A226507 4*B(n+4) - (4*n+15)*B(n+3) + (n^2+8*n+9)*B(n+2) - (4*n+3)*B(n+1) + n*B(n), where B(i) are the Bell numbers A000110.

0, 0, 0, 1, 16, 177, 1726, 15912, 143148, 1279939, 11504326, 104686659, 968808308, 9144180028, 88184565504, 869867691833, 8781919559956, 90765497635245, 960434143555986, 10403548856756708, 115336464546432180, 1308260884070774299, 15177980646442995698, 180036437138753006607, 2182526416321158803528

Offset: 0

Author

N. J. A. Sloane, Jun 10 2013

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• B. Chern, P. Diaconis, D. M. Kane, and R. C. Rhoades, Closed expressions for averages of set partition statistics, arXiv:1304.4309 [math.CO], 2013.

Crossrefs

Cf. A000110, A200580.

Programs

Magma

[4*Bell(n+4)-(4*n+15)*Bell(n+3)+(n^2+8*n+9)*Bell(n+2)-(4*n+3)*Bell(n+1)+n*Bell(n): n in [0..30]]; // Vincenzo Librandi, Jul 16 2013

Maple

A000110 := proc(n) option remember; if n <= 1 then 1 else add( binomial(n-1, i)*A000110(n-1-i), i=0..n-1); fi; end;
B:=A000110;
f:=n->4*B(n+4) - (4*n+15)*B(n+3) + (n^2+8*n+9)*B(n+2) - (4*n+3)*B(n+1) + n*B(n);
  [seq(f(n),n=0..30)];

Mathematica

Table[4 BellB[n+4] - (4 n + 15) BellB[n + 3] + (n^2 + 8 n + 9) BellB[n+2] - (4 n + 3) BellB[n+1] + n BellB[n],{n, 0, 30}] (* Vincenzo Librandi, Jul 16 2013 *)

Formula

a(n) ~ n^4 * Bell(n) / LambertW(n)^2 * (1 - 4/LambertW(n) + 4/LambertW(n)^2). - Vaclav Kotesovec, Jul 28 2021