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Triangle read by rows: T(n, k) = (-1)^k*binomial(n, k) * A050446(n, n - k).
2

%I #9 Jun 13 2024 02:04:05

%S 1,2,-1,6,-6,1,30,-42,15,-1,190,-340,186,-32,1,1547,-3355,2460,-700,

%T 65,-1,15106,-38430,35295,-14140,2355,-126,1,173502,-506114,558285,

%U -289520,71295,-7413,238,-1,2286648,-7520040,9681700,-6174224,2033920,-328384,22204,-440,1

%N Triangle read by rows: T(n, k) = (-1)^k*binomial(n, k) * A050446(n, n - k).

%H Guoce Xin and Yueming Zhong, <a href="https://arxiv.org/abs/2201.02376">Proving some conjectures on Kekulé numbers for certain benzenoids by using Chebyshev polynomials</a>, arXiv:2201.02376 [math.CO], 2022.

%F Row sums are the Euler numbers A000111.

%e Triangle starts:

%e [0] 1;

%e [1] 2, -1;

%e [2] 6, -6, 1;

%e [3] 30, -42, 15, -1;

%e [4] 190, -340, 186, -32, 1;

%e [5] 1547, -3355, 2460, -700, 65, -1;

%e [6] 15106, -38430, 35295, -14140, 2355, -126, 1;

%e [7] 173502, -506114, 558285, -289520, 71295, -7413, 238, -1;

%p T := (n, k) -> (-1)^k*binomial(n, k) * A050446(n, n - k):

%p for n from 0 to 7 do print(seq(T(n, k), k=0..n)) od;

%Y Cf. A050446, A373659 (column 0), A000111 (row sums), A373658 (alternating row sums).

%K sign,tabl

%O 0,2

%A _Peter Luschny_, Jun 12 2024