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A376871 a(n) = Sum_{k=0..n} n^k * hypergeom([-k, k - n], [1], 2).

%I A376871 #11 Oct 13 2024 02:08:05
%S A376871 1,2,11,88,941,12546,200479,3735264,79524793,1905008050,50720779691,
%T A376871 1486111590360,47524305052069,1647275572867666,61522053792814679,
%U A376871 2463133651846231936,105244572157172848369,4780359272226823337250,230016032074517010618403,11688053305141450955275800
%N A376871 a(n) = Sum_{k=0..n} n^k * hypergeom([-k, k - n], [1], 2).
%F A376871 a(n) is the Delannoy polynomial D(n, x) evaluated at x = n.
%F A376871 a(n) ~ exp(2) * n^n. - _Vaclav Kotesovec_, Oct 13 2024
%p A376871 a := n -> add(A008288(n, k)*n^k, k=0..n):
%p A376871 seq(a(n), n=0..19);
%t A376871 Unprotect[Power]; Power[0, 0] = 1; Protect[Power];
%t A376871 a[n_] := Sum[n^k Hypergeometric2F1[-k, k - n, 1, 2], {k, 0, n}];
%t A376871 Table[a[n], {n, 0, 19}]
%Y A376871 D(n, 1) = A000128(n + 1), D(n, -1) = A056594(n).
%Y A376871 Cf. A008288.
%K A376871 nonn
%O A376871 0,2
%A A376871 _Peter Luschny_, Oct 12 2024