A390311 a(n) = n!*(4^(n+1) - 3^(n+1)).

1, 7, 74, 1050, 18744, 404040, 10221840, 297234000, 9776027520, 359079557760, 14577459321600, 648479351520000, 31381569962265600, 1641769520520729600, 92356064213443430400, 5560127550699321600000, 356748820743606939648000, 24304853343006640115712000, 1752429851448311972192256000

Offset

0,2

References

Miklos Bona, Introduction to Enumerative and Analytic Combinatorics, CRC Press, 2025, pp. 142-144.

Links

Robert Israel, Table of n, a(n) for n = 0..364

Formula

a(n) = Sum_{i=0..n} binomial(n,i)*3^i*i!*4^(n-i)*(n - i)!.

E.g.f.: 1/((1 - 3*x)*(1 - 4*x)).

D-finite with recurrence: 12*a(n)*(n + 1)*(n + 2) - 7*(n + 2)*a(n + 1) + a(n + 2) = 0. - Robert Israel, Apr 01 2026

Maple

f:= gfun:-rectoproc({12*a(n)*(n + 1)*(n + 2) - 7*(n + 2)*a(n + 1) + a(n + 2), a(0)=1, a(1) = 7}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Apr 01 2026

Mathematica

a[n_]:=n!(4^(n+1)-3^(n+1)); Array[a, 19, 0]

Crossrefs

Cf. A000142, A000244, A000583, A005061, A007318.

Sequence in context: A386772 A341330 A379205 * A266305 A098118 A097821

Adjacent sequences: A390308 A390309 A390310 * A390312 A390313 A390314

Keyword

nonn,easy

Author

Stefano Spezia, Mar 27 2026

Status

approved