A366037 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 5 * A(x)).

0, 1, 10, 160, 3110, 67155, 1548526, 37346040, 930513870, 23765376580, 618871054120, 16370119905880, 438628647940730, 11880264846822610, 324739360804852980, 8946782070689651280, 248184394985913218910, 6926162613387923126700, 194320992885495965332600, 5477763483026946993808960, 155070883903415687652796120

Offset

0,3

Comments

Reversion of g.f. for 4-dimensional figurate numbers A002419 (with signs).

Links

Table of n, a(n) for n=0..20.

Eric Weisstein's World of Mathematics, Series Reversion

Formula

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(5*n,n-k-1) * 5^k for n > 0.

a(n) ~ sqrt((5168 - 869*sqrt(34)) / (17*Pi)) * (22 - sqrt(34))^(5*n) / (2 * n^(3/2) * 3^(3*n + 3/2) * 5^(4*n + 1) * (11*sqrt(34) - 62)^n). - Vaclav Kotesovec, Sep 27 2023

Mathematica

nmax = 20; A[_] = 0; Do[A[x_] = x (1 + A[x])^5/(1 - 5 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 5 x)/(1 + x)^5, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[5 n, n - k - 1] 5^k, {k, 0, n - 1}], {n, 1, 20}]]

Crossrefs

Cf. A002294, A002419, A064089, A365668, A365755, A365818, A366017, A366035, A366036.

Sequence in context: A180881 A106261 A112125 * A090374 A034724 A234283

Adjacent sequences: A366034 A366035 A366036 * A366038 A366039 A366040

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 26 2023

Status

approved