A373657 Triangle read by rows: Coefficients of the polynomials P(n, x) * EP(n, x), where P denote the signed Pascal polynomials and EP the Eulerian polynomials A173018.

1, -1, 1, 1, -1, -1, 1, -1, -1, 8, -8, 1, 1, 1, 7, -27, 19, 19, -27, 7, 1, -1, -21, 54, 54, -276, 276, -54, -54, 21, 1, 1, 51, -25, -675, 1650, -1002, -1002, 1650, -675, -25, 51, 1, -1, -113, -372, 3436, -5125, -5013, 21216, -21216, 5013, 5125, -3436, 372, 113, 1

Offset: 0

Author

Peter Luschny, Jun 15 2024

Keywords

• sign
• tabf

Examples

			Triangle starts:
[0] [ 1]
[1] [-1,   1]
[2] [ 1,  -1,  -1,    1]
[3] [-1,  -1,   8,   -8,    1,     1]
[4] [ 1,   7, -27,   19,   19,   -27,     7,    1]
[5] [-1, -21,  54,   54, -276,   276,   -54,  -54,   21,   1]
[6] [ 1,  51, -25, -675, 1650, -1002, -1002, 1650, -675, -25, 51, 1]
		

Links

• S. Tanimoto, A new approach to signed Eulerian numbers, arXiv:math/0602263 [math.CO], 2006. (see p. 7)

Crossrefs

Cf. A173018, A049061, A101842, A000007 (row sums).

Programs

Maple

PolyProd := proc(P, Q, len) local ep, eq, epq, CL, n, k;
ep := (n, x) -> simplify(add(Q(n, k)*x^k, k = 0..n)):
eq := (n, x) -> simplify(add(P(n, k)*x^k, k = 0..n)):
epq := (n, x) -> expand(ep(n, x) * eq(n, x)):
CL := p -> PolynomialTools:-CoefficientList(p, x);
seq(CL(epq(n, x)), n = 0..len); ListTools:-Flatten([%]) end:
PolyProd((n, k) -> (-1)^(n-k)*binomial(n, k), combinat:-eulerian1, 7);