A176627 Triangle T(n, k) = 12^(k*(n-k)), read by rows.

1, 1, 1, 1, 12, 1, 1, 144, 144, 1, 1, 1728, 20736, 1728, 1, 1, 20736, 2985984, 2985984, 20736, 1, 1, 248832, 429981696, 5159780352, 429981696, 248832, 1, 1, 2985984, 61917364224, 8916100448256, 8916100448256, 61917364224, 2985984, 1

Offset

0,5

Links

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

Formula

T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, k) = Product_{j=1..n} (q*(3*q - 1)/2)^j and q = 3.

T(n, k, q) = (binomial(3*q, 2)/3)^(k*(n-k)) with q = 3.

T(n, k, m) = (m+2)^(k*(n-k)) with m = 10. - G. C. Greubel, Jun 30 2021

Example

Triangle begins as:
  1;
  1,       1;
  1,      12,           1;
  1,     144,         144,             1;
  1,    1728,       20736,          1728,             1;
  1,   20736,     2985984,       2985984,         20736,           1;
  1,  248832,   429981696,    5159780352,     429981696,      248832,       1;
  1, 2985984, 61917364224, 8916100448256, 8916100448256, 61917364224, 2985984, 1;

Mathematica

(* First program *)
T[n_, k_, q_]= (Binomial[3*q, 2]/3)^(k*(n-k));
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 30 2021 *)
(* Second program *)
With[{m=10}, Table[(m+2)^(k*(n-k)), {n, 0, 12}, {k, 0, n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)

Prog

(Magma) [(12)^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 30 2021
(Sage) flatten([[(12)^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

Crossrefs

Cf. A000326,

Cf. A118190 (q=2), this sequence (q=3), A176631 (q=4).

Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), this sequence (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).

Sequence in context: A166962 A022175 A340427 * A015129 A172376 A289673

Adjacent sequences: A176624 A176625 A176626 * A176628 A176629 A176630

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 22 2010

Extensions

Edited by G. C. Greubel, Jun 30 2021

Status

approved