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A325137 Triangle T(n, k) = [x^n] (n + k + x)!/(k + x)! for 0 <= k <= n, read by rows.

%I A325137 #8 Apr 13 2019 08:09:21
%S A325137 1,1,1,2,5,1,6,26,12,1,24,154,119,22,1,120,1044,1175,355,35,1,720,
%T A325137 8028,12154,5265,835,51,1,5040,69264,133938,77224,17360,1687,70,1,
%U A325137 40320,663696,1580508,1155420,342769,46816,3066,92,1
%N A325137 Triangle T(n, k) = [x^n] (n + k + x)!/(k + x)! for 0 <= k <= n, read by rows.
%C A325137 Sister triangle of A307419.
%F A325137 T(n, k) = Sum_{j=0..n-k} binomial(j+k, k)*|Stirling1(n, j+k)|*(k+1)^j.
%e A325137 Triangle starts:
%e A325137 [0]      1
%e A325137 [1]      1,       1
%e A325137 [2]      2,       5,        1
%e A325137 [3]      6,      26,       12,        1
%e A325137 [4]     24,     154,      119,       22,       1
%e A325137 [5]    120,    1044,     1175,      355,      35,       1
%e A325137 [6]    720,    8028,    12154,     5265,     835,      51,      1
%e A325137 [7]   5040,   69264,   133938,    77224,   17360,    1687,     70,    1
%e A325137 [8]  40320,  663696,  1580508,  1155420,  342769,   46816,   3066,   92,   1
%e A325137 [9] 362880, 6999840, 19978308, 17893196, 6687009, 1197273, 109494, 5154, 117, 1
%e A325137    A000142, A001705,  A001712,  A001718, A001724, ...
%p A325137 T := (n, k) -> add(binomial(j+k, k)*(k+1)^j*abs(Stirling1(n, j+k)), j=0..n-k);
%p A325137 seq(seq(T(n,k), k=0..n), n=0..8);
%p A325137 # Note that for n > 16 Maple fails (at least in some versions) to compute the
%p A325137 # terms properly. Inserting 'simplify' or numerical evaluation might help.
%p A325137 A325137Row := proc(n) local ogf, ser; ogf := (n, k) -> (n+k+x)!/(k+x)!;
%p A325137 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 A325137 Row sums: A325138.
%Y A325137 Columns are: A000142, A001705, A001712, A001718, A001724.
%Y A325137 Cf. A307419.
%K A325137 nonn,tabl
%O A325137 0,4
%A A325137 _Peter Luschny_, Apr 13 2019