OFFSET
0,5
COMMENTS
We propose to call this sequence the 'Ward cycle numbers' and sequence A269939 the 'Ward set numbers'. - Peter Luschny, Nov 25 2022
LINKS
Peter Luschny, The P-transform.
A. Mansuy, Preordered forests, packed words and contraction algebras, J. Algebra 411 (2014) 259-311, section 4.4.
N. M. Temme, Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters, Integral Transforms and Special Functions, 2021.
N. M. Temme and E. J. M. Veling, Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a, b and z, arXiv:2202.12857 [math.CA], 2022.
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)!.
EXAMPLE
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]
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 *)
PROG
(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))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 27 2016
EXTENSIONS
Name corrected after notice from Ed Veling by Peter Luschny, Jun 14 2022
STATUS
approved