A088996 Triangle T(n,k) read by rows, given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938.

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

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

Diagonals give A000007, A000142, A067318.

Cf. A001147 (row sums), A048994, A059364, A084938.

Sequence in context: A363027 A235712 A154852 * A211888 A350601 A293783

Adjacent sequences: A088993 A088994 A088995 * A088997 A088998 A088999

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Dec 01 2003, Aug 17 2007

Status

approved