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

0,5

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) = 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

Example

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;

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 *)

Prog

(Magma) [10^(k*(n-k)): k in [0..n], n in [0..12]]; // 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

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).

Sequence in context: A160562 A176243 A022173 * A172378 A015124 A156767

Adjacent sequences: A158114 A158115 A158116 * A158118 A158119 A158120

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 12 2009

Extensions

Edited by G. C. Greubel, Jun 30 2021

Status

approved