editing

approved

(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]}

approved

editing

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Nov 12 2005

nonn,tabl,new

Paul D . Hanna (pauldhanna(AT)juno.com), Nov 12 2005

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.

nonn,tabl,new

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

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, 72, 12, 1, 464, 1680, 2194, 1366, 480, 104, 14, 1, 1818, 7656, 11294, 7976, 3222, 828, 142, 16, 1, 8122, 39256, 64152, 50186, 22590, 6568, 1316, 186, 18, 1

0,2

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.

Triangle begins:

1;

2,1;

4,4,1;

8,12,6,1;

18,36,26,8,1;

46,116,108,46,10,1;

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

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

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

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

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

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

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

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

(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]}

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

nonn,tabl

Paul D Hanna (pauldhanna(AT)juno.com), Nov 12 2005

approved