A113368 - OEIS

%I #5 Jun 13 2017 23:21:16

%S 1,0,1,0,2,1,0,5,4,1,0,19,22,6,1,0,113,166,51,8,1,0,966,1671,561,92,

%T 10,1,0,10958,21510,7726,1324,145,12,1,0,156700,341463,129406,23010,

%U 2575,210,14,1,0,2727794,6496923,2572892,471724,53935,4434,287,16,1

%N Triangle, read by rows, given by the product Q^-1*P^2, where the triangular matrices involved are P = A113340 and Q = A113350.

%C Matrix product Q^-1*P^2 = SHIFT_DOWN_RIGHT(Q). Compare to the matrix product Q^2*P^-1 = SHIFT_LEFT_UP(P), as given by triangle A113369.

%e The product Q^-1*P^2 forms a triangle that begins:

%e 1;

%e 0,1;

%e 0,2,1;

%e 0,5,4,1;

%e 0,19,22,6,1;

%e 0,113,166,51,8,1;

%e 0,966,1671,561,92,10,1;

%e 0,10958,21510,7726,1324,145,12,1;

%e 0,156700,341463,129406,23010,2575,210,14,1;

%e 0,2727794,6496923,2572892,471724,53935,4434,287,16,1; ...

%e Compare Q^-1*P^2 to Q (A113350) which begins:

%e 1;

%e 2,1;

%e 5,4,1;

%e 19,22,6,1;

%e 113,166,51,8,1;

%e 966,1671,561,92,10,1;

%e 10958,21510,7726,1324,145,12,1; ...

%o (PARI) T(n,k)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B);(A^(2*k))[n-k+1,1]

%Y Cf. A113340, A113350, A113369 (Q^2*P^-1).

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Nov 12 2005