login
A366014
G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 2 * A(x)).
5
0, 1, 6, 54, 580, 6873, 86688, 1141500, 15512220, 215928900, 3063184410, 44124882750, 643692232404, 9490176205006, 141184118174640, 2116751269990968, 31951313566227228, 485159929343783532, 7405637373574690968, 113572576254948487800, 1749075343256441443320
OFFSET
0,3
COMMENTS
Reversion of g.f. for pentagonal pyramidal numbers (with signs).
LINKS
Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
Eric Weisstein's World of Mathematics, Series Reversion
FORMULA
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(4*n,n-k-1) * 2^k for n > 0.
MATHEMATICA
nmax = 20; A[_] = 0; Do[A[x_] = x (1 + A[x])^4/(1 - 2 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 2 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] 2^k, {k, 0, n - 1}], {n, 1, 20}]]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 26 2023
STATUS
approved