%I #33 Feb 17 2021 03:50:56

%S 1,1,3,19,185,2426,39907,788019,18130401,475697854,14004694451,

%T 456820603086,16343563014649,636020474595988,26736885607750515,

%U 1207031709414024451,58225055056545820545,2988064457570991780854,162517551565531508113699,9336340704734213892357498

%N a(n) = P(n, 1, -2*n-1, 1-2*n)/(n+1), P the Jacobi polynomial.

%H Alois P. Heinz, <a href="/A242369/b242369.txt">Table of n, a(n) for n = 0..350</a>

%F a(n) = 2F1([1-n, -n], [2], n), 2F1 the hypergeometric function.

%F a(n) = Sum_{j=0..n-1} ( binomial(n,j)^2*(n-j)/(j+1)*n^(j-1) ), for n>0.

%F a(n) ~ (sqrt(n)+1)^(2*n+1)/(2*sqrt(Pi)*(n+1/2)^(9/4)). - _Peter Luschny_, Nov 17 2014

%p a := n -> `if`(n=0,1, add(binomial(n,j)^2*(n-j)/(j+1)*n^(j-1), j=0..n-1)); seq(a(n), n=0..20);

%t Table[JacobiP[n, 1, -2*n-1, 1-2*n]/(n+1), {n, 0, 20}]

%o (Sage)

%o def A242369(n): return 1 if n==0 else sum( binomial(n, j)^2*(n-j)*n^(j-1)/(j+1) for j in [0..n-1])

%o [A242369(n) for n in [0..20]] # _G. C. Greubel_, Feb 16 2021

%o (Magma)

%o A242369:= func< n | n eq 0 select 1 else (&+[Binomial(n, j)^2*(n-j)*n^(j-1)/(j+1): j in [0..n-1]]) >;

%o [A242369(n): n in [0..30]]; // _G. C. Greubel_, Feb 16 2021

%Y Main diagonal of A008550, A243631.

%Y Cf. A204057.

%K nonn

%O 0,3

%A _Peter Luschny_, Jun 08 2014