A366017 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 5 * A(x)).

0, 1, 9, 132, 2365, 47169, 1005564, 22431720, 517122117, 12222124035, 294569159313, 7212098118888, 178877944712844, 4484938858752940, 113488477622130600, 2894560146756466320, 74335973069605120725, 1920587845828953301479, 49886703842977713177723, 1301959618949870922531300, 34123873581608909988904245

Offset: 0

Author

Ilya Gutkovskiy, Sep 26 2023

Keywords

• nonn

Comments

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

Links

• Eric Weisstein's World of Mathematics, Series Reversion

Crossrefs

Cf. A002293, A002414, A064089, A365754, A365818, A366014, A366015, A366016.

Programs

Mathematica

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^4/(1 - 5 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 5 x)/(1 + x)^4, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[4 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(4*n,n-k-1) * 5^k for n > 0.