A365818 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 5 * A(x)).

0, 1, 8, 107, 1760, 32298, 634128, 13034247, 276943568, 6033834950, 134069957840, 3026476515790, 69213144181888, 1600157697995092, 37337615574348960, 878166685063548639, 20797051344280763184, 495509950454603339310, 11869278747340342255440, 285669061791469915886250, 6904850429493240677872320

Offset: 0

Author

Ilya Gutkovskiy, Sep 25 2023

Keywords

• nonn

Comments

Reversion of g.f. for octagonal numbers (with signs).

Links

• Eric Weisstein's World of Mathematics, Octagonal Number
• Eric Weisstein's World of Mathematics, Series Reversion

Crossrefs

Cf. A000567, A064089, A113207, A179848, A263843, A365816, A365817.

Programs

Mathematica

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

Formula

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

a(n) ~ 3^(3/2) * 2^(n - 1/2) * (154 + 31*sqrt(31))^n / (sqrt((2821 + 506*sqrt(31))*Pi) * n^(3/2) * 5^(2*n)). - Vaclav Kotesovec, Sep 26 2023