OFFSET
0,5
COMMENTS
Consider the k-fold Cartesian products CP(n,k) of the vector A(n) = [1, 2, 3, ..., n].
An element of CP(n,k) is a n-tuple T_t of the form T_t = [i_1, i_2, i_3, ..., i_k] with t=1, .., n^k.
We count members T of CP(n,k) which satisfy some condition delta(T_t), so delta(.) is an indicator function which attains values of 1 or 0 depending on whether T_t is to be counted or not; the summation sum_{CP(n,k)} delta(T_t) over all elements T_t of CP produces the count.
For the triangle here we have delta(T_t) = 0 if for any two i_j, i_(j+1) in T_t one has i_j = i_(j+1): T(n,k) = Sum_{CP(n,k)} delta(T_t) = Sum_{CP(n,k)} delta(i_j = i_(j+1)).
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
T(n, k) = (n-1)^(k-1) + (n-1)^k = n*A079901(n-1,k-1), k > 0.
Sum_{k=0..n} T(n,k) = (n*(n-1)^n - 2)/(n-2), n > 2.
EXAMPLE
Array, A(n, k) = n*(n-1)^(k-1) for n > 1, A(n, k) = 1 otherwise, begins as:
1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012;
1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012;
1, 2, 2, 2, 2, 2, 2, 2, 2, ... A040000;
1, 3, 6, 12, 24, 48, 96, 192, 384, ... A003945;
1, 4, 12, 36, 108, 324, 972, 2916, 8748, ... A003946;
1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, ... A003947;
1, 6, 30, 150, 750, 3750, 18750, 93750, 468750, ... A003948;
1, 7, 42, 252, 1512, 9072, 54432, 326592, 1959552, ... A003949;
1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, ... A003950;
1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, ... A003951;
1, 10, 90, 810, 7290, 65610, 590490, 5314410, 47829690, ... A003952;
1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, ............. A003953;
1, 12, 132, 1452, 15972, 175692, 1932612, 21258732, ............. A003954;
1, 13, 156, 1872, 22464, 269568, 3234816, 38817792, ............. A170732;
... ;
The triangle begins as:
1
1, 1;
1, 2, 2;
1, 3, 6, 12;
1, 4, 12, 36, 108;
1, 5, 20, 80, 320, 1280;
1, 6, 30, 150, 750, 3750, 18750;
1, 7, 42, 252, 1512, 9072, 54432, 326592;
1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344;
...;
T(3,3) = 12 counts the triples (1,2,1), (1,2,3), (1,3,1), (1,3,2), (2,1,2), (2,1,3), (2,3,1), (2,3,2), (3,1,2), (3,1,3), (3,2,1), (3,2,3) out of a total of 3^3 = 27 triples in the CP(3,3).
MATHEMATICA
A158497[n_, k_]:= If[n<2 || k==0, 1, n*(n-1)^(k-1)];
Table[A158497[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 18 2025 *)
PROG
(Magma)
A158497:= func< n, k | k le 1 select n^k else n*(n-1)^(k-1) >;
[A158497(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 18 2025
(SageMath)
def A158497(n, k): return n^k if k<2 else n*(n-1)^(k-1)
print(flatten([[A158497(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 18 2025
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Thomas Wieder, Mar 20 2009
EXTENSIONS
Edited by R. J. Mathar, Mar 31 2009
More terms added by G. C. Greubel, Mar 18 2025
STATUS
approved