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A365817
G.f. A(x) satisfies: A(x) = x * (1 + A(x))^3 / (1 - 4 * A(x)).
5
0, 1, 7, 80, 1119, 17437, 290532, 5066364, 91311055, 1687341227, 31797227631, 608727899936, 11805599569092, 231454163924700, 4579765707561240, 91340133073920420, 1834295500622405295, 37059418988408887015, 752741444501505866325, 15362331852042084534240, 314860558967057266779495
OFFSET
0,3
COMMENTS
Reversion of g.f. for heptagonal numbers (with signs).
LINKS
Eric Weisstein's World of Mathematics, Heptagonal 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(3*n,n-k-1) * 4^k for n > 0.
a(n) ~ 5 * (81 + 21*sqrt(21))^n / (sqrt((427 + 93*sqrt(21))*Pi) * n^(3/2) * 2^(3*n + 3/2)). - Vaclav Kotesovec, Sep 26 2023
MATHEMATICA
nmax = 20; A[_] = 0; Do[A[x_] = x (1 + A[x])^3/(1 - 4 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 4 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] 4^k, {k, 0, n - 1}], {n, 1, 20}]]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 25 2023
STATUS
approved