cp's OEIS Frontend

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.

%I A220884 #26 Feb 14 2023 16:45:40
%S A220884 1,1,2,1,6,8,2,24,58,37,6,120,444,504,204,24,720,3708,6388,4553,1318,
%T A220884 120,5040,33984,81136,87296,44176,9792,720,40320,341136,1064124,
%U A220884 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 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.
%C A220884 Related to Stirling numbers A008275, A008277.
%H A220884 Alois P. Heinz, <a href="/A220884/b220884.txt">Rows n = 0..141, flattened</a>
%H A220884 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 A220884 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 A220884 Triangle begins:
%e A220884 [1]
%e A220884 [1]
%e A220884 [2, 1]
%e A220884 [6, 8, 2]
%e A220884 [24, 58, 37, 6]
%e A220884 [120, 444, 504, 204, 24]
%e A220884 [720, 3708, 6388, 4553, 1318, 120]
%e A220884 [5040, 33984, 81136, 87296, 44176, 9792, 720]
%e A220884 ...
%p A220884 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
%p A220884          expand(mul((n+1-k)*x+k, k=2..n))):
%p A220884 seq(T(n), n=0..10);  # _Alois P. Heinz_, Nov 29 2015
%t A220884 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 A220884 Cf. A220883, A008275, A008277.
%Y A220884 Row sums give A000272(n+1).
%Y A220884 Columns k=0-1 give A000142, A002538(n-1).
%K A220884 nonn,tabf
%O A220884 0,3
%A A220884 _N. J. A. Sloane_, Dec 29 2012
%E A220884 T(0,0)=1 prepended by _Alois P. Heinz_, Nov 29 2015