A366035 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^5 / (1 - 3 * A(x)).

0, 1, 8, 98, 1440, 23389, 404712, 7314724, 136476912, 2608808180, 50828498336, 1005682252458, 20152470321984, 408149824237302, 8341496306085040, 171812412714350280, 3562961488550366480, 74328284438252301996, 1558783863783469298016, 32844108784368485209320, 694957689921176181019520

Offset: 0

Author

Ilya Gutkovskiy, Sep 26 2023

Keywords

• nonn

Comments

Reversion of g.f. for 4-dimensional figurate numbers A002417 (with signs).

Links

• Eric Weisstein's World of Mathematics, Series Reversion

Crossrefs

Cf. A002294, A002417, A064087, A365668, A365755, A365816, A366015, A366036, A366037.

Programs

Mathematica

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^5/(1 - 3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 3 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] 3^k, {k, 0, n - 1}], {n, 1, 20}]]

Formula

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(5*n,n-k-1) * 3^k for n > 0.

a(n) ~ 2^(4*n - 1) * 5^(5*n + 1/2) / (sqrt(Pi) * n^(3/2) * 3^(7*n + 5/2)). - Vaclav Kotesovec, Sep 27 2023