0,5

A(n,m) is the number of ways to pair the elements of two sets (with respectively n and m elements), where each element of either set may be paired with zero or one elements of the other set; number of n x m matrices of zeros and ones with at most one one in each row and column. E.g. A(2,2)=7 because we can pair {A,B} with {C,D} as {AB,CD}, {AC,BD}, {AC,B,D}, {AD,B,C}, {BC,A,D}, {BD,A,C}, or {A,B,C,D}. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 06 2006

E.g.f.: exp(x)/(1-y-xy)=Sum_{i, j} A(i, j) y^j x^i/i!.

A(i, j) = A(i-1, j)+j*A(i-1, j-1)+(i==0) = A(j, i).

T(n, k)=sum{j=0..k, C(n, k-j)*k!/j!}=sum{j=0..k, (k-j)!*C(k, j)C(n, k-j)}; - Paul Barry (pbarry(AT)wit.ie), Nov 14 2005

A(i,j) = sum_k C(i,k)*C(j,k)*k!. E.g.f. sum_{i,j} a(i,j)*x^i/i!*y^j/j! = e^{x+y+xy}. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 06 2006

The LDU factorization of this array, formatted as a square array, is P * D * transpose(P), where P is Pascal's triangle A007318 and D = diag(0!, 1!, 2!, ... ). Compare with A099597. - Peter Bala (pbala(AT)toucansurf.com), Nov 06 2007

Antidiagonals: 1; 1,1; 1,2,1; 1,3,3,1; 1,4,7,4,1; ...

(PARI) A(i, j)=if(i<0|j<0, 0, i!*polcoeff(exp(x+x*O(x^i))*(1+x)^j, i))

(PARI) A(i, j)=if(i<0|j<0, 0, i!*polcoeff(exp(x/(1-x)+x*O(x^i))*(1-x)^(i-j-1), i))

(PARI) A(i, j)=local(M); if(i<0|j<0, 0, M=matrix(j+1, j+1, n, m, if(n==m, 1, if(n==m+1, m))); (M^i)[j+1, ]*vectorv(j+1, n, 1)) /* Michael Somos Jul 03 2004 */

Row sums give A081124.

Main diagonal is A002720.

Cf. A099597.

Sequence in context: A108350 A086617 A094526 this_sequence A101515 A028657 A053534

Adjacent sequences: A088696 A088697 A088698 this_sequence A088700 A088701 A088702

nonn,tabl

Michael Somos, Oct 08 2003