A220884 Triangle read by rows: row n gives coefficients of expansion of Product_{k=2..n} ((n+1-k)*x+k), starting with lowest power.

1, 1, 2, 1, 6, 8, 2, 24, 58, 37, 6, 120, 444, 504, 204, 24, 720, 3708, 6388, 4553, 1318, 120, 5040, 33984, 81136, 87296, 44176, 9792, 720, 40320, 341136, 1064124, 1582236, 1203921, 463860, 82332, 5040, 362880, 3733920, 14602320, 28328480, 29724000, 17164320, 5270480, 773280, 40320, 3628800, 44339040, 210852936, 512539012, 700870638, 557061609, 255644668, 64621692, 8026416, 362880

Offset: 0

Author

N. J. A. Sloane, Dec 29 2012

Keywords

• nonn
• tabf

Comments

Related to Stirling numbers A008275, A008277.

Examples

			Triangle begins:
[1]
[1]
[2, 1]
[6, 8, 2]
[24, 58, 37, 6]
[120, 444, 504, 204, 24]
[720, 3708, 6388, 4553, 1318, 120]
[5040, 33984, 81136, 87296, 44176, 9792, 720]
...
		

Links

• Alois P. Heinz, Rows n = 0..141, flattened
• Irfan Durmić, Alex Han, Pamela E. Harris, Rodrigo Ribeiro, and Mei Yin, Probabilistic Parking Functions, arXiv:2211.00536 [math.CO], 2022.
• Jean-Christophe Novelli and Jean-Yves Thibon, Duplicial algebras and Lagrange inversion, arXiv preprint arXiv:1209.5959 [math.CO], 2012-2013.

Crossrefs

Cf. A220883, A008275, A008277.

Row sums give A000272(n+1).

Columns k=0-1 give A000142, A002538(n-1).

Programs

Maple

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
         expand(mul((n+1-k)*x+k, k=2..n))):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 29 2015

Mathematica

row[n_] := CoefficientList[Product[((n+1-k)*x+k), {k, 2, n}], x]; Table[ row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 17 2016 *)

Extensions

T(0,0)=1 prepended by Alois P. Heinz, Nov 29 2015