cp's OEIS Frontend

A171150 Triangle related to T(x,2x).

%I A171150 #19 Jan 31 2023 14:19:30
%S A171150 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 A171150 63,1,35,245,651,1029,903,441,127,1,70,504,1736,3108,3612,2568,1016,
%U A171150 255,1,126,1134,4116,9324,12636,11556,6921,2295,511,1,252,2310,10290,25080,42120,46035,34605,17930,5110,1023,1
%N A171150 Triangle related to T(x,2x).
%C A171150 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 A171150 This triangle gives the coefficients of Sum_{k=0..n} T(n,k) where y=2x.
%C A171150 T_(0,0) = A053121, T_(1,2) = A039599, T_(2,4) = A124575.
%C A171150 First column of T_(x,2x) is given by A126222.
%H A171150 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 A171150 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 A171150 Triangle begins:
%e A171150    1;
%e A171150    1,  1;
%e A171150    2,  3,  1;
%e A171150    3,  9,  7,  1;
%e A171150    6, 20, 28, 15,  1;
%e A171150   10, 50, 85, 75, 31,  1;
%e A171150   ...
%Y A171150 Cf. A000012, A000225, A058877, A126222.
%Y A171150 Row sums give A000984.
%K A171150 nonn,tabl
%O A171150 0,4
%A A171150 _Philippe Deléham_, Dec 04 2009
%E A171150 More terms from _Alois P. Heinz_, Jan 31 2023