A171150 - OEIS

%I #19 Jan 31 2023 14:19:30

%S 1,1,1,2,3,1,3,9,7,1,6,20,28,15,1,10,50,85,75,31,1,20,105,255,294,186,

%T 63,1,35,245,651,1029,903,441,127,1,70,504,1736,3108,3612,2568,1016,

%U 255,1,126,1134,4116,9324,12636,11556,6921,2295,511,1,252,2310,10290,25080,42120,46035,34605,17930,5110,1023,1

%N Triangle related to T(x,2x).

%C Let the triangle T_(x,y)=T defined by T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1.

%C This triangle gives the coefficients of Sum_{k=0..n} T(n,k) where y=2x.

%C T_(0,0) = A053121, T_(1,2) = A039599, T_(2,4) = A124575.

%C First column of T_(x,2x) is given by A126222.

%H M. Barnabei, F. Bonetti, and M. Silimbani, <a href="http://puma.dimai.unifi.it/21_2/1_Barnabei_Bonetti_Silimbani.pdf">The Eulerian numbers on restricted centrosymmetric permutations</a>, PU. M. A. Vol. 21 (2010), No. 2, pp. 99-118 (see Table p. 118, with additional zeros); see <a href="https://arxiv.org/abs/0910.2376">also</a>, arXiv:0910.2376 [math.CO], 2009.

%F Sum_{k=0..n} T(n,k)*x^k = A000007(n), A001405(n), A000984(n), A133158(n) for x = -1, 0, 1, 2 respectively.

%e Triangle begins:

%e    1;

%e    1,  1;

%e    2,  3,  1;

%e    3,  9,  7,  1;

%e    6, 20, 28, 15,  1;

%e   10, 50, 85, 75, 31,  1;

%e   ...

%Y Cf. A000012, A000225, A058877, A126222.

%Y Row sums give A000984.

%K nonn,tabl

%O 0,4

%A _Philippe Deléham_, Dec 04 2009

%E More terms from _Alois P. Heinz_, Jan 31 2023