A158117 Triangle T(n, k) = 10^(k*(n-k)), read by rows.

1, 1, 1, 1, 10, 1, 1, 100, 100, 1, 1, 1000, 10000, 1000, 1, 1, 10000, 1000000, 1000000, 10000, 1, 1, 100000, 100000000, 1000000000, 100000000, 100000, 1, 1, 1000000, 10000000000, 1000000000000, 1000000000000, 10000000000, 1000000, 1

Offset: 0

Author

Roger L. Bagula, Mar 12 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      10,           1;
  1,     100,         100,             1;
  1,    1000,       10000,          1000,             1;
  1,   10000,     1000000,       1000000,         10000,           1;
  1,  100000,   100000000,    1000000000,     100000000,      100000,       1;
  1, 1000000, 10000000000, 1000000000000, 1000000000000, 10000000000, 1000000, 1;
		

Links

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

Crossrefs

Cf. A007318 (q=0), A118180 (q=1), A158116 (q=2), this sequence (q=3), A176639 (q=4), A176643 (q=5), A176641 (q=6).

Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), this sequence (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).

Programs

Magma

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

Mathematica

(* First program *)
T[n_, k_, q_]= Binomial[q+2,2](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=8}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)

Sage

flatten([[10^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

Formula

T(n, k, q) = c(n,q)/(c(k,q)*c(n-k,q)) where c(n, k) = binomial(q+2, 2)^binomial(n+1, 2), c(n, 0) = n!, and q = 3.

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

T(n, k, m) = (m+2)^(k*(n-k)) with m = 8. - G. C. Greubel, Jun 30 2021

Extensions

Edited by G. C. Greubel, Jun 30 2021