A350290 - OEIS

%I #11 Jul 27 2022 08:09:01

%S 1,1,-3,-2,21,-4,-150,155,1029,-2072,-6468,22056,34122,-208857,

%T -106249,1816958,-639067,-14629264,17635800,108117620,-239571684,

%U -711876496,2628772968,3825823888,-25582846134,-10997156129,227594431035,-98360217830,-1864646227185

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

%F a(n) = (-1)^(n-1)*n^2*hypergeom([1 - n, 1 - n, n + 1], [3/2, 2], -1/4) for n >= 1.

%F D-finite with recurrence 4*n*(2*n-1)*(9789*n-26254)*a(n) +2*(28924*n^3-27550*n^2-236727*n+284748)*a(n-1) +2*(342172*n^3-1352012*n^2+1027500*n+356439)*a(n-2) -2*(n-3)*(43143*n^2-783097*n+1918735)*a(n-3) -5*(5116*n-30173)*(n-3)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jul 27 2022

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

%p seq(a(n), n = 0..28);

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*binomial(n+k-1, n-k)); \\ _Michel Marcus_, Mar 07 2022

%Y Cf. A109081, A161798, A262440, A262441, A262442.

%K sign

%O 0,3

%A _Peter Luschny_, Mar 07 2022