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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Diagonals give A000007, A000142; A000142, A067318.

Row sums give A001147.

Sum_{k=0..n} (-1)^k*T(n,k) = (-1)^n.

LINKS

Table of n, a(n) for n=0..44.

Trevor Hyde, Liminal reciprocity and factorization statistics, arXiv:1803.08438 [math.NT], 2018.

FORMULA

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). - Vladeta Jovovic, Dec 15 2004

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

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, 7}, {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..6): [A088996(n, k) for k in (0..n)]  # Peter Luschny, May 12 2013

CROSSREFS

Cf. A000007, A000142, A001147, A067318, A084938, A059364.

Sequence in context: A161014 A235712 A154852 * A211888 A293783 A274541

Adjacent sequences:  A088993 A088994 A088995 * A088997 A088998 A088999

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Dec 01 2003, Aug 17 2007

STATUS

approved

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Last modified August 21 06:54 EDT 2019. Contains 326162 sequences. (Running on oeis4.)