A225591 a(n) = B(n+3) - 6*B(n+2) + 8*B(n+1)*B(n+1) - B(n), where the B(i) are Bell numbers (A000110).

0, 16, 160, 1686, 21276, 328498, 6149136, 137105016, 3577543452, 107601726030, 3683660206080, 142035221781402, 6113719409724768, 291540411275223912, 15300594717301253800, 878667035554110785662, 54932693182800769213284

Offset: 0

Author

Michel Marcus, Jun 19 2013

Keywords

• nonn

Links

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

Crossrefs

Cf. A005493, A226506 (see Prop 3.1 (i) in Chern et al. link).

Programs

Magma

[Bell(n+3)-6*Bell(n+2)+8*Bell(n+1)*Bell(n+1)-Bell(n): n in [0..20]]; // Vincenzo Librandi, Jul 16 2013

Mathematica

Table[BellB[n+3] - 6 BellB[n+2] + 8 BellB[n+1] BellB[n+1] - BellB[n], {n, 0, 20}] (* Vincenzo Librandi, Jul 16 2013 *)
#[[4]]-6#[[3]]+8#[[2]]^2-#[[1]]&/@Partition[BellB[Range[0,20]],4,1] (* Harvey P. Dale, Nov 01 2016 *)

PARI

B(n) = if (n<=1, return (1), return (sum(i=0, n-1, binomial(n-1, i)*B(n-1-i))))
a(n) = B(n+3) - 6*B(n+2) + 8*B(n+1)*B(n+1) - B(n)