A134484 Triangle, read by rows, where T(n,k) = 2^[n*(n-1) - k*(k-1)] * binomial(n,k) for n>=k>=0.

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

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: A367416 A295086 A331331 * 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