A136467 Triangle T, read by rows, where column 0 of T^m equals C(m*2^(n-1), n) as n=0,1,2,3,..., for all m.

1, 1, 1, 1, 4, 1, 4, 32, 16, 1, 70, 848, 576, 64, 1, 4368, 75648, 62208, 9216, 256, 1, 906192, 22313216, 21169152, 3792896, 143360, 1024, 1, 621216192, 21827627008, 23212261376, 4793434112, 223215616, 2228224, 4096, 1, 1429702652400, 71889350288384, 83889221697536, 19373156990976, 1055047811072, 13257146368, 34865152, 16384, 1, 11288510714272000, 812123016027521024, 1022353118549770240, 258404578922332160, 15923445036482560, 238096880762880, 803108552704, 549453824, 65536, 1

Offset

0,5

Comments

Column 0 of T^(n+1) = row n 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 = A136467.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..495

Formula

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

T(n,k) is divisible by 2^((n-k)*k) for n>=k>=0.

Example

Triangle T begins:
1;
1, 1;
1, 4, 1;
4, 32, 16, 1;
70, 848, 576, 64, 1;
4368, 75648, 62208, 9216, 256, 1;
906192, 22313216, 21169152, 3792896, 143360, 1024, 1;
621216192, 21827627008, 23212261376, 4793434112, 223215616, 2228224, 4096, 1;
1429702652400, 71889350288384, 83889221697536, 19373156990976, 1055047811072, 13257146368, 34865152, 16384, 1;
11288510714272000, 812123016027521024, 1022353118549770240, 258404578922332160, 15923445036482560, 238096880762880, 803108552704, 549453824, 65536, 1; ...
Column 0 of T^m is given by: [T^m](n,0) = C(m*2^(n-1), n) for n>=0.
Matrix square T^2 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; ...
in which column 0 equals [T^2](n,0) = C(2^n, n) for n>=0.
Matrix cube T^3 begins:
1;
3, 1;
15, 12, 1;
220, 288, 48, 1;
10626, 19696, 4800, 192, 1;
1712304, 4213376, 1333504, 76800, 768, 1;
927048304, 2927926016, 1133186048, 83992576, 1216512, 3072, 1;
1708566412608, 6784661682176, 3094826778624, 278193635328, 5216272384, 19267584, 12288, 1; ...
in which column 0 equals [T^3](n,0) = C(3*2^(n-1), n) for n>=0.
Matrix 4th power T^4 begins:
1;
4, 1;
28, 16, 1;
560, 512, 64, 1;
35960, 45888, 8448, 256, 1;
7624512, 12731904, 3066880, 135168, 1024, 1;
5423611200, 11434738688, 3390050304, 193953792, 2146304, 4096, 1;
13161885792000, 34243130728448, 12032434503680, 841005662208, 12133597184, 34078720, 16384, 1; ...
in which column 0 equals [T^4](n,0) = C(4*2^(n-1), n) for n>=0.
Matrix 5th power T^5 begins:
1;
5, 1;
45, 20, 1;
1140, 800, 80, 1;
91390, 88720, 13120, 320, 1;
24040016, 30268800, 5881600, 209920, 1280, 1;
21193254160, 33353694464, 8005555200, 372858880, 3338240, 5120, 1;
63815149590720, 122539734714368, 34967493738496, 1998561607680, 23429775360, 53084160, 20480, 1; ...
in which column 0 equals [T^5](n,0) = C(5*2^(n-1), n) for n>=0.

Prog

(PARI) {T(n, k) = my(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)[n+1, k+1]}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. columns: A136465, A136468, A136469; A136470 (matrix square); A136462.

Sequence in context: A066808 A363935 A033918 * A079188 A076810 A303144

Adjacent sequences: A136464 A136465 A136466 * A136468 A136469 A136470

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 31 2007

Status

approved