A363744 -id:A363744 - cp's OEIS frontend

A365030 E.g.f. satisfies A(x) = exp(x * (1 + x * A(x))^3).

1, 1, 7, 55, 709, 11761, 243181, 6054763, 175803097, 5847578785, 219175994521, 9144024668131, 420340277237365, 21111584238219697, 1150333949592549541, 67589878866533749531, 4260172601206280708401, 286737199114729515029569

Offset: 0

Seiichi Manyama, Aug 17 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..363

Cf. A125500, A363744.

Cf. A364938.

Array[#!*Sum[ (# - k + 1)^(k - 1)*Binomial[3 k, # - k]/k!, {k, 0, #}] &, 18, 0] (* Michael De Vlieger, Aug 18 2023 *)

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

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

A372202 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + x * A(x)^(1/2))^2 ).

Original entry on oeis.org

1, 2, 12, 92, 1024, 14192, 241624, 4855832, 112887520, 2981919392, 88274138344, 2896196131688, 104341759873168, 4096112838853232, 174063938788299928, 7961816811743462648, 390072804802178286016, 20381525361707872355648, 1131437006755346662551496

Offset: 0

Seiichi Manyama, Apr 21 2024

nonn

Cf. A363744, A372200.

a(n, r=2, s=2, t=0, u=1) = r*n!*sum(k=0, n, (t*k+u*(n-k)+r)^(k-1)*binomial(s*k, n-k)/k!);

E.g.f.: A(x) = B(x)^2 where B(x) is the e.g.f. of A363744.

If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(s*k,n-k)/k!.

cp's OEIS Frontend

A365030 E.g.f. satisfies A(x) = exp(x * (1 + x * A(x))^3).

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