A337967 - OEIS

%I #14 Oct 06 2020 07:02:06

%S 1,1,0,-1,-1,0,0,-4,0,0,-1,11,11,-1,0,-1,0,66,0,-1,0,1,57,-302,-302,

%T 57,1,0,0,120,0,-2416,0,120,0,0,1,-247,-4293,15619,15619,-4293,-247,1,

%U 0,1,0,-14608,0,156190,0,-14608,0,1,0,-1,-1013,47840,455192,-1310354,-1310354,455192,47840,-1013,-1,0

%N Triangle read by rows, application of the transformation A337966 to Euler's triangle A173018. T(n, k) for 0 <= k <= n.

%C Row sums divided by 2^floor(n/2) are the Euler up/down numbers A000111 with signs.

%F T(n, k) = A173018(n, k)*A337966(n, k).

%e Triangle starts:

%e [0]  1

%e [1]  1,    0

%e [2] -1,   -1,      0

%e [3]  0,   -4,      0,     0

%e [4] -1,   11,     11,    -1,      0

%e [5] -1,    0,     66,     0,     -1,     0

%e [6]  1,   57,   -302,  -302,     57,     1,      0

%e [7]  0,  120,      0, -2416,      0,   120,      0,  0

%e [8]  1, -247,  -4293, 15619,  15619, -4293,   -247,  1,  0

%e [9]  1,    0, -14608,     0, 156190,     0, -14608,  0,  1,  0

%e .

%e A000111(4) = 5 = -1 + 11 + 11 - 1 = 20/4 = A001250(4)/2.

%e A000111(5) = 16 = -1 + 66 - 1 = 64/4 = A001250(5)/2.

%p U := (n, k) -> combinat:-eulerian1(n, k):

%p Trow := n -> seq(coeff(A337966(n, x, U), z, k), k=0..n):

%p seq(lprint([n], Trow(n)), n=0..9);

%Y Cf. A337966, A173018, A000111, A001250, A337616 (row sums).

%K sign,tabl

%O 0,8

%A _Peter Luschny_, Oct 04 2020