A322943 a(n) = n! [x^n] -exp(-1/(3*(x - 1)^3) - 1/3)/(x - 1).

1, 2, 9, 60, 513, 5286, 63417, 865824, 13229505, 223336458, 4123226601, 82559530692, 1780580892609, 41125146159150, 1012187976013593, 26434618529133096, 729843662368002177, 21233024209964157714, 649022529915336217545, 20789723945673232443468, 696253958136289126229121

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = (4*n - 2)*a(n-1) - 3*(n - 1)*(2*n - 3)*a(n-2) + (n - 1)*(n - 2)*(4*n - 9)*a(n-3) - (n - 2)*(n - 1)*(n - 3)^2*a(n-4) for n >= 4.

a(n) ~ exp(-1/4 + 5*n^(1/4)/24 + sqrt(n)/2 + 4*n^(3/4)/3 - n) * n^(n + 1/8) / 2 * (1 + 1357/(2560*n^(1/4))). - Vaclav Kotesovec, Jan 02 2019

Maple

a := proc(n) option remember; local e, b, c, d;
e := n -> 4*n-2; b := n -> 3*(n-1)*(2*n-3);
c := n -> (n-1)*(n-2)*(4*n-9); d := n -> (n-2)*(n-1)*(n-3)^2;
if n < 4 then return [1, 2, 9, 60][n+1] fi;
e(n)*a(n-1) - b(n)*a(n-2) + c(n)*a(n-3) - d(n)*a(n-4) end:
seq(a(n), n=0..20);

Prog

(Sage) # uses[riordan_square from A321620]
R = riordan_square((1 - 3*x)^(-1/3), 24, True).inverse()
[sum([(-1)^(n-k)*c for k, c in enumerate(R.row(n))]) for n in (0..23)]

Crossrefs

Row sums of A322944.

Cf. A322944, A321620, A321965.

Sequence in context: A120970 A339360 A111558 * A168449 A001193 A161391

Adjacent sequences: A322940 A322941 A322942 * A322944 A322945 A322946

Keyword

nonn

Author

Peter Luschny, Jan 02 2019

Status

approved