A269940 Triangle read by rows, T(n, k) = Sum_{m=0..k} (-1)^(m + k)*binomial(n + k, n + m) * |Stirling1(n + m, m)|, for n >= 0, 0 <= k <= n.

1, 0, 1, 0, 2, 3, 0, 6, 20, 15, 0, 24, 130, 210, 105, 0, 120, 924, 2380, 2520, 945, 0, 720, 7308, 26432, 44100, 34650, 10395, 0, 5040, 64224, 303660, 705320, 866250, 540540, 135135, 0, 40320, 623376, 3678840, 11098780, 18858840, 18288270, 9459450, 2027025

Offset: 0

Author

Peter Luschny, Mar 27 2016

Keywords

• nonn
• tabl

Comments

We propose to call this sequence the 'Ward cycle numbers' and sequence A269939 the 'Ward set numbers'. - Peter Luschny, Nov 25 2022

Examples

			Triangle T(n,k) starts:
  [1]
  [0,   1]
  [0,   2,      3]
  [0,   6,     20,     15]
  [0,  24,    130,    210,    105]
  [0,  120,   924,   2380,   2520,    945]
  [0,  720,  7308,  26432,  44100,  34650,  10395]
  [0, 5040, 64224, 303660, 705320, 866250, 540540, 135135]
		

Links

• Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 6.
• Peter Luschny, The P-transform.
• Anthony Mansuy, Preordered forests, packed words and contraction algebras, J. Algebra 411 (2014) 259-311, section 4.4.
• Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 2.
• Nico M. Temme, Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters, Integral Transforms and Special Functions, 2021.
• Nico M. Temme and Ed J. M. Veling, Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z, arXiv:2202.12857 [math.CA], 2022.
• Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Crossrefs

Variants: A111999, A259456.

Cf. A269939 (Stirling2 counterpart), A268438, A032188 (row sums).

Programs

Maple

T := (n, k) -> add((-1)^(m+k)*binomial(n+k,n+m)*abs(Stirling1(n+m, m)), m=0..k):
seq(print(seq(T(n, k), k=0..n)), n=0..6);
# Alternatively:
T := proc(n, k) option remember;
    `if`(k=0, k^n,
    `if`(k<=0 or k>n, 0,
    (n+k-1)*(T(n-1, k)+T(n-1, k-1)))) end:
for n from 0 to 6 do seq(T(n, k), k=0..n) od;

Mathematica

T[n_, k_] := Sum[(-1)^(m+k)*Binomial[n+k, n+m]*Abs[StirlingS1[n+m, m]], {m, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 12 2022 *)

Sage

T = lambda n, k: sum((-1)^(m+k)*binomial(n+k, n+m)*stirling_number1(n+m, m) for m in (0..k))
for n in (0..7): print([T(n, k) for k in (0..n)])

Sage

# uses[PtransMatrix from A269941]
PtransMatrix(8, lambda n: n/(n+1), lambda n, k: (-1)^k*falling_factorial(n+k,n))

Formula

T(n,k) = (-1)^k*FF(n+k,n)*P[n,k](n/(n+1)) where P is the P-transform and FF the falling factorial function. For the definition of the P-transform see the link.

T(n,k) = A268438(n,k)*FF(n+k,n)/(2*n)!.

Extensions

Name corrected after notice from Ed Veling by Peter Luschny, Jun 14 2022