OFFSET
0,6
COMMENTS
We propose to call this sequence the 'Ward set numbers' and sequence A269940 the 'Ward cycle numbers'. - Peter Luschny, Nov 25 2022
LINKS
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.
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)
EXAMPLE
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;
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 *)
PROG
(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))
(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
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 26 2016
STATUS
approved