A366037 - OEIS

%I #8 Sep 27 2023 04:36:51

%S 0,1,10,160,3110,67155,1548526,37346040,930513870,23765376580,

%T 618871054120,16370119905880,438628647940730,11880264846822610,

%U 324739360804852980,8946782070689651280,248184394985913218910,6926162613387923126700,194320992885495965332600,5477763483026946993808960,155070883903415687652796120

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

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SeriesReversion.html">Series Reversion</a>

%F 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.

%F 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

%t 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]

%t CoefficientList[InverseSeries[Series[x (1 - 5 x)/(1 + x)^5, {x, 0, 20}], x], x]	

%t 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}]]

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Sep 26 2023