cp's OEIS Frontend

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

%I A323207 #8 Feb 26 2019 03:58:01
%S A323207 1,2,4,10,33,141,752,4825,36027,305132,2879840,29909421,338479429,
%T A323207 4139716658,54339861530,761150445734,11322139144239,178116143657889,
%U A323207 2952831190016238,51423702126549166,938126972940647197,17883424301972473339
%N A323207 a(n) = Sum_{k=0..n} hypergeometric([-k, k + 1], [-k - 1], n - k).
%F A323207 a(n) = Sum_{k=0..n} A323206(n-k, k).
%F A323207 a(n) = Sum_{k=0..n} Sum_{j=0..k} A238762(2*j, 2*k)*(n-k)^j.
%F A323207 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).
%p A323207 # The function ballot is defined in A238762.
%p A323207 A323207 := n -> add(add(ballot(2*j, 2*k)*(n-k)^j, j=0..k), k=0..n):
%p A323207 seq(A323207(n), n=0..21);
%t A323207 a[n_] := Sum[Hypergeometric2F1[-k, k + 1, -k - 1, n - k], {k, 0, n}];
%t A323207 Table[a[n], {n, 0, 21}]
%Y A323207 Cf. A323206, A238762.
%K A323207 nonn
%O A323207 0,2
%A A323207 _Peter Luschny_, Feb 25 2019