OFFSET
0,6
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Trevor Hyde, Liminal reciprocity and factorization statistics, arXiv:1803.08438 [math.NT], 2018.
FORMULA
Sum_{k=0..n} (-1)^k*T(n,k) = (-1)^n.
From Vladeta Jovovic, Dec 15 2004: (Start)
E.g.f.: (1-y-y*x)^(-1/(1+x)).
Sum_{k=0..n} T(n, k)*x^k = Product_{k=1..n} (k*x+k-1). (End)
T(n, k) = n*T(n-1, k-1) + (n-1)*T(n-1, k); T(0, 0) = 1, T(0, k) = 0 if k > 0, T(n, k) = 0 if k < 0. - Philippe Deléham, May 22 2005
Sum_{k=0..n} T(n,k)*x^(n-k) = A019590(n+1), A000012(n), A000142(n), A001147(n), A007559(n), A007696(n), A008548(n), A008542(n), A045754(n), A045755(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, respectively. Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000007(n), A001147(n), A008544(n), A008545(n), A008546(n), A008543(n), A049209(n), A049210(n), A049211(n), A049212(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Aug 10 2007
T(n, k) = Sum_{j=0..n} (-1)^j*binomial(j, n-k)*StirlingS1(n, n-j). - G. C. Greubel, Feb 23 2022
EXAMPLE
Triangle begins:
1;
0, 1;
0, 1, 2;
0, 2, 7, 6;
0, 6, 29, 46, 24;
0, 24, 146, 329, 326, 120;
0, 120, 874, 2521, 3604, 2556, 720;
0, 720, 6084, 21244, 39271, 40564, 22212, 5040;
0, 5040, 48348, 197380, 444849, 598116, 479996, 212976, 40320;
...
MATHEMATICA
T[n_, k_]:= T[n, k]= Sum[(-1)^(n-i)*Binomial[i, k] StirlingS1[n+1, n+1-i], {i, 0, n}]; {{1}}~Join~Table[Abs@ T[n, k], {n, 0, 10}, {k, n+1, 0, -1}] (* Michael De Vlieger, Jun 19 2018 *)
PROG
(Sage)
def A088996(n, k): return add((-1)^(n-i)*binomial(i, k)*stirling_number1(n+1, n+1-i) for i in (0..n))
for n in (0..10): [A088996(n, k) for k in (0..n)] # Peter Luschny, May 12 2013
(Magma)
A088996:= func< n, k | (&+[(-1)^j*Binomial(j, n-k)*StirlingFirst(n, n-j): j in [0..n]]) >;
[A088996(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 23 2022
CROSSREFS
KEYWORD
AUTHOR
Philippe Deléham, Dec 01 2003, Aug 17 2007
STATUS
approved