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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. 9
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 (list; table; graph; refs; listen; history; text; internal format)
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: A143461 A066808 A033918 * A079188 A076810 A303144

Adjacent sequences:  A136464 A136465 A136466 * A136468 A136469 A136470

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 31 2007

STATUS

approved

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Last modified July 23 12:35 EDT 2019. Contains 325254 sequences. (Running on oeis4.)