|
%I #18 Mar 28 2020 05:28:51
%S 1,0,1,0,8,3,0,90,120,15,0,1344,3640,1680,105,0,25200,110880,107100,
%T 25200,945,0,570240,3617460,5815040,2910600,415800,10395,0,15135120,
%U 128576448,303963660,256736480,78828750,7567560,135135
%N Triangle read by rows, T(n,k) = C(2*n,n+k)*Sum_{m=0..k} (-1)^(m+k)*C(n+k,n+m)* Stirling1(n+m,m), 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) = ((-1)^k*(2*n)!/(k!*(n-k)!))*P[n,k](n/(n+1)) where P is the P-transform. The P-transform is defined in the link.
%F T(n,k) = A269940*binomial(2*n,n+k).
%F T(n,k) = A268438(n,k)/(k!*(n-k)!).
%F T(n,1) = n*(2*n)!/(n+1)! for n>=1 (cf. A092956).
%F T(n,n) = (2*n-1)!! = A001147(n) for n>=0.
%e [1]
%e [0, 1]
%e [0, 8, 3]
%e [0, 90, 120, 15]
%e [0, 1344, 3640, 1680, 105]
%e [0, 25200, 110880, 107100, 25200, 945]
%e [0, 570240, 3617460, 5815040, 2910600, 415800, 10395]
%p # The function PTrans is defined in A269941.
%p A268440_row := n -> PTrans(n, n->n/(n+1), (n,k) -> (-1)^k*(2*n)!/(k!*(n-k)!)):
%p seq(print(A268440_row(n)), n=0..8);
%o (Sage)
%o A268440 = lambda n, k: binomial(2*n,n+k)*sum((-1)^(m+k)*binomial(n+k,n+m)* stirling_number1(n+m, m) for m in (0..k))
%o for n in (0..7): print([A268440(n, m) for m in (0..n)])
%o (Sage) # uses[PtransMatrix from A269941]
%o # Alternatively
%o PtransMatrix(7, lambda n: n/(n+1), lambda n,k: (-1)^k*factorial(2*n)/ (factorial(k)*factorial(n-k)))
%Y Cf. A001147, A092956, A268438, A268439, A269940, A269941.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Mar 08 2016
|
