A268435 - OEIS

%I #17 Jul 12 2019 15:35:59

%S 1,0,1,0,2,1,0,12,6,1,0,120,84,12,1,0,1680,1800,300,20,1,0,30240,

%T 52080,10800,780,30,1,0,665280,1905120,505680,42000,1680,42,1,0,

%U 17297280,84490560,29211840,2857680,126000,3192,56,1

%N 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.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform">The P-transform</a>.

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

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

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

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

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

%F 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).

%e [1]

%e [0,      1]

%e [0,      2,       1]

%e [0,     12,       6,      1]

%e [0,    120,      84,     12,     1]

%e [0,   1680,    1800,    300,    20,    1]

%e [0,  30240,   52080,  10800,   780,   30,  1]

%e [0, 665280, 1905120, 505680, 42000, 1680, 42, 1]

%p T := (n, k) -> pochhammer(n-k+1,n-k)*Stirling2(n,k):

%p for n from 0 to 9 do seq(T(n,k), k=0..n) od;

%p # Alternatively:

%p T := proc(n,k) option remember;

%p   `if`( n=k, 1,

%p   `if`( k=0, 0,

%p    k*(4*(n-k)-2)*T(n-1,k)+T(n-1,k-1))) end:

%p for n from 0 to 7 do seq(T(n,k),k=0..n) od;

%t T[n_, k_] := Pochhammer[n-k+1, n-k] StirlingS2[n, k];

%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 12 2019 *)

%o (Sage)

%o A268435 = lambda n,k: rising_factorial(n-k+1,n-k)*stirling_number2(n,k)

%o [[A268435(n,k) for k in (0..n)] for n in range(8)]

%Y Cf. A048993, A268436, A268437, A268438.

%K nonn,tabl

%O 0,5

%A _Peter Luschny_, Mar 07 2016