A059364 Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.

1, 2, 1, 6, 7, 2, 24, 46, 29, 6, 120, 326, 329, 146, 24, 720, 2556, 3604, 2521, 874, 120, 5040, 22212, 40564, 39271, 21244, 6084, 720, 40320, 212976, 479996, 598116, 444849, 197380, 48348, 5040, 362880, 2239344, 6023772, 9223012, 8788569

Offset: 1

Author

Vladeta Jovovic, Jan 28 2001

Keywords

• easy
• nonn
• tabl

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

Examples

			[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!!.
		

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.

Crossrefs

Cf. A001147, A059340.

Programs

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 *)

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

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