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A134484 Triangle, read by rows, where T(n,k) = 2^[n*(n-1) - k*(k-1)] * binomial(n,k) for n>=k>=0. 3
1, 1, 1, 4, 8, 1, 64, 192, 48, 1, 4096, 16384, 6144, 256, 1, 1048576, 5242880, 2621440, 163840, 1280, 1, 1073741824, 6442450944, 4026531840, 335544320, 3932160, 6144, 1, 4398046511104, 30786325577728, 23089744183296, 2405181685760, 37580963840, 88080384, 28672, 1, 72057594037927936, 576460752303423488, 504403158265495552, 63050394783186944, 1231453023109120, 3848290697216, 1879048192, 131072, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Has similar matrix power formulas as those for triangle A134049.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..495, of rows 0..30 of the flattened triangle.

FORMULA

[T^(2^m)](n,k) = (2^m)^(n-k) * 2^[n*(n-1) - k*(k-1)] * C(n,k) for n>=k>=0 ; this is the formula for the matrix power T^(2^m) at row n and column k.

Matrix log is given by: [log(T)](n+1,n) = (n+1)*4^n for n>=0 along a secondary diagonal with zeros elsewhere.

EXAMPLE

Matrix powers of triangle T also satisfy:

(1) [T^(2^m)](n,k) = T(n+m,k+m)/(2^m)^(n-k) for n>=k>=0;

(2) [T^( 1/2^(n-1) )](n,k) = (2^k)^(n-k) * C(n,k) for n>=k>=0;

compare to the formulas for matrix powers of triangle A134049.

Triangle T begins:

1;

1, 1;

4, 8, 1;

64, 192, 48, 1;

4096, 16384, 6144, 256, 1;

1048576, 5242880, 2621440, 163840, 1280, 1;

1073741824, 6442450944, 4026531840, 335544320, 3932160, 6144, 1;

4398046511104, 30786325577728, 23089744183296, 2405181685760, 37580963840, 88080384, 28672, 1; ...

Matrix log of this triangle begins:

0;

1, 0;

0, 8, 0;

0, 0, 48, 0;

0, 0, 0, 256, 0;

0, 0, 0, 0, 1280, 0;

0, 0, 0, 0, 0, 6144, 0; ...

a single nonzero diagonal given by [log(T)](n+1,n) = (n+1)*4^n.

Matrix square of this triangle begins:

1;

2, 1;

16, 16, 1;

512, 768, 96, 1;

65536, 131072, 24576, 512, 1;

33554432, 83886080, 20971520, 655360, 2560, 1; ...

where [T^2](n,k) = 2^(n-k) * 2^[n*(n-1) - k*(k-1)] * C(n,k) for n>=k>=0.

Matrix 4th power of this triangle begins:

1;

4, 1;

64, 32, 1;

4096, 3072, 192, 1;

1048576, 1048576, 98304, 1024, 1;

1073741824, 1342177280, 167772160, 2621440, 5120, 1; ...

where [T^4](n,k) = 4^(n-k) * 2^[n*(n-1) - k*(k-1)] * C(n,k) for n>=k>=0.

Matrix 8th power of this triangle begins:

1;

8, 1;

256, 64, 1;

32768, 12288, 384, 1;

16777216, 8388608, 393216, 2048, 1;

34359738368, 21474836480, 1342177280, 10485760, 10240, 1; ...

where [T^8](n,k) = 8^(n-k) * 2^[n*(n-1) - k*(k-1)] * C(n,k) for n>=k>=0.

Matrix square-root of this triangle begins:

1;

1/2, 1;

1, 4, 1; <== row 2: [T^(1/2^1)](2,k) = (2^k)^(2-k)*C(2,k), k=0..2

8, 48, 24, 1;

256, 2048, 1536, 128, 1;

32768, 327680, 327680, 40960, 640, 1;

16777216, 201326592, 251658240, 41943040, 983040, 3072, 1; ...

Matrix 4th root of this triangle begins:

1;

1/4, 1;

1/4, 2, 1;

1, 12, 12, 1; <== row 3: [T^(1/2^2)](3,k) = (2^k)^(3-k)*C(3,k), k=0..3

16, 256, 384, 64, 1;

1024, 20480, 40960, 10240, 320, 1;

262144, 6291456, 15728640, 5242880, 245760, 1536, 1; ...

Matrix 8th root of this triangle begins:

1;

1/8, 1;

1/16, 1, 1;

1/8, 3, 6, 1;

1, 32, 96, 32, 1; <== row 4: [T^(1/2^3)](4,k) = (2^k)^(4-k)*C(4,k), k=0..4

32, 1280, 5120, 2560, 160, 1;

4096, 196608, 983040, 655360, 61440, 768, 1; ...

PROG

(PARI) {T(n, k)=2^(n*(n-1) - k*(k-1))*binomial(n, k)}

for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

(PARI) /* Matrix Power T^(2^m): */

{T(n, k, m)=2^(m*(n-k))*2^(n*(n-1) - k*(k-1))*binomial(n, k)}

for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A134049; A134485 (row sums).

Sequence in context: A294830 A248415 A295086 * A244641 A274192 A021958

Adjacent sequences:  A134481 A134482 A134483 * A134485 A134486 A134487

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Oct 28 2007

EXTENSIONS

Entry revised by Paul D. Hanna, Jun 24 2016

STATUS

approved

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Last modified September 20 03:51 EDT 2019. Contains 327210 sequences. (Running on oeis4.)