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

0,3

Comments

Related to Stirling numbers A008275, A008277.

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.

Example

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]
...

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 *)

Crossrefs

Cf. A220883, A008275, A008277.

Row sums give A000272(n+1).

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

Sequence in context: A113374 A136470 A363747 * A217877 A138510 A026215

Adjacent sequences: A220881 A220882 A220883 * A220885 A220886 A220887

Keyword

nonn,tabf

Author

N. J. A. Sloane, Dec 29 2012

Extensions

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

Status

approved