A132228 -id:A132228 - cp's OEIS frontend

A331660 E.g.f. A(x) satisfies: d/dx A(x) = 1 + (1/(1 - x)) * A(x/(1 - x)).

1, 1, 5, 32, 280, 3280, 49480, 927560, 21037640, 566134160, 17803754560, 646052181520, 26757321804880, 1252934215973600, 65791336312915520, 3846554938702140320, 248841434876849499040, 17713758333248102781760, 1380631354206969100115200

Offset: 1

Ilya Gutkovskiy, Jan 23 2020

nonn

Cf. A001063, A110083, A132228, A331661.

terms = 20; A[] = 0; Do[A[x] = Normal[Integrate[1 + 1/(1 - x) A[x/(1 - x) + O[x]^(terms + 1)], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]! // Rest
a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k]^2 k! a[n - k - 1], {k, 0, n - 2}]; Table[a[n], {n, 1, 20}]

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

A346186 a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n-1,k)^3 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 2, 12, 132, 2664, 86328, 4257504, 302249232, 29749088160, 3929593244256, 678729769735680, 149951093054923584, 41593439995316826624, 14254785976456164378240, 5952723682598023085466624, 2992204352832467277009072384, 1791038631707465961597691673088

Offset: 0

Ilya Gutkovskiy, Jul 09 2021

nonn

Cf. A000142, A132228, A181543, A346185, A346187.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k]^3 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]

cp's OEIS Frontend

A331660 E.g.f. A(x) satisfies: d/dx A(x) = 1 + (1/(1 - x)) * A(x/(1 - x)).

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