OFFSET
1,2
COMMENTS
If T(0,0) = 0 is prepended, then row sums give A001788.
REFERENCES
Eugene Jahnke and Fritz Emde, Table of Functions with Formulae and Curves, Dover Publications, 1945, p. 32.
FORMULA
Let p(x;n) = (x + 1)^n. Then row n are the coefficients in the expansion of p''(x;n) - x*p'(x;n) + n*p(x;n) = n*(x + n)*(x + 1)^(n - 2).
From Franck Maminirina Ramaharo, Oct 02 2018: (Start)
T(n,1) = A000290(n).
T(n,2) = A011379(n).
T(n,3) = 3*A002417(n-2).
T(n,n-2) = A046092(n-1).
T(n,n-3) = 9*A000292(n-2).
G.f.: y*(x*y - y - 1)/(x*y + y - 1)^3. (End)
EXAMPLE
Triangle begins:
n\k| 0 1 2 3 4 6 7 8 9
-------------------------------------------------
1 | 1
2 | 4 2
3 | 9 12 3
4 | 16 36 24 4
5 | 25 80 90 40 5
6 | 36 150 240 180 60 6
7 | 49 252 525 560 315 84 7
8 | 64 392 1008 1400 1120 504 112 8
9 | 81 576 1764 3024 3150 2016 756 144 9
... reformatted. - Franck Maminirina Ramaharo, Oct 02 2018
MATHEMATICA
Table[CoefficientList[n*(x + n)*(x + 1)^(n - 2), x], {n, 1, 12}]//Flatten
PROG
(Maxima) T(n, k) := n*(binomial(n - 2, k - 1) + n*binomial(n - 2, k))$
tabl(nn) := for n:1 thru nn do print(makelist(T(n, k), k, 0, n - 1))$ /* Franck Maminirina Ramaharo, Oct 02 2018 */
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Dec 13 2009
EXTENSIONS
Edited and new name by Franck Maminirina Ramaharo, Oct 02 2018
STATUS
approved