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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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.

LINKS

Table of n, a(n) for n=0..44.

EXAMPLE

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.

PROG

(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]

CROSSREFS

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

Cf. A136555.

Sequence in context: A337514 A054505 A132610 * A164792 A132318 A078089

Adjacent sequences: A132622 A132623 A132624 * A132626 A132627 A132628

KEYWORD

nonn,tabl

AUTHOR

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

STATUS

approved

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Last modified February 2 23:07 EST 2023. Contains 360024 sequences. (Running on oeis4.)