A136470 Triangle T, read by rows, where column 0 of T^m = {C(m*2^n, n), n>=0} for all m.

1, 2, 1, 6, 8, 1, 56, 128, 32, 1, 1820, 6048, 2176, 128, 1, 201376, 912128, 419328, 34816, 512, 1, 74974368, 449708544, 249300992, 26198016, 548864, 2048, 1, 94525795200, 739136655360, 477013868544, 59943682048, 1604059136, 8650752, 8192, 1, 409663695276000, 4132411271661568, 3028532448264192, 439222754869248, 14159357935616, 98723430400, 136839168, 32768, 1, 6208116950265950720, 80121787455478857728, 65415571433959456768, 10679727629898088448, 399723620798038016, 3391703461396480, 6141702569984, 2172649472, 131072, 1

Offset

0,2

Comments

Column 0 of T^(n+1) = row 2n+1 of square array A136462 defined by: A136462(n,k) = C((n+1)*2^(k-1), k); T^n denotes the n-th matrix power of this triangle T = A136470.

Links

Table of n, a(n) for n=0..54.

Formula

Equals the matrix square of triangle A136467.

Diagonals: T(n+1,n) = 2*4^n; T(n+2,n) = 2*8^n*(2^(n+2) + n-1).

Example

Triangle T begins:
1;
2, 1;
6, 8, 1;
56, 128, 32, 1;
1820, 6048, 2176, 128, 1;
201376, 912128, 419328, 34816, 512, 1;
74974368, 449708544, 249300992, 26198016, 548864, 2048, 1;
94525795200, 739136655360, 477013868544, 59943682048, 1604059136, 8650752, 8192, 1;
409663695276000, 4132411271661568, 3028532448264192, 439222754869248, 14159357935616, 98723430400, 136839168, 32768, 1; ...
Column 0 of T^m is given by: [T^m](n,0) = C(m*2^n, n) for n>=0.

Prog

(PARI) {T(n, k)=local(M=matrix(n+1, n+1, r, c, binomial(r*2^(c-2), c-1)), P); P=matrix(n+1, n+1, r, c, binomial((r+1)*2^(c-2), c-1)); ((P~*M~^-1)^2)[n+1, k+1]}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. columns: A014070, A136471, A136472; A136467 (matrix square-root); A136462.

Sequence in context: A190015 A112007 A113374 * A363747 A220884 A217877

Adjacent sequences: A136467 A136468 A136469 * A136471 A136472 A136473

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 31 2007

Status

approved