A337967 Triangle read by rows, application of the transformation A337966 to Euler's triangle A173018. T(n, k) for 0 <= k <= n.

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, 57, 1, 0, 0, 120, 0, -2416, 0, 120, 0, 0, 1, -247, -4293, 15619, 15619, -4293, -247, 1, 0, 1, 0, -14608, 0, 156190, 0, -14608, 0, 1, 0, -1, -1013, 47840, 455192, -1310354, -1310354, 455192, 47840, -1013, -1, 0

Offset

0,8

Comments

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

Links

Table of n, a(n) for n=0..65.

Formula

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

Example

Triangle starts:
[0]  1
[1]  1,    0
[2] -1,   -1,      0
[3]  0,   -4,      0,     0
[4] -1,   11,     11,    -1,      0
[5] -1,    0,     66,     0,     -1,     0
[6]  1,   57,   -302,  -302,     57,     1,      0
[7]  0,  120,      0, -2416,      0,   120,      0,  0
[8]  1, -247,  -4293, 15619,  15619, -4293,   -247,  1,  0
[9]  1,    0, -14608,     0, 156190,     0, -14608,  0,  1,  0
.
A000111(4) = 5 = -1 + 11 + 11 - 1 = 20/4 = A001250(4)/2.
A000111(5) = 16 = -1 + 66 - 1 = 64/4 = A001250(5)/2.

Maple

U := (n, k) -> combinat:-eulerian1(n, k):
Trow := n -> seq(coeff(A337966(n, x, U), z, k), k=0..n):
seq(lprint([n], Trow(n)), n=0..9);

Crossrefs

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

Sequence in context: A134832 A123163 A194794 * A317448 A292900 A177893

Adjacent sequences: A337964 A337965 A337966 * A337968 A337969 A337970

Keyword

sign,tabl

Author

Peter Luschny, Oct 04 2020

Status

approved