cp's OEIS Frontend

A382405 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n+k,k) * 2^(n-k).

%I A382405 #24 Jun 22 2025 00:16:14
%S A382405 1,4,34,352,4006,48184,600916,7687936,100240198,1326277144,
%T A382405 17753591164,239915864896,3267780399196,44805617380528,
%U A382405 617844108170344,8561667414341632,119151750609504838,1664497333624420888,23330380347342383404,327990673915214512192,4623496960858710060916
%N A382405 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n+k,k) * 2^(n-k).
%C A382405 Diagonal of the rational function 1 / (1 - x - x*y - y*z - 2*x*z - 2*x*y*z).
%F A382405 a(n) = 2^n * hypergeom([-n, -n, n+1], [1, 1], 1/2).
%F A382405 From _Peter Bala_, May 23 2025: (Start)
%F A382405 a(n) = Sum_{k = 0..n} (-1)^(n-k) * binomial(n, k)*binomial(2*k, n)^2.
%F A382405 (11*n - 16)*n^2*a(n) = 2*(77*n^3 - 189*n^2 + 132*n - 30)*a(n-1) + 4*(33*n^3 - 114*n^2 + 124*n - 40)*a(n-2) + 4*(11*n - 5)*(n - 2)^2*a(n-3) with a(0) = 1, a(1) = 4 and a(2) = 34. (End)
%F A382405 a(n) ~ sqrt((55 + (22*(7513 - 183*sqrt(33)))^(1/3) + (22*(7513 + 183*sqrt(33)))^(1/3)) / 33) * ((14 + (1/3)*(95958 - 1782*sqrt(33))^(1/3) + (2*(1777 + 33*sqrt(33)))^(1/3)) / 3)^n / (2*Pi*n). - _Vaclav Kotesovec_, Jun 07 2025
%p A382405 seq(simplify(2^n*hypergeom([-n, -n, n+1], [1, 1], 1/2)), n = 0..20); # _Peter Bala_, May 23 2025
%t A382405 Table[Sum[Binomial[n, k]^2 Binomial[n + k, k] 2^(n - k), {k, 0, n}], {n, 0, 20}]
%t A382405 Table[2^n HypergeometricPFQ[{-n, -n, n + 1}, {1, 1}, 1/2], {n, 0, 20}]
%t A382405 Table[SeriesCoefficient[1/(1 - x - x y - y z - 2 x z - 2 x y z), {x, 0, n}, {y, 0, n}, {z, 0, n}], {n, 0, 20}]
%Y A382405 Cf. A001850, A005258, A069835, A274671, A382642, A382848.
%K A382405 nonn,easy
%O A382405 0,2
%A A382405 _Ilya Gutkovskiy_, Apr 08 2025