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
Examples
[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]
Links
- Peter Luschny, The P-transform.
Programs
-
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 *) -
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)]
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).