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Triangle read by rows: row n gives coefficients of expansion of Product_{k=2..n} ((n+1-k)*x+k), starting with lowest power.
3

%I #26 Feb 14 2023 16:45:40

%S 1,1,2,1,6,8,2,24,58,37,6,120,444,504,204,24,720,3708,6388,4553,1318,

%T 120,5040,33984,81136,87296,44176,9792,720,40320,341136,1064124,

%U 1582236,1203921,463860,82332,5040,362880,3733920,14602320,28328480,29724000,17164320,5270480,773280,40320,3628800,44339040,210852936,512539012,700870638,557061609,255644668,64621692,8026416,362880

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

%C Related to Stirling numbers A008275, A008277.

%H Alois P. Heinz, <a href="/A220884/b220884.txt">Rows n = 0..141, flattened</a>

%H Irfan Durmić, Alex Han, Pamela E. Harris, Rodrigo Ribeiro, and Mei Yin, <a href="https://arxiv.org/abs/2211.00536">Probabilistic Parking Functions</a>, arXiv:2211.00536 [math.CO], 2022.

%H Jean-Christophe Novelli and Jean-Yves Thibon, <a href="http://arxiv.org/abs/1209.5959">Duplicial algebras and Lagrange inversion</a>, arXiv preprint arXiv:1209.5959 [math.CO], 2012-2013.

%e Triangle begins:

%e [1]

%e [1]

%e [2, 1]

%e [6, 8, 2]

%e [24, 58, 37, 6]

%e [120, 444, 504, 204, 24]

%e [720, 3708, 6388, 4553, 1318, 120]

%e [5040, 33984, 81136, 87296, 44176, 9792, 720]

%e ...

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(

%p expand(mul((n+1-k)*x+k, k=2..n))):

%p seq(T(n), n=0..10); # _Alois P. Heinz_, Nov 29 2015

%t 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 *)

%Y Cf. A220883, A008275, A008277.

%Y Row sums give A000272(n+1).

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

%K nonn,tabf

%O 0,3

%A _N. J. A. Sloane_, Dec 29 2012

%E T(0,0)=1 prepended by _Alois P. Heinz_, Nov 29 2015