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

Author

Paul D. Hanna, Dec 31 2007

Keywords

• nonn
• tabl

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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(""))

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.