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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.

%I A268435 #17 Jul 12 2019 15:35:59
%S A268435 1,0,1,0,2,1,0,12,6,1,0,120,84,12,1,0,1680,1800,300,20,1,0,30240,
%T A268435 52080,10800,780,30,1,0,665280,1905120,505680,42000,1680,42,1,0,
%U A268435 17297280,84490560,29211840,2857680,126000,3192,56,1
%N 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.
%H A268435 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform">The P-transform</a>.
%F A268435 T(n,k) = binomial(n,k)*Sum_{i=0..k} binomial(k,i)*A268437(n-k,i).
%F A268435 T(n,k) = binomial(-k,-n)*Sum_{i=0..n-k} binomial(-n,i)*A268438(n-k,i).
%F A268435 T(n,k) = 4^(n-k)*Gamma(n-k+1/2)*A048993(n,k)/sqrt(Pi).
%F A268435 T(n,1) = (2*n-2)!/(n-1)! for n>=1.
%F A268435 T(n,n-1) = (n-1)*n for n>=1.
%F A268435 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 A268435 [1]
%e A268435 [0,      1]
%e A268435 [0,      2,       1]
%e A268435 [0,     12,       6,      1]
%e A268435 [0,    120,      84,     12,     1]
%e A268435 [0,   1680,    1800,    300,    20,    1]
%e A268435 [0,  30240,   52080,  10800,   780,   30,  1]
%e A268435 [0, 665280, 1905120, 505680, 42000, 1680, 42, 1]
%p A268435 T := (n, k) -> pochhammer(n-k+1,n-k)*Stirling2(n,k):
%p A268435 for n from 0 to 9 do seq(T(n,k), k=0..n) od;
%p A268435 # Alternatively:
%p A268435 T := proc(n,k) option remember;
%p A268435   `if`( n=k, 1,
%p A268435   `if`( k=0, 0,
%p A268435    k*(4*(n-k)-2)*T(n-1,k)+T(n-1,k-1))) end:
%p A268435 for n from 0 to 7 do seq(T(n,k),k=0..n) od;
%t A268435 T[n_, k_] := Pochhammer[n-k+1, n-k] StirlingS2[n, k];
%t A268435 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 12 2019 *)
%o A268435 (Sage)
%o A268435 A268435 = lambda n,k: rising_factorial(n-k+1,n-k)*stirling_number2(n,k)
%o A268435 [[A268435(n,k) for k in (0..n)] for n in range(8)]
%Y A268435 Cf. A048993, A268436, A268437, A268438.
%K A268435 nonn,tabl
%O A268435 0,5
%A A268435 _Peter Luschny_, Mar 07 2016