A344556 - OEIS

%I #7 May 24 2021 23:34:04

%S 1,1,5,34,315,3741,54531,944035,18934763,431773963,11030464423,

%T 312023972228,9680623848325,326823162461823,11926991260987869,

%U 467837288974848642,19628089812933434547,877052336082168698715,41581946832665768549007,2084818230218269733957646

%N a(n) = [x^n] 2 / (1 - (2*n - 1)*x + sqrt(1 - 2*x - 3*x^2)).

%C Main diagonal of A344567.

%F a(n) = Sum_{j=0..n} (n - 1)^j*binomial(n, j)*hypergeom([(j - n)/2, (j - n + 1)/2], [j + 2], 4).

%F a(n) ~ n^n * (1 + 1/n + 1/(2*n^2) - 4/(3*n^3) - 119/(24*n^4) - 1249/(120*n^5) - ...). - _Vaclav Kotesovec_, May 24 2021

%p aList := proc(len) 2 / (1 - (2*n - 1)*x + sqrt(1 - 2*x - 3*x^2));

%p seq(coeff(series(%, x, len+2), x, n), n = 0..len) end: aList(19);

%t Unprotect[Power]; 0^0 := 1;

%t a[n_] := Sum[(n-1)^j Binomial[n, j] Hypergeometric2F1[(j - n)/2, (j - n + 1)/2, j + 2, 4], {j, 0, n}]; Table[a[n], {n, 0, 19}]

%Y Cf. A344567.

%K nonn

%O 0,3

%A _Peter Luschny_, May 24 2021