OFFSET
0,2
FORMULA
a(n) = [x^n] (1/x) * Series_Reversion( x * (1 - n * x) / (1 + x)^n ).
a(n) ~ phi^(3*n + 3/2) * exp(n/phi^2 + 1/(2*phi)) * n^(n - 3/2) / (5^(1/4) * sqrt(2*Pi)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Sep 27 2023
MAPLE
A366038 := proc(n)
add(binomial(n+k, k)*binomial(n*(n+1), n-k)*n^k, k=0..n) ;
%/(n+1) ;
end proc:
seq(A366038(n), n=0..80) ; # R. J. Mathar, Oct 24 2024
MATHEMATICA
Unprotect[Power]; 0^0 = 1; Table[1/(n + 1) Sum[Binomial[n + k, k] Binomial[n (n + 1) , n - k] n^k, {k, 0, n}], {n, 0, 16}]
Table[Binomial[n (n + 1), n] Hypergeometric2F1[-n, n + 1, n^2 + 1, -n]/(n + 1), {n, 0, 16}]
Table[SeriesCoefficient[(1/x) InverseSeries[Series[x (1 - n x)/(1 + x)^n, {x, 0, n + 1}], x], {x, 0, n}], {n, 0, 16}]
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Ilya Gutkovskiy, Sep 26 2023
STATUS
approved