A217392 Alternating sums of the squares of the ordered Bell numbers (number of preferential arrangements) A000670.

1, 0, 9, 160, 5465, 287216, 21643273, 2214984576, 295720862649, 49933547619472, 10404630591819497, 2622531836368780832, 786513638108085303193, 276793205620647080017968, 112961387008976003691598281, 52917386659933341334644891328, 28203267311410367019573922744697

Offset: 0

Author

Emanuele Munarini, Oct 02 2012

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000670, A006957, A005649, A217388, A217389, A217391.

Programs

Magma

A000670:=func;
[&+[(-1)^(n-k)*A000670(k)^2: k in [0..n]]: n in [0..14]]; // Bruno Berselli, Oct 03 2012

Mathematica

t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[(-1)^(n-k)t[k]^2, {k, 0, n}], {n, 0, 100}]

Maxima

t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum((-1)^(n-k)*t(k)^2,k,0,n),n,0,40);

PARI

for(n=0,30, print1(sum(k=0,n, (-1)^(n-k)*(sum(j=0,k, j!*stirling(k,j,2)))^2), ", ")) \\ G. C. Greubel, Feb 07 2018

Formula

a(n) = sum((-1)^(n-k)*t(k)^2, k=0..n), where t = A000670 (ordered Bell numbers).

a(n) ~ (n!)^2 / (4 * (log(2))^(2*n+2)). - Vaclav Kotesovec, Nov 08 2014