0,5

Determinant of n X n upper left submatrix is 3^[n(n-1)(n-2)/6]. Related to partitions of numbers into powers of 3 (see A078122). Triangle A078122 shifts left one column under matrix cube. Column 1 is A078125, which equals row sums of A078122; column 2 is A078124, which equals row sums of A078122^2.

T(n,k) = T(n,k-1) + T(n-1,3*k) for n>0, k>0, with T(0,n)=T(n,0)=1 for n>=0.

Recurrence T(n,k) = T(n,k-1) + T(n-1,3*k) is illustrated by:

T(3,3) = T(3,2) + T(2,9) = 93 + 145 = 238;

T(4,3) = T(4,2) + T(3,9) = 1632 + 4195 = 5827;

T(5,3) = T(5,2) + T(4,9) = 68457 + 273925 = 342382.

Rows of this table begin:

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...;

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, ...;

1, 23, 93, 238, 485, 861, 1393, 2108, 3033, 4195, 5621, ...;

1, 239, 1632, 5827, 15200, 32856, 62629, 109082, 177507, 273925,...;

1, 5828, 68457, 342382, 1144664, 3013980, 6769672, 13570796, ...;

1, 342383, 7112055, 50110483, 215155493, 690729981, 1828979530, ...;

1, 50110484, 1879090014, 18757984045, 103674882878, 406279238154,..;

1, 18757984046, 1287814075131, 18318289003447, 130648799730635, ...;

Triangle A078122 begins:

1;

1, 1;

1, 3, 1;

1, 12, 9, 1;

1, 93, 117, 27, 1;

1, 1632, 3033, 1080, 81, 1;

1, 68457, 177507, 86373, 9801, 243, 1; ...

where row sums form column 1 of this table A125790,

and column k of A078122 equals column 3^k-1 of this table A125790.

Matrix square A078122^2 begins:

1;

2, 1;

5, 6, 1;

23, 51, 18, 1;

239, 861, 477, 54, 1;

5828, 32856, 25263, 4347, 162, 1; ...

where row sums form column 2 of this table A125790,

and column 0 of A078122^2 forms column 1 of this table A125790.

(PARI) {T(n, k, p=0, q=3)=local(A=Mat(1), B); if(n<p|p<0, 0, for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i|j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return((A^(k+1))[n+1, p+1]))}

Cf. A078122; columns: A078125, A078124, A125801, A125802, A125803; A125804 (diagonal), A125805 (antidiagonal sums); related table: A125800 (q=2).

Sequence in context: A078920 A117396 A125860 this_sequence A076241 A139347 A050145

Adjacent sequences: A125797 A125798 A125799 this_sequence A125801 A125802 A125803

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 10 2006