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A366015
G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 3 * A(x)).
5
0, 1, 7, 76, 995, 14433, 223300, 3611016, 60305787, 1032115315, 18007816255, 319110233104, 5727667197044, 103913426353324, 1902498385538520, 35106179258551632, 652236828560562987, 12190651925663309175, 229059610932456616501, 4324334144117016053500, 81983637468108446363755
OFFSET
0,3
COMMENTS
Reversion of g.f. for hexagonal pyramidal numbers (with signs).
LINKS
Eric Weisstein's World of Mathematics, Hexagonal 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) * 3^k for n > 0.
MATHEMATICA
nmax = 20; A[_] = 0; Do[A[x_] = x (1 + A[x])^4/(1 - 3 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 3 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] 3^k, {k, 0, n - 1}], {n, 1, 20}]]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 26 2023
STATUS
approved