A132625 Triangle T, read by rows, where row n+1 of T = row n of T^(2^n) with appended '1' for n>=0 with T(0,0)=1.

1, 1, 1, 2, 1, 1, 14, 4, 1, 1, 336, 60, 8, 1, 1, 25836, 2960, 248, 16, 1, 1, 6251504, 454072, 24800, 1008, 32, 1, 1, 4838830976, 216266368, 7603952, 202944, 4064, 64, 1, 1, 12344615283200, 328381917376, 7190266752, 124427232, 1641856, 16320, 128, 1, 1

Offset: 0

Author

Paul D. Hanna, Aug 25 2007, Jan 07 2008

Keywords

• nonn
• tabl

Comments

Let R_{n} equal row n of square array A136555, where A136555(n,k) = C(2^k + n-1, k); this triangle transforms rows of A136555: T * R_{n} = R_{n+1} for n>=0.

Examples

			Triangle begins:
1;
1, 1;
2, 1, 1;
14, 4, 1, 1;
336, 60, 8, 1, 1;
25836, 2960, 248, 16, 1, 1;
6251504, 454072, 24800, 1008, 32, 1, 1;
4838830976, 216266368, 7603952, 202944, 4064, 64, 1, 1;
12344615283200, 328381917376, 7190266752, 124427232, 1641856, 16320, 128, 1, 1; ...
GENERATE T FROM MATRIX POWERS OF T.
Matrix power T^4 begins:
1;
4, 1;
14, 4, 1; <-- row 3 of T
96, 22, 4, 1;
1941, 316, 38, 4, 1;
129206, 14185, 1140, 70, 4, 1; ...
where row 3 of T = row 2 of T^(2^2) with appended '1'.
Matrix power T^8 begins:
1;
8, 1;
44, 8, 1;
336, 60, 8, 1; <-- row 4 of T
6062, 872, 92, 8, 1;
345596, 35734, 2712, 156, 8, 1; ...
where row 4 of T = row 3 of T^(2^3) with appended '1'.
Matrix power T^16 begins:
1;
16, 1;
152, 16, 1;
1504, 184, 16, 1;
25836, 2960, 248, 16, 1; <-- row 5 of T
1197304, 109500, 7408, 376, 16, 1; ...
where row 5 of T = row 4 of T^(2^4) with appended '1'.
Alternate generating method:
RoW 3: start with '1' followed by (2^2 - 1) zeros;
take partial sums and append (2^1 - 1) zero;
take partial sums twice more:
(1), 0, 0, 0;
1, 1, 1, (1), 0;
1, 2, 3, 4, (4);
1, 3, 6, 10, (14);
the final nonzero terms form row 3: [14, 4, 1, 1].
Row 4: start with '1' followed by (2^3 - 1) zeros;
take partial sums and append (2^2 - 1) zeros;
take partial sums and append (2^1 - 1) zero;
take partial sums twice more:
(1), 0, 0, 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1, (1), 0, 0, 0;
1, 2, 3, 4, 5, 6, 7, 8, 8, 8, (8), 0;
1, 3, 6, 10, 15, 21, 28, 36, 44, 52, 60, (60);
1, 4, 10, 20, 35, 56, 84, 120, 164, 216, 276, (336);
the final nonzero terms form row 4: [336, 60, 8, 1, 1].
Continuing in this way produces all the rows of this triangle.
		

Crossrefs

Cf. variants: A101479, A132610, A132615; columns: A132626, A132627.

Cf. A136555.

Programs

PARI

T(n, k)=local(A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j]=(A^(2^(i-2)))[i-1, j]); )); A=B); return( ((A)[n+1, k+1]))

PARI

/* Generate using partial sums method (faster) */ T(n, k)=local(A=vector(n+1), p); A[1]=1; for(j=1, n-k, p=2^n-2^(n-j)-j; A=Vec((Polrev(A)+x*O(x^p))/(1-x))); A[p+1]

PARI

/* As Row Transformation of Square Array A136555(n,k) = C(2^k + n-1, k): */ T(n,k)=local(M=matrix(n+2,n+2,r,c,binomial(2^(c-1)+r-2,c-1)), N=matrix(n+1,n+1,r,c,M[r,c]),P=matrix(n+1,n+1,r,c,M[r+1,c]),R=P~*N~^-1); R[n+1,k+1]