A366038 - OEIS

%I #11 Oct 24 2024 05:48:33

%S 1,2,25,658,27193,1548526,112916830,10062563610,1061196371665,

%T 129369938790070,17909387604206371,2776290021986848588,

%U 476539253976442601735,89736215305419802692184,18395742890606906720656524,4078527943680251523126851306,972490249766494185823234587681

%N a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(n*(n+1),n-k) * n^k.

%F a(n) = [x^n] (1/x) * Series_Reversion( x * (1 - n * x) / (1 + x)^n ).

%F 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

%p A366038 := proc(n)

%p     add(binomial(n+k,k)*binomial(n*(n+1),n-k)*n^k,k=0..n) ;

%p     %/(n+1) ;

%p end proc:

%p seq(A366038(n),n=0..80) ; # _R. J. Mathar_, Oct 24 2024

%t 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}]

%t Table[Binomial[n (n + 1), n] Hypergeometric2F1[-n, n + 1, n^2 + 1, -n]/(n + 1), {n, 0, 16}]

%t Table[SeriesCoefficient[(1/x) InverseSeries[Series[x (1 - n x)/(1 + x)^n, {x, 0, n + 1}], x], {x, 0, n}], {n, 0, 16}]

%Y Cf. A006318, A064063, A135861, A291699, A292798, A365816, A366012, A366016, A366037.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Sep 26 2023