a(n, q=1, r=3, s=0, t=1, u=1) = q*n!*sum(k=0, n, (r*n+(s-r)*k+q)^(k-1)*binomial((r*u+1)*n+((s-r)*u+t-1)*k+q*u-1, n-k)/k!);

a(n) = n! * Sum_{k=0..n} (3*n-3*k+1)^(k-1) * binomial(4*n-3*k,n-k)/k!.

cp's OEIS Frontend

A380726 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2 / (1 - xA(x)^2)) / (1 - xA(x)^2).

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A380915 E.g.f. A(x) satisfies A(x) = exp(x / (1 - xA(x)^3)) / (1 - xA(x)^3).

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cp's OEIS Frontend

A380726 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2 / (1 - x*A(x)^2)) / (1 - x*A(x)^2).

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A380915 E.g.f. A(x) satisfies A(x) = exp(x / (1 - x*A(x)^3)) / (1 - x*A(x)^3).

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A380726 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2 / (1 - xA(x)^2)) / (1 - xA(x)^2).

A380915 E.g.f. A(x) satisfies A(x) = exp(x / (1 - xA(x)^3)) / (1 - xA(x)^3).