Search: id:A088699
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%I A088699
%S A088699 1,1,1,1,2,1,1,3,3,1,1,4,7,4,1,1,5,13,13,5,1,1,6,21,34,21,6,1,1,7,31,73,
%T A088699 73,31,7,1,1,8,43,136,209,136,43,8,1,1,9,57,229,501,501,229,57,9,1,1,10,
%U A088699 73,358,1045,1546,1045,358,73,10,1,1,11,91,529,1961,4051,4051,1961
%N A088699 Array read by antidiagonals of coefficients of generating function exp(x)/
               (1-y-xy).
%C A088699 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
%C A088699 Compare with A086885. [From Peter Bala (pbala(AT)toucansurf.com), Sep 
               17 2008]
%F A088699 E.g.f.: exp(x)/(1-y-xy)=Sum_{i, j} A(i, j) y^j x^i/i!.
%F A088699 A(i, j) = A(i-1, j)+j*A(i-1, j-1)+(i==0) = A(j, i).
%F A088699 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
%F A088699 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
%F A088699 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
%e A088699 Antidiagonals: 1; 1,1; 1,2,1; 1,3,3,1; 1,4,7,4,1; ...
%o A088699 (PARI) A(i,j)=if(i<0|j<0,0,i!*polcoeff(exp(x+x*O(x^i))*(1+x)^j,i))
%o A088699 (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))
%o A088699 (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 
               */
%Y A088699 Row sums give A081124.
%Y A088699 Main diagonal is A002720.
%Y A088699 Cf. A099597.
%Y A088699 Sequence in context: A108350 A086617 A094526 this_sequence A101515 A028657 
               A053534
%Y A088699 Adjacent sequences: A088696 A088697 A088698 this_sequence A088700 A088701 
               A088702
%K A088699 nonn,tabl
%O A088699 0,5
%A A088699 Michael Somos, Oct 08 2003

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