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Triangle, read by rows, equal to the matrix square of A113983.
7

%I #6 Jun 13 2017 23:38:18

%S 1,2,1,4,4,1,8,12,6,1,18,36,26,8,1,46,116,108,46,10,1,136,416,468,248,

%T 72,12,1,464,1680,2194,1366,480,104,14,1,1818,7656,11294,7976,3222,

%U 828,142,16,1,8122,39256,64152,50186,22590,6568,1316,186,18,1

%N Triangle, read by rows, equal to the matrix square of A113983.

%F G.f.: A(x, y) = ( (1-x*y)*GF(A113983) - 1/(1-x) )/(x^2*y) (cf. A113983). T(n, 0) = T(n-2, 0) + T(n-1, 1) + 2.

%e Triangle begins:

%e 1;

%e 2,1;

%e 4,4,1;

%e 8,12,6,1;

%e 18,36,26,8,1;

%e 46,116,108,46,10,1;

%e 136,416,468,248,72,12,1;

%e 464,1680,2194,1366,480,104,14,1;

%e 1818,7656,11294,7976,3222,828,142,16,1;

%e 8122,39256,64152,50186,22590,6568,1316,186,18,1;

%e 41076,225348,402072,342584,168296,53816,12056,1968,236,20,1; ...

%e Notice that T(n+1,0) = T(n,1) + T(n-1,0) + 2:

%e T(7,0) = 464 = T(6,1) + T(5,0) = 416 + 46 + 2;

%e T(8,0) = 1818 = T(7,1) + T(6,0) = 1680 + 136 + 2.

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

%Y Cf. A113989 (column 0), A113990 (column 1), A113991 (column 2), A113992 (column 3); A113983, A113993.

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Nov 12 2005