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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.

%I A373657 #8 Jun 16 2024 04:46:47
%S A373657 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,
%T A373657 -276,276,-54,-54,21,1,1,51,-25,-675,1650,-1002,-1002,1650,-675,-25,
%U A373657 51,1,-1,-113,-372,3436,-5125,-5013,21216,-21216,5013,5125,-3436,372,113,1
%N 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.
%H A373657 S. Tanimoto, <a href="http://arXiv.org/abs/math.CO/0602263">A new approach to signed Eulerian numbers</a>, arXiv:math/0602263 [math.CO], 2006. (see p. 7)
%e A373657 Triangle starts:
%e A373657 [0] [ 1]
%e A373657 [1] [-1,   1]
%e A373657 [2] [ 1,  -1,  -1,    1]
%e A373657 [3] [-1,  -1,   8,   -8,    1,     1]
%e A373657 [4] [ 1,   7, -27,   19,   19,   -27,     7,    1]
%e A373657 [5] [-1, -21,  54,   54, -276,   276,   -54,  -54,   21,   1]
%e A373657 [6] [ 1,  51, -25, -675, 1650, -1002, -1002, 1650, -675, -25, 51, 1]
%p A373657 PolyProd := proc(P, Q, len) local ep, eq, epq, CL, n, k;
%p A373657 ep := (n, x) -> simplify(add(Q(n, k)*x^k, k = 0..n)):
%p A373657 eq := (n, x) -> simplify(add(P(n, k)*x^k, k = 0..n)):
%p A373657 epq := (n, x) -> expand(ep(n, x) * eq(n, x)):
%p A373657 CL := p -> PolynomialTools:-CoefficientList(p, x);
%p A373657 seq(CL(epq(n, x)), n = 0..len); ListTools:-Flatten([%]) end:
%p A373657 PolyProd((n, k) -> (-1)^(n-k)*binomial(n, k), combinat:-eulerian1, 7);
%Y A373657 Cf. A173018, A049061, A101842, A000007 (row sums).
%K A373657 sign,tabf
%O A373657 0,10
%A A373657 _Peter Luschny_, Jun 15 2024