A268436 Triangle read by rows, T(n,k) = RF(n-k+1, n-k) * S1(n,k) where RF denotes the rising factorial and S1 the Stirling cycle numbers, for n>=0 and 0<=k<=n.

1, 0, 1, 0, 2, 1, 0, 24, 6, 1, 0, 720, 132, 12, 1, 0, 40320, 6000, 420, 20, 1, 0, 3628800, 460320, 27000, 1020, 30, 1, 0, 479001600, 53343360, 2728320, 88200, 2100, 42, 1, 0, 87178291200, 8693879040, 397111680, 11371920, 235200, 3864, 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)*A268438(n-k,i).

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

T(n,k) = 4^(n-k)*Gamma(n-k+1/2)*A132393(n,k)/sqrt(Pi).

T(n,1) = (2*n-2)! 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; and otherwise (n-1)*(4*(n-k)-2)*T(n-1,k) + T(n-1,k-1).

Example

[1]
[0,         1]
[0,         2,        1]
[0,        24,        6,       1]
[0,       720,      132,      12,     1]
[0,     40320,     6000,     420,    20,    1]
[0,   3628800,   460320,   27000,  1020,   30,  1]
[0, 479001600, 53343360, 2728320, 88200, 2100, 42, 1]

Maple

T := (n, k) -> pochhammer(n-k+1, n-k)*abs(Stirling1(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,
   (n-1)*(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] Abs[StirlingS1[n, k]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2019 *)

Prog

(Sage)
A268436 = lambda n, k: rising_factorial(n-k+1, n-k)*stirling_number1(n, k)
[[A268436(n, k) for k in (0..n)] for n in range(8)]

Crossrefs

Cf. A132393, A268435, A268437, A268438.

Sequence in context: A009829 A202697 A369117 * A249570 A051652 A077019

Adjacent sequences: A268433 A268434 A268435 * A268437 A268438 A268439

Keyword

nonn,tabl

Author

Peter Luschny, Mar 07 2016

Status

approved