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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.

%I A268436 #18 Jul 12 2019 15:42:03
%S A268436 1,0,1,0,2,1,0,24,6,1,0,720,132,12,1,0,40320,6000,420,20,1,0,3628800,
%T A268436 460320,27000,1020,30,1,0,479001600,53343360,2728320,88200,2100,42,1,
%U A268436 0,87178291200,8693879040,397111680,11371920,235200,3864,56,1
%N 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.
%H A268436 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform">The P-transform</a>.
%F A268436 T(n,k) = binomial(n,k)*Sum_{i=0..k} binomial(k,i)*A268438(n-k,i).
%F A268436 T(n,k) = binomial(-k,-n)*Sum_{i=0..n-k} binomial(-n,i)*A268437(n-k,i).
%F A268436 T(n,k) = 4^(n-k)*Gamma(n-k+1/2)*A132393(n,k)/sqrt(Pi).
%F A268436 T(n,1) = (2*n-2)! for n>=1.
%F A268436 T(n,n-1) = (n-1)*n for n>=1.
%F A268436 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).
%e A268436 [1]
%e A268436 [0,         1]
%e A268436 [0,         2,        1]
%e A268436 [0,        24,        6,       1]
%e A268436 [0,       720,      132,      12,     1]
%e A268436 [0,     40320,     6000,     420,    20,    1]
%e A268436 [0,   3628800,   460320,   27000,  1020,   30,  1]
%e A268436 [0, 479001600, 53343360, 2728320, 88200, 2100, 42, 1]
%p A268436 T := (n,k) -> pochhammer(n-k+1, n-k)*abs(Stirling1(n,k)):
%p A268436 for n from 0 to 9 do seq(T(n,k), k=0..n) od;
%p A268436 # Alternatively:
%p A268436 T := proc(n,k) option remember;
%p A268436   `if`( n=k, 1,
%p A268436   `if`( k=0, 0,
%p A268436    (n-1)*(4*(n-k)-2)*T(n-1,k)+T(n-1,k-1))) end:
%p A268436 for n from 0 to 7 do seq(T(n,k),k=0..n) od;
%t A268436 T[n_, k_] := Pochhammer[n-k+1, n-k] Abs[StirlingS1[n, k]];
%t A268436 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 12 2019 *)
%o A268436 (Sage)
%o A268436 A268436 = lambda n,k: rising_factorial(n-k+1,n-k)*stirling_number1(n,k)
%o A268436 [[A268436(n,k) for k in (0..n)] for n in range(8)]
%Y A268436 Cf. A132393, A268435, A268437, A268438.
%K A268436 nonn,tabl
%O A268436 0,5
%A A268436 _Peter Luschny_, Mar 07 2016