A277362 Self-convolution of a(n)/4^n gives factorials (A000142).

1, 2, 14, 164, 2646, 53852, 1316364, 37467080, 1215510118, 44249471916, 1785942489700, 79150848980216, 3821494523507708, 199668288426274968, 11225643465179779544, 675769562728962818448, 43370783734391689628294, 2956329387192674856638668

Offset: 0

Author

Vladimir Reshetnikov, Oct 10 2016

Keywords

• nonn

Comments

Self-convolution of a(n) gives A047053.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..360

Crossrefs

Cf. A000142, A000302, A047053.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
     (n!*4^n-add(a(k)*a(n-k), k=1..n-1))/2)
    end:
seq(a(n), n=0...20);  # Alois P. Heinz, Oct 12 2016

Mathematica

With[{n = 20}, Sqrt[Sum[k! (4 x)^k, {k, 0, n - 1}] + O[x]^n][[3]]]
CoefficientList[Series[Sqrt[-Gamma[0, -1/(4*x)]/(x*E^(1/(4*x)))]/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 27 2021 *)

Formula

Sum_{k=0..n} a(k)/4^k * a(n-k)/4^(n-k) = n!.

a(n) ~ 2^(2*n-1) * n!. - Vaclav Kotesovec, Oct 27 2021