A382405 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n+k,k) * 2^(n-k).
1, 4, 34, 352, 4006, 48184, 600916, 7687936, 100240198, 1326277144, 17753591164, 239915864896, 3267780399196, 44805617380528, 617844108170344, 8561667414341632, 119151750609504838, 1664497333624420888, 23330380347342383404, 327990673915214512192, 4623496960858710060916
Offset: 0
Programs
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Maple
seq(simplify(2^n*hypergeom([-n, -n, n+1], [1, 1], 1/2)), n = 0..20); # Peter Bala, May 23 2025
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Mathematica
Table[Sum[Binomial[n, k]^2 Binomial[n + k, k] 2^(n - k), {k, 0, n}], {n, 0, 20}] Table[2^n HypergeometricPFQ[{-n, -n, n + 1}, {1, 1}, 1/2], {n, 0, 20}] 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}]
Formula
a(n) = 2^n * hypergeom([-n, -n, n+1], [1, 1], 1/2).
From Peter Bala, May 23 2025: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k) * binomial(n, k)*binomial(2*k, n)^2.
(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)
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
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