OFFSET
0,8
COMMENTS
If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018
T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020
LINKS
T. D. Noe, Rows n = 0..50 of triangle, flattened
FORMULA
E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004
E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006
EXAMPLE
Rows begin:
[1, 0, 0, 0, 0, 0, 0, 0, ...],
[1, 1, 1, 1, 1, 1, 1, 1, ...],
[1, 2, 4, 8, 16, 32, 64, 128, ...],
[1, 3, 9, 27, 81, 243, 729, 2187, ...],
[1, 4, 16, 64, 256, 1024, 4096, 16384, ...],
[1, 5, 25, 125, 625, 3125, 15625, 78125, ...],
[1, 6, 36, 216, 1296, 7776, 46656, 279936, ...],
[1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...
MATHEMATICA
Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten
PROG
(PARI) T(n, k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017
(Magma) [[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018
CROSSREFS
Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
Cf. A099555.
KEYWORD
AUTHOR
EXTENSIONS
More terms from David W. Wilson
Edited by Paul D. Hanna, Oct 22 2004
STATUS
approved