A323207 a(n) = Sum_{k=0..n} hypergeometric([-k, k + 1], [-k - 1], n - k).

1, 2, 4, 10, 33, 141, 752, 4825, 36027, 305132, 2879840, 29909421, 338479429, 4139716658, 54339861530, 761150445734, 11322139144239, 178116143657889, 2952831190016238, 51423702126549166, 938126972940647197, 17883424301972473339

Offset: 0

Author

Peter Luschny, Feb 25 2019

Keywords

• nonn

Crossrefs

Cf. A323206, A238762.

Programs

Maple

# The function ballot is defined in A238762.
A323207 := n -> add(add(ballot(2*j, 2*k)*(n-k)^j, j=0..k), k=0..n):
seq(A323207(n), n=0..21);

Mathematica

a[n_] := Sum[Hypergeometric2F1[-k, k + 1, -k - 1, n - k], {k, 0, n}];
Table[a[n], {n, 0, 21}]

Formula

a(n) = Sum_{k=0..n} A323206(n-k, k).

a(n) = Sum_{k=0..n} Sum_{j=0..k} A238762(2*j, 2*k)*(n-k)^j.

a(n) = Sum_{k=0..n} Sum_{j=0..n-k} (binomial(2*(n-k)-j, n-k) - binomial(2*(n-k)-j, n-k+1))*k^(n-k-j).