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Triangle read by rows: denominators of polynomials from A000142: P(0,x) = 1, P(n,x) = 1/n! + x*Sum_{i=0..n-1} P(n-i-1)/i!. Numerators are A152650.
7

%I #21 Jul 11 2022 14:49:32

%S 1,1,1,2,1,1,6,1,1,1,24,3,2,1,1,120,3,2,1,1,1,720,15,8,3,2,1,1,5040,

%T 45,40,3,6,1,1,1,40320,315,80,15,24,1,2,1,1,362880,315,560,45,24,1,6,

%U 1,1,1,3628800,2835,4480,315,144,5,24,3,2,1,1

%N Triangle read by rows: denominators of polynomials from A000142: P(0,x) = 1, P(n,x) = 1/n! + x*Sum_{i=0..n-1} P(n-i-1)/i!. Numerators are A152650.

%C a(n) is the last sequence of a trio with, first, A141412 and, second, A142048 (denominators).

%H Vincenzo Librandi, <a href="/A152656/b152656.txt">Rows n = 0..100, flattened</a>

%e Contribution from _Vincenzo Librandi_, Dec 16 2012: (Start)

%e Triangle begins:

%e 1,

%e 1, 1,

%e 2, 1, 1,

%e 6, 1, 1, 1,

%e 24, 3, 2, 1, 1,

%e 120, 3, 2, 1, 1, 1,

%e 720, 15, 8, 3, 2, 1, 1,

%e 5040, 45, 40, 3, 6, 1, 1, 1,

%e 40320, 315, 80, 15, 24, 1, 2, 1, 1,

%e 362880, 315, 560, 45, 24, 1, 6, 1, 1, 1,

%e 3628800, 2835, 4480, 315, 144, 5, 24, 3, 2, 1, 1,

%e ...

%e First column: A000142; second column: A049606. (End)

%t ClearAll[u, p]; u[n_] := 1/n!; p[0][x_] := u[0]; p[n_][x_] := p[n][x] = u[n] + x*Sum[u[i]*p[n-i-1][x] , {i, 0, n-1}] // Expand; row[n_] := CoefficientList[p[n][x], x]; Table[row[n], {n, 0, 10}] // Flatten // Denominator (* _Jean-François Alcover_, Oct 02 2012 *)

%Y Cf. A152650, A142048, A142412,

%K nonn,tabl

%O 0,4

%A _Paul Curtz_, Dec 10 2008

%E More terms from _Jean-François Alcover_, Oct 02 2012