OFFSET
0,8
COMMENTS
Row sums = (n^n-n)/(n-1)^2 = A058128(n).
Column k without leading zeros is the k-th exponential (also called binomial) convolution of the sequence {A000272(n+1)} = {A232006(n+1, 1)}, for n >= 0, with e.g.f. LamberW(-x)/(-x), where LambertW is the principal branch of the Lambert W-function. This is also the row polynomial P(n, x) of the unsigned triangle A137452, evaluated at x = k. - Wolfdieter Lang, Apr 24 2023
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Proposition 5.3.2.
LINKS
G. C. Greubel, Rows n=0..75 of triangle, flattened
Chad Casarotto, Graph Theory and Cayley's Formula, 2006
Alan D. Sokal, A remark on the enumeration of rooted labeled trees, arXiv:1910.14519 [math.CO], 2019.
Marc van Leeuwen, I am stuck with a combinatoric problem... Math Stackexchange, Answer May 14 2017.
Eric Weisstein's World of Mathematics, Lambert W-function
Wikipedia, Lambert W function
FORMULA
T(n, k) = k*n^(n-k-1).
T(n, k) = Sum_{i=0..n-k} T(n-1, k-1+i)*C(n-k,i), T(0, 0) = 1, T(n, 0) = 0 when n >= 1.
From Wolfdieter Lang, Apr 24 2023: (Start)
E.g.f. for {T(n+k, k)}_{n>=0} is (LambertW(-x)/(-x))^k, for k >= 0.
T(n, k) = Sum_{m=0..n-k} |A137452(n-k, m)|*k^m, for n >= 0 and k = 0..n. That is, T(n, n) = 1, for n >= 0, and T(n, k) = Sum_{m=1..n-k} binomial(n-k-1, m-1)*(n-k)^(n-k-m)*k^m, for k = 0..n-1 and n >= k+1. (End)
EXAMPLE
The triangle begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 0 1
2: 0 1 1
3: 0 3 2 1
4: 0 16 8 3 1
5: 0 125 50 15 4 1
6: 0 1296 432 108 24 5 1
7: 0 16807 4802 1029 196 35 6 1
8: 0 262144 65536 12288 2048 320 48 7 1
9: 0 4782969 1062882 177147 26244 3645 486 63 8 1
10: 0 100000000 20000000 3000000 400000 50000 6000 700 80 9 1
... Reformatted by Wolfdieter Lang, Apr 24 2023
MATHEMATICA
Prepend[Table[Table[k n^(n-k-1), {k, 0, n}], {n, 1, 8}], {1}]//Grid
PROG
(PARI) {T(n, k) = if( k<0 || k>n, 0, n^(n-k-1))}; /* Michael Somos, May 15 2017 */
CROSSREFS
KEYWORD
AUTHOR
Geoffrey Critzer, Nov 16 2013
STATUS
approved