A325137 - OEIS

%I #8 Apr 13 2019 08:09:21

%S 1,1,1,2,5,1,6,26,12,1,24,154,119,22,1,120,1044,1175,355,35,1,720,

%T 8028,12154,5265,835,51,1,5040,69264,133938,77224,17360,1687,70,1,

%U 40320,663696,1580508,1155420,342769,46816,3066,92,1

%N Triangle T(n, k) = [x^n] (n + k + x)!/(k + x)! for 0 <= k <= n, read by rows.

%C Sister triangle of A307419.

%F T(n, k) = Sum_{j=0..n-k} binomial(j+k, k)*|Stirling1(n, j+k)|*(k+1)^j.

%e Triangle starts:

%e [0]      1

%e [1]      1,       1

%e [2]      2,       5,        1

%e [3]      6,      26,       12,        1

%e [4]     24,     154,      119,       22,       1

%e [5]    120,    1044,     1175,      355,      35,       1

%e [6]    720,    8028,    12154,     5265,     835,      51,      1

%e [7]   5040,   69264,   133938,    77224,   17360,    1687,     70,    1

%e [8]  40320,  663696,  1580508,  1155420,  342769,   46816,   3066,   92,   1

%e [9] 362880, 6999840, 19978308, 17893196, 6687009, 1197273, 109494, 5154, 117, 1

%e    A000142, A001705,  A001712,  A001718, A001724, ...

%p T := (n, k) -> add(binomial(j+k, k)*(k+1)^j*abs(Stirling1(n, j+k)), j=0..n-k);

%p seq(seq(T(n,k), k=0..n), n=0..8);

%p # Note that for n > 16 Maple fails (at least in some versions) to compute the

%p # terms properly. Inserting 'simplify' or numerical evaluation might help.

%p A325137Row := proc(n) local ogf, ser; ogf := (n, k) -> (n+k+x)!/(k+x)!;

%p ser := (n, k) -> series(ogf(n,k),x,k+2); seq(coeff(ser(n,k),x,k), k=0..n) end: seq(A325137Row(n), n=0..8);

%Y Row sums: A325138.

%Y Columns are: A000142, A001705, A001712, A001718, A001724.

%Y Cf. A307419.

%K nonn,tabl

%O 0,4

%A _Peter Luschny_, Apr 13 2019