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A337967
Triangle read by rows, application of the transformation A337966 to Euler's triangle A173018. T(n, k) for 0 <= k <= n.
4
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.
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
KEYWORD
sign,tabl
AUTHOR
Peter Luschny, Oct 04 2020
STATUS
approved