A269939 Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.

1, 0, 1, 0, 1, 3, 0, 1, 10, 15, 0, 1, 25, 105, 105, 0, 1, 56, 490, 1260, 945, 0, 1, 119, 1918, 9450, 17325, 10395, 0, 1, 246, 6825, 56980, 190575, 270270, 135135, 0, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025

Offset: 0

Author

Peter Luschny, Mar 26 2016

Keywords

• nonn
• tabl

Comments

We propose to call this sequence the 'Ward set numbers' and sequence A269940 the 'Ward cycle numbers'. - Peter Luschny, Nov 25 2022

Examples

			Triangle starts:
  1;
  0, 1;
  0, 1,   3;
  0, 1,  10,   15;
  0, 1,  25,  105,   105;
  0, 1,  56,  490,  1260,    945;
  0, 1, 119, 1918,  9450,  17325,  10395;
  0, 1, 246, 6825, 56980, 190575, 270270, 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.
• Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
• Marko Riedel, Math Stackexchange, Upper Stirling numbers and Ward numbers.
• Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See p. 2.
• Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 1, 12.

Crossrefs

Variants: A134991 (main entry for this triangle), A181996.

Row sums are A000311.

Alternating row sums are signed factorials A133942.

Cf. A269940 (Stirling1 counterpart), A268437.

Programs

Maple

# first version
A269939 := (n,k) -> add((-1)^(m+k)*binomial(n+k,n+m)*Stirling2(n+m, m), m=0..k):
seq(seq(A269939(n,k), k=0..n), n=0..8);
# Alternatively:
T := proc(n,k) option remember;
    `if`(k=0 and n=0, 1,
    `if`(k<=0 or k>n, 0,
    k*T(n-1,k)+(n+k-1)*T(n-1,k-1))) end:
for n from 0 to 6 do seq(T(n,k),k=0..n) od;
# simple, third version
T := (n,k)->  (n+k)!*coeftayl((exp(z)-z-1)^k/k!, z=0, n+k); # Marko Riedel, Apr 14 2016

Mathematica

Table[Sum[(-1)^(m + k) Binomial[n + k, n + m] StirlingS2[n + m, m], {m, 0, k}], {n, 0, 8}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 15 2016 *)

PARI

T(n) = {[Vecrev(Pol(p)) | p<-Vec(serlaplace(1/((1+y)*(1 + lambertw(-y/(1+y)*exp((x-y)/(1+y) + O(x*x^n)))))))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 14 2022

Sage

T = lambda n,k: sum((-1)^(m+k)*binomial(n+k,n+m)*stirling_number2(n+m,m) for m in (0..k))
for n in (0..6): print([T(n,k) for k in (0..n)])

Sage

# uses[PtransMatrix from A269941]
PtransMatrix(8, lambda n: 1/(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](1/(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) = A268437(n,k)*FF(n+k,n)/(2*n)!.

T(n,k) = (n+k)! [z^{n+k}] (exp(z)-z-1)^k/k!. - Marko Riedel, Apr 14 2016

From Fabián Pereyra, Jan 12 2022: (Start)

T(n,k) = k*T(n-1,k) + (n+k-1)*T(n-1,k-1) for n > 0, T(0,0) = 1, T(n,0) = 0 for n > 0. (See the second Maple program.)

E.g.f.: A(x,t) = 1/((1+t)*(1 + W(-t/(1+t)*exp((x-t)/(1+t))))), where W(x) is the Lambert W-function.

T(n,k) = Sum_{j=0..k} E2(n,j)*binomial(n-j,k-j), where E2(n,k) are the second-order Eulerian numbers A340556.

T(n,k) = Sum_{j=k..n} (-1)^(n-j)*A112486(n,j)*binomial(j,k). (End)