A303224 a(0)=0, a(1)=1; for n>1, a(n) = n*a(n-1) - 3*a(n-2).

0, 1, 2, 3, 6, 21, 108, 693, 5220, 44901, 433350, 4632147, 54285714, 691817841, 9522592632, 140763435957, 2223647197416, 37379712048201, 666163875275370, 12544974494087427, 248900998255922430, 5189286039892108749, 113417589882858625188, 2593036709186072053077, 61892628250817153398284

Offset: 0

Author

Bruno Berselli, Apr 20 2018

Keywords

• nonn

Comments

a(n) is divisible by 3^floor(n/3).

Links

• Colin Barker, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A058798: a(n) = n*a(n-1) - a(n-2).

Cf. A222470: a(n) = n*a(n-1) - 2*a(n-2), without 0.

Cf. A222472: a(n) = n*a(n-1) + 3*a(n-2), without 0.

Cf. A221913.

Programs

Mathematica

RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == n a[n - 1] - 3 a[n - 2]}, a, {n, 0, 30}]
Flatten[{0, Table[n!*HypergeometricPFQ[{1/2 - n/2, 1 - n/2}, {2, 1 - n, -n}, -12], {n, 1, 25}]}] (* Vaclav Kotesovec, Apr 20 2018 *)
Round[Table[-2 I^n 3^(n/2) (BesselI[1 + n, -2 I Sqrt[3]] BesselK[1, -2 I Sqrt[3]] + (-1)^n BesselI[1, 2 I Sqrt[3]] BesselK[1 + n, -2 I Sqrt[3]]), {n, 0, 25}]] (* Vaclav Kotesovec, Apr 20 2018 *)

PARI

a=vector(30); a[1]=0; a[2]=1; for(n=3, #a, a[n]=(n-1)*a[n-1]-3*a[n-2]); a

Formula

From Peter Bala, Apr 20 2018: (Start)

a(n) = Sum_{k = 0..floor((n-1)/2)} (-3)^k*binomial(n-k,k+1)*binomial(n-k-1,k)*(n-2*k-1)!.

a(n)/n! ~ BesselJ(1, 2*sqrt(3)) / sqrt(3). (End)

a(n) = -2 * i^n * 3^(n/2) * (BesselI(1+n, -2*i*sqrt(3)) * BesselK(1,-2*i*sqrt(3)) + (-1)^n * BesselI(1, 2*i*sqrt(3)) * BesselK(1+n, -2*i*sqrt(3))), where i is the imaginary unit. - Vaclav Kotesovec, Apr 20 2018