A373433 -id:A373433 - cp's OEIS frontend

A373434 Triangle read by rows: Coefficients of the Eulerian polynomials EC(n, x)*EZ(n, x), where EC denote the classical Eulerian and EZ the zig-zag Eulerian polynomials.

1, 1, 1, 1, 1, 5, 5, 1, 1, 14, 45, 45, 14, 1, 1, 33, 255, 671, 671, 255, 33, 1, 1, 71, 1131, 6311, 14446, 14446, 6311, 1131, 71, 1, 1, 146, 4420, 46571, 206932, 427370, 427370, 206932, 46571, 4420, 146, 1, 1

Offset: 0

Peter Luschny, Jun 04 2024

There are various conventions for indexing Eulerian numbers. The one used here determines that all corresponding polynomials have p(n, 0) = 1. This applies equally to the classical Eulerian polynomials with coefficients A173018, the Eulerian zig-zag polynomials with coefficients A205497, and the polynomials here.

			Triangle T(n, k) starts:
  [0] 1;
  [1] 1;
  [2] 1,  1;
  [3] 1,  5,    5,    1;
  [4] 1, 14,   45,   45,    14,     1;
  [5] 1, 33,  255,  671,   671,   255,   33,    1;
  [6] 1, 71, 1131, 6311, 14446, 14446, 6311, 1131, 71,  1;
  ...
Written as polynomials P(n, x):
  [0] 1;
  [1] 1;
  [2] 1 +    x;
  [3] 1 +  5*x +   5*x^2 +     x^3;
  [4] 1 + 14*x +  45*x^2 +  45*x^3 +  14*x^4 +     x^5;
  [5] 1 + 33*x + 255*x^2 + 671*x^3 + 671*x^4 + 255*x^5 + 33*x^6 + x^7;
  ...
P(3, x) = A205497(3, x) * A173018(3, x) = (1 + x) * (1 + 4*x + x^2) = 1 + 5*x + 5*x^2 + x^3.

Peter Luschny, Illustrating the polynomials.

Cf. A173018 (Eulerian), A205497 (Eulerian zig-zag), A373433 (row sums).

# Using the recurrence by Kyle Petersen from A205497.
R := proc(n) option remember; local F; if n = 0 then 1/(1 - q*x) else F := R(n - 1); simplify(p/(p - q)*(subs({p = q, q = p}, F) - subs(p = q, F))) fi end:
EZ := (n, x) -> ifelse(n < 3, 1, expand(simplify(subs({p = 1, q = 1}, R(n))*(1 - x)^(n + 1)) / x^2)):
EC := (n, x) -> local k; simplify(add(combinat:-eulerian1(n, k)*x^k, k = 0..n)):
EZC := (n, x) -> expand(EZ(n, x) * EC(n, x)):
Trow := n -> local k; if n < 2 then [1] elif n = 2 then [1, 1] else [seq(coeff(EZC(n, x), x, k), k = 0..2*n-3)] fi:
seq(print(EZC(n, x)), n = 0..6); seq(print(Trow(n)), n = 0..6);

R[n_] := R[n] = Module[{F}, If[n == 0, 1/(1 - q*x), F = R[n - 1]; Simplify[p/(p - q)*(ReplaceAll[F, {p -> q, q -> p}] - ReplaceAll[F, p -> q])]]];
EZ[n_, x_] := If[n < 3, 1, Expand[Simplify[ReplaceAll[R[n], {p -> 1, q -> 1}]*(1 - x)^(n + 1)] / x^2]];
eulerian1[n_, k_] := If[n == 0, 1, Sum[(-1)^j*Binomial[n + 1, j]*(k + 1 - j)^n, {j, 0, k + 1}]];
EC[n_, x_] :=  Sum[eulerian1[n, k]*x^k, {k, 0, n}];
EZC [n_, x_] := Expand[EZ[n, x] * EC[n, x]];
Trow[n_] := CoefficientList[EZC[n, x], x];
Table[Trow[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jun 07 2024, after Peter Luschny's Maple program *)

A373427 Triangle read by rows: Coefficients of the polynomials SC(n, x) * EZ(n, x), where SC denote the Stirling cycle polynomials and EZ the Eulerian zig-zag polynomials A205497.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 5, 4, 1, 0, 6, 29, 45, 30, 9, 1, 0, 24, 218, 553, 629, 366, 112, 17, 1, 0, 120, 1954, 7781, 13409, 12136, 6270, 1894, 326, 29, 1, 0, 720, 20484, 125968, 313715, 407297, 308286, 143725, 42124, 7683, 830, 47, 1

Offset: 0

Peter Luschny, Jun 07 2024

			Tracing the computation:
0: [1] *          [1] =                    [1]
1: [1] *          [0,  1] =                [0,  1]
2: [1] *          [0,  1, 1] =             [0,  1,   1]
3: [1, 1] *       [0,  2, 3, 1] =          [0,  2,   5,   4,   1]
4: [1, 3, 1] *    [0,  6, 11, 6, 1] =      [0,  6,  29,  45,  30,   9,   1]
5: [1, 7, 7, 1] * [0, 24, 50, 35, 10, 1] = [0, 24, 218, 553, 629, 366, 112,17,1]

Peter Luschny, Illustrating the polynomials.

Cf. A132393 (Stirling cycle), A205497 (zig-zag Eulerian), A373433 (row sums).

Cf. A373429, A373428, A373432.

EZP((n, k) -> abs(Stirling1(n, k)), 7);  # Using function EZP from A373432.

cp's OEIS Frontend

A373434 Triangle read by rows: Coefficients of the Eulerian polynomials EC(n, x)*EZ(n, x), where EC denote the classical Eulerian and EZ the zig-zag Eulerian polynomials.

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