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A344469 Triangle read by rows: T(n, k) (0 <= k <= n) = [x^k] x^n * n! * [t^n] x*(1 + t)/(x*exp(-t) - t).

%I A344469 #20 Sep 03 2023 10:15:40
%S A344469 1,1,2,2,6,3,6,24,24,4,24,120,180,80,5,120,720,1440,1080,240,6,720,
%T A344469 5040,12600,13440,5670,672,7,5040,40320,120960,168000,107520,27216,
%U A344469 1792,8,40320,362880,1270080,2177280,1890000,774144,122472,4608,9
%N A344469 Triangle read by rows: T(n, k) (0 <= k <= n) = [x^k] x^n * n! * [t^n] x*(1 + t)/(x*exp(-t) - t).
%C A344469 Related to the Lambert W-function, see Cohen, Corollary 2.4.
%H A344469 Henri Cohen, <a href="https://arxiv.org/abs/2012.11698">Lambert W-Function Branch Identities</a>, arXiv:2012.11698v2 [math.CV], 2020-2021.
%e A344469 Triangle starts:
%e A344469 [0] 1;
%e A344469 [1] 1,     2;
%e A344469 [2] 2,     6,      3;
%e A344469 [3] 6,     24,     24,      4;
%e A344469 [4] 24,    120,    180,     80,      5;
%e A344469 [5] 120,   720,    1440,    1080,    240,     6;
%e A344469 [6] 720,   5040,   12600,   13440,   5670,    672,    7;
%e A344469 [7] 5040,  40320,  120960,  168000,  107520,  27216,  1792,   8;
%e A344469 [8] 40320, 362880, 1270080, 2177280, 1890000, 774144, 122472, 4608, 9.
%p A344469 gf := x*(1+t)/(x*exp(-t)-t): ser := series(gf,t,12):
%p A344469 seq(seq(coeff(expand(x^n*n!*coeff(ser,t,n)),x,k),k=0..n),n=0..8);
%t A344469 (* rows[n], n[0..oo] *)
%t A344469 n=12;r={};For[k=0,k<n+1,k++,AppendTo[r,(n!)*((n-k+1)^(k-1))*(n+1)/(k!)]];r
%t A344469 (* columns[k], k[0..oo] *)
%t A344469 k=3;c={};For[n=k,n<13+k,n++,AppendTo[c,(n!)*((n-k+1)^(k-1))*(n+1)/(k!)]];c
%t A344469 (* sequence *)
%t A344469 s={};For[n=0,n<13,n++,For[k=0,k<n+1,k++,AppendTo[s,(n!)*((n-k+1)^(k-1))*(n+1)/(k!)]]];s
%t A344469 (* _Detlef Meya_, Jul 31 2023 *)
%Y A344469 Cf. A305990 (row sums), A009306 (alternating row sums).
%K A344469 nonn,tabl
%O A344469 0,3
%A A344469 _Peter Luschny_, May 20 2021