A350309 a(n) = (n+2)*a(n-1) + (n+1)*(A003422(n) - 4)/6 for n > 0 with a(0) = 1.

1, 2, 7, 35, 215, 1535, 12455, 113255, 1141415, 12632615, 152341415, 1988514215, 27934434215, 420236744615, 6740662856615, 114841743944615, 2071122598472615, 39418302548552615, 789563088403016615, 16603426141551176615, 365724864314899016615, 8421063413387754056615

Offset: 0

Author

Mikhail Kurkov, Dec 24 2021

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445

Crossrefs

Cf. A003422, A006157 (first differences).

Programs

Magma

[n le 2 select n else ((n-1)*(2*n+3)*Self(n-1) - n*(2*n-1)*Self(n-2))/(2*n-3): n in [1..30]]; // G. C. Greubel, Jan 07 2025

Mathematica

a[n_]:= a[n]= If[n<2, n+1, (n*(2*n+5)*a[n-1] - (n+1)*(2*n+1)*a[n-2])/(2*n-1)]; (* a = A350309 *)
Table[a[n], {n,0,30}] (* G. C. Greubel, Jan 07 2025 *)

PARI

lf(n) = sum(k=0, n-1, k!); \\ A003422
a(n) = if (n, (n+2)*a(n-1) + (n+1)*(lf(n) - 4)/6, 1); \\ Michel Marcus, Jan 11 2022

Python

from sage.all import * # remove for SageMath
@CachedFunction # a = A350309
def a(n): return n+1 if n<2 else (n*(2*n+5)*a(n-1) - (n+1)*(2*n+1)*a(n-2))//(2*n-1)
print([a(n) for n in range(31)]) # G. C. Greubel, Jan 07 2025

Formula

Recurrence: (2*n-1)*a(n) = n*(2*n+5)*a(n-1) - (n+1)*(2*n+1)*a(n-2). - Vaclav Kotesovec, Nov 21 2024