OFFSET
1,2
COMMENTS
Sum_{k=0..n-1} T(n,k)=(2*n-1)!!.
Alternating row sums = 1. - Gerald McGarvey, Aug 06 2006
Essentially triangle given by [1,1,2,2,3,3,4,4,5,5,6,6,...] DELTA [0,1,1,2,2,3,3,4,4,5,5,...] = [1;1,0;2,1,0;6,7,2,0;24,46,29,6,0;...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 20 2006
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Grzegorz Rzadkowski and M. Urlinska, A Generalization of the Eulerian Numbers, arXiv preprint arXiv:1612.06635 [math.CO], 2016-2017.
FORMULA
For n>1, T(n,k) = (n-1)*T(n-1,k-1) + n*T(n-1,k) (assuming any T(i,j) outside the triangle = 0). - Gerald McGarvey, Aug 06 2006
EXAMPLE
[1],
[2, 1],
[6, 7, 2],
[24, 46, 29, 6],
[120, 326, 329, 146, 24],
[720, 2556, 3604, 2521, 874, 120], ...
2+1=3!!, 6+7+2=5!!, 24+46+29+6=7!!, 120+326+329+146+24=9!!.
MATHEMATICA
Table[Sum[Abs[StirlingS1[n, n - j]]*Binomial[j, k], {j, 0, n}], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* G. C. Greubel, Jan 08 2017 *)
PROG
(PARI) T(n, k)=if(n<1, 0, n!*polcoeff(polcoeff((1-x-x*y+x*O(x^n))^(-1/(1+y)), n), k))
(Sage)
def A059364(n, k): return add(stirling_number1(n, n-i)*binomial(i, k) for i in (0..n))
for n in (1..5): [A059364(n, k) for k in (0..n-1)] # Peter Luschny, May 12 2013
CROSSREFS
KEYWORD
AUTHOR
Vladeta Jovovic, Jan 28 2001
STATUS
approved