1,2

a(n) = (1/(4n!)) * Sum_{r, s>=0} (rs)_n / 2^(r+s) }, where (m)_n is the falling factorial m * (m-1) * ... * (m-n+1). [Maia and Mendez]

Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, p. 435 (IV, 4. Mitteilungen zur Lehre vom Transfiniten, VIII Nr. 13), Springer, Berlin.

P. J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394.

M. Maia and M. Mendez, On the arithmetic product of combinatorial species

a(n) = (Sum s(n, k) * P(k)^2)/n!, where P(n) is the number of labeled total preorders on {1, ..., n} (A000670), s are signed Stirling numbers of the first kind.

G.f.: Sum_{m>=0,n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1+x)^j-1)^m. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 25 2006

Inverse binomial transform of A007322. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 17 2006

G.f.: Sum_{n>=0} 1/(2-(1+x)^n)/2^(n+1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 23 2006

a(2)=4:

[1 1] [1] [1 0] [0 1]

..... [1] [0 1] [1 0]

(GAP) P:=function(n) return Sum([1..n], x->Stirling2(n, x)*Factorial(x)); end;

(GAP) F:=function(n) return Sum([1..n], x->(-1)^(n-x)*Stirling1(n, x)*P(x)^2)/Factorial(n); end;

Cf. A000670 (the sequence (P(n)).

Cf. A049311 (row and column permutations allowed).

Sequence in context: A073840 A024249 A007145 this_sequence A099021 A136229 A138419

Adjacent sequences: A101367 A101368 A101369 this_sequence A101371 A101372 A101373

easy,nonn

Peter J. Cameron (p.j.cameron(AT)qmul.ac.uk), Jan 14 2005

Cantor reference from Rainer Rosenthal, Apr 10 2007