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

0,3

Links

Table of n, a(n) for n=0..24.

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)

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);

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 *)

Crossrefs

Cf. A099174, A344500.

Sequence in context: A052978 A351511 A151023 * A151024 A151025 A333799

Adjacent sequences: A344498 A344499 A344500 * A344502 A344503 A344504

Keyword

nonn

Author

Peter Luschny, May 22 2021

Status

approved