A185109 a(0)=2; for n > 0, a(n) = (n+2)*a(n-1) + 1.

2, 7, 29, 146, 877, 6140, 49121, 442090, 4420901, 48629912, 583558945, 7586266286, 106207728005, 1593115920076, 25489854721217, 433327530260690, 7799895544692421, 148198015349156000, 2963960306983120001, 62243166446645520022

Offset: 0

Author

Olivier Gérard, Nov 02 2012

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Jia Huang and Erkko Lehtonen, Associative-commutative spectra for some varieties of groupoids, arXiv:2401.15786 [math.CO], 2024. See p. 10.

Programs

Magma

[n le 1 select 2 else (n+1)*Self(n-1)+1: n in [1..20]]; // Vincenzo Librandi, Dec 22 2012

Mathematica

RecurrenceTable[{a[0]==2, a[n]==(n+2)*a[n-1] + 1}, a, {n, 20}] (* Vincenzo Librandi, Dec 23 2012 *)

Formula

a(n) = e*Gamma(n+3,1)-(3/2)*(n+2)!, where Gamma(a,x) is the incomplete gamma function. [Bruno Berselli, Dec 24 2012]

a(n) = (n+3)*a(n-1) - (n+1)*a(n-2). - Vaclav Kotesovec, Aug 13 2013

a(n) ~ (exp(1)-3/2)*sqrt(2*Pi)*exp(-n)*n^(n+5/2). - Vaclav Kotesovec, Aug 13 2013

Extensions

Edited by Vincenzo Librandi and N. J. A. Sloane, Dec 24 2012