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

0,5

Comments

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

Links

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

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

Variants: A111999, A259456.

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

Sequence in context: A089134 A349776 A305622 * A350463 A341339 A084257

Adjacent sequences: A269937 A269938 A269939 * A269941 A269942 A269943

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