A344501 a(n) = Sum_{k=0..n} binomial(n, k)*HT(n, k) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*HT(n, k), where HT(n, k) is the Hermite triangle A099174.
1, 1, 2, 10, 40, 176, 916, 4852, 27350, 163270, 1009396, 6504356, 43400512, 298682320, 2118282440, 15433768456, 115345136566, 882900083222, 6910879999420, 55255039432300, 450744068706896, 3747796352076736, 31734090674951512, 273414453918459800, 2395202886317347900
Offset: 0
Keywords
Programs
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Maple
a := proc(n) add((if n - j mod 2 = 0 then binomial(n, j)*2^((j - n)/2)*n!/(j!*((n - j)/2)!) else 0 fi), j = 0..n) end: seq(a(n), n = 0..24);
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Mathematica
Table[n! * Sum[Binomial[n, 2*j] / (2^j * (n - 2*j)! * j!), {j, 0, n/2}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 21 2024 *)
Formula
a(n) = Sum_{j=0..n} even(n - j)*binomial(n, j)*2^((j - n)/2)*n!/(j!*((n - j)/2)!), where even(k) = 1 if k is even and otherwise 0.
From Vaclav Kotesovec, Apr 21 2024: (Start)
Recurrence: n*(9*n - 13)*a(n) = (3*n - 4)*(9*n - 5)*a(n-1) + (18*n^3 - 89*n^2 + 131*n - 56)*a(n-2) + (54*n^3 - 219*n^2 + 261*n - 92)*a(n-3) - (n-3)^2*(n-1)*(9*n - 4)*a(n-4).
a(n) ~ n^(n/2 - 3/8) / (2^(3/2) * sqrt(Pi) * exp(n/2 - 2*n^(3/4) + 3*sqrt(n)/4 - 5*n^(1/4)/16 + 1/8)) * (1 + 5351/(5120*n^(1/4))). (End)