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A366038
a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(n*(n+1),n-k) * n^k.
0
1, 2, 25, 658, 27193, 1548526, 112916830, 10062563610, 1061196371665, 129369938790070, 17909387604206371, 2776290021986848588, 476539253976442601735, 89736215305419802692184, 18395742890606906720656524, 4078527943680251523126851306, 972490249766494185823234587681
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}]
KEYWORD
nonn,changed
AUTHOR
Ilya Gutkovskiy, Sep 26 2023
STATUS
approved