A158116 Triangle T(n,k) = 6^(k*(n-k)), read by rows.

1, 1, 1, 1, 6, 1, 1, 36, 36, 1, 1, 216, 1296, 216, 1, 1, 1296, 46656, 46656, 1296, 1, 1, 7776, 1679616, 10077696, 1679616, 7776, 1, 1, 46656, 60466176, 2176782336, 2176782336, 60466176, 46656, 1, 1, 279936, 2176782336, 470184984576, 2821109907456, 470184984576, 2176782336, 279936, 1

Offset: 0

Author

Roger L. Bagula, Mar 12 2009

Keywords

• nonn
• tabl

Examples

			Triangle starts:
  1;
  1,     1;
  1,     6,        1;
  1,    36,       36,          1;
  1,   216,     1296,        216,          1;
  1,  1296,    46656,      46656,       1296,        1;
  1,  7776,  1679616,   10077696,    1679616,     7776,     1;
  1, 46656, 60466176, 2176782336, 2176782336, 60466176, 46656, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), this sequence (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).

Cf. this sequence (q=2), A176639 (q=3), A176641 (q=4).

Programs

Magma

[6^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 30 2021

Mathematica

With[{m=4}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)

PARI

T(n,k) = 6^(k*(n-k));
for (n=0,11,for (k=0,n, print1(T(n,k),", "));print();); \\ Joerg Arndt, Feb 21 2014

Sage

flatten([[6^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

Formula

T(n,k) = 6^(k*(n-k)). - Tom Edgar, Feb 20 2014

T(n,k) = (1/n)*(6^(n-k)*k*T(n-1,k-1) + 6^k*(n-k)*T(n-1,k)). - Tom Edgar, Feb 20 2014

From G. C. Greubel, Jun 30 2021: (Start)

T(n, k, m) = (m+2)^(k*(n-k)) with m = 4.

T(n, k, q) = binomial(2*q, 2)^(k*(n-k)) with q = 2. (End)

Extensions

Overall edit and new name by Tom Edgar and Joerg Arndt, Feb 21 2014