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

%I #20 Sep 03 2023 10:15:40

%S 1,1,2,2,6,3,6,24,24,4,24,120,180,80,5,120,720,1440,1080,240,6,720,

%T 5040,12600,13440,5670,672,7,5040,40320,120960,168000,107520,27216,

%U 1792,8,40320,362880,1270080,2177280,1890000,774144,122472,4608,9

%N 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 Related to the Lambert W-function, see Cohen, Corollary 2.4.

%H Henri Cohen, <a href="https://arxiv.org/abs/2012.11698">Lambert W-Function Branch Identities</a>, arXiv:2012.11698v2 [math.CV], 2020-2021.

%e Triangle starts:

%e [0] 1;

%e [1] 1, 2;

%e [2] 2, 6, 3;

%e [3] 6, 24, 24, 4;

%e [4] 24, 120, 180, 80, 5;

%e [5] 120, 720, 1440, 1080, 240, 6;

%e [6] 720, 5040, 12600, 13440, 5670, 672, 7;

%e [7] 5040, 40320, 120960, 168000, 107520, 27216, 1792, 8;

%e [8] 40320, 362880, 1270080, 2177280, 1890000, 774144, 122472, 4608, 9.

%p gf := x*(1+t)/(x*exp(-t)-t): ser := series(gf,t,12):

%p seq(seq(coeff(expand(x^n*n!*coeff(ser,t,n)),x,k),k=0..n),n=0..8);

%t (* rows[n], n[0..oo] *)

%t n=12;r={};For[k=0,k<n+1,k++,AppendTo[r,(n!)*((n-k+1)^(k-1))*(n+1)/(k!)]];r

%t (* columns[k], k[0..oo] *)

%t k=3;c={};For[n=k,n<13+k,n++,AppendTo[c,(n!)*((n-k+1)^(k-1))*(n+1)/(k!)]];c

%t (* sequence *)

%t 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 (* _Detlef Meya_, Jul 31 2023 *)

%Y Cf. A305990 (row sums), A009306 (alternating row sums).

%K nonn,tabl

%O 0,3

%A _Peter Luschny_, May 20 2021