A268435 Triangle read by rows, T(n,k) = RF(n-k+1,n-k)*S2(n,k) where RF denotes the rising factorial and S2 the Stirling set numbers, for n>=0 and 0<=k<=n.

1, 0, 1, 0, 2, 1, 0, 12, 6, 1, 0, 120, 84, 12, 1, 0, 1680, 1800, 300, 20, 1, 0, 30240, 52080, 10800, 780, 30, 1, 0, 665280, 1905120, 505680, 42000, 1680, 42, 1, 0, 17297280, 84490560, 29211840, 2857680, 126000, 3192, 56, 1

Offset

0,5

Links

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

Peter Luschny, The P-transform.

Formula

T(n,k) = binomial(n,k)*Sum_{i=0..k} binomial(k,i)*A268437(n-k,i).

T(n,k) = binomial(-k,-n)*Sum_{i=0..n-k} binomial(-n,i)*A268438(n-k,i).

T(n,k) = 4^(n-k)*Gamma(n-k+1/2)*A048993(n,k)/sqrt(Pi).

T(n,1) = (2*n-2)!/(n-1)! for n>=1.

T(n,n-1) = (n-1)*n for n>=1.

Recurrence: T(n,k) = 1 if k=n; 0 if k=0; otherwise k*(4*(n-k)-2)*T(n-1,k)+T(n-1,k-1).

Example

[1]
[0,      1]
[0,      2,       1]
[0,     12,       6,      1]
[0,    120,      84,     12,     1]
[0,   1680,    1800,    300,    20,    1]
[0,  30240,   52080,  10800,   780,   30,  1]
[0, 665280, 1905120, 505680, 42000, 1680, 42, 1]

Maple

T := (n, k) -> pochhammer(n-k+1, n-k)*Stirling2(n, k):
for n from 0 to 9 do seq(T(n, k), k=0..n) od;
# Alternatively:
T := proc(n, k) option remember;
  `if`( n=k, 1,
  `if`( k=0, 0,
   k*(4*(n-k)-2)*T(n-1, k)+T(n-1, k-1))) end:
for n from 0 to 7 do seq(T(n, k), k=0..n) od;

Mathematica

T[n_, k_] := Pochhammer[n-k+1, n-k] StirlingS2[n, k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)

Prog

(Sage)
A268435 = lambda n, k: rising_factorial(n-k+1, n-k)*stirling_number2(n, k)
[[A268435(n, k) for k in (0..n)] for n in range(8)]

Crossrefs

Cf. A048993, A268436, A268437, A268438.

Sequence in context: A010107 A324429 A119830 * A039910 A352399 A129467

Adjacent sequences: A268432 A268433 A268434 * A268436 A268437 A268438

Keyword

nonn,tabl

Author

Peter Luschny, Mar 07 2016

Status

approved