A394446 Triangle T(n,k), where row n lists the coefficients of the n-th palindromic polynomial p(n) defined by a recurrence of order 2.

1, 1, 3, 1, 1, 7, 13, 7, 1, 1, 12, 45, 69, 45, 12, 1, 1, 18, 108, 296, 411, 296, 108, 18, 1, 1, 25, 215, 890, 2015, 2633, 2015, 890, 215, 25, 1, 1, 33, 381, 2180, 7110, 14148, 17739, 14148, 7110, 2180, 381, 33, 1, 1, 42, 623, 4662, 20391, 56238, 101941, 124029, 101941, 56238, 20391, 4662, 623, 42, 1

Offset

0,3

Comments

This should be the h^* vectors of some reflexive polytopes.

Links

Alois P. Heinz, Rows n = 1..100, flattened

Formula

T(n,k) = [x^k] p(n) with p(0) = 1; p(1) = 1+3*x+x^2; (n+2)*p(n) = p(1)*(2*n+1)*p(n-1)-(1+x+x^2)^2*(n-1)*p(n-2).

Example

The first few polynomials are:
  1,
  x^2 + 3*x + 1,
  x^4 + 7*x^3 + 13*x^2 + 7*x + 1,
  x^6 + 12*x^5 + 45*x^4 + 69*x^3 + 45*x^2 + 12*x + 1.
Triangle T(n,k) begins:
  1;
  1,  3,   1;
  1,  7,  13,   7,    1;
  1, 12,  45,  69,   45,   12,    1;
  1, 18, 108, 296,  411,  296,  108,  18,   1;
  1, 25, 215, 890, 2015, 2633, 2015, 890, 215, 25, 1;
  ...

Maple

b:= proc(n) option remember; `if`(n=0, 1, `if`(n=1, 1+3*x+x^2,
      expand((b(1)*(2*n+1)*b(n-1)-(1+x+x^2)^2*(n-1)*b(n-2))/(n+2))))
    end:
T:= n-> coeffs(b(n)):
seq(T(n), n=0..7);  # Alois P. Heinz, Mar 20 2026

Mathematica

b[n_] := b[n] = If[n == 0, 1, If[n == 1, 1+3*x + x^2, Expand[(b[1]*(2*n+1)*b[n-1] - (1+x+x^2)^2*(n-1)*b[n- 2])/(n+2)]]];
T[n_] := CoefficientList[b[n], x];
Table[T[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Jun 14 2026, after Alois P. Heinz *)

Prog

(SageMath)
@cached_function
def a(n):
    R = PolynomialRing(ZZ, 'x')
    if n == 0:
        return R.one()
    p5 = R([1, 3, 1])
    if n == 1:
        return p5
    p9 = R([1, 1, 1])**2
    return (p5*(2*n+1)*a(n-1)-p9*(n-1)*a(n-2)) / (n + 2)

Crossrefs

Row sums give A059231(n+1).

Main diagonal gives A128079.

Columns k=0-1 give: A000012, A055998.

Sequence in context: A353532 A049290 A297191 * A147990 A134567 A131932

Adjacent sequences: A394443 A394444 A394445 * A394447 A394448 A394449

Keyword

nonn,tabf

Author

F. Chapoton, Mar 20 2026

Status

approved