A388426 Triangle of palindromic polynomials.

1, 1, 1, 1, 5, 1, 1, 16, 16, 1, 1, 43, 121, 43, 1, 1, 106, 666, 666, 106, 1, 1, 249, 3089, 6649, 3089, 249, 1, 1, 568, 12916, 51976, 51976, 12916, 568, 1, 1, 1271, 50481, 348471, 641281, 348471, 50481, 1271, 1, 1, 2806, 188398, 2106622, 6488230, 6488230, 2106622, 188398, 2806, 1

Offset

0,5

Comments

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

Links

Table of n, a(n) for n=0..54.

Formula

E.g.f.: (s-1)/(s-exp((s-1)*x))*exp(s*x*(s-1)/(s-exp((s-1)*x))).

Example

Triangle begins:
  [1],
  [1, 1],
  [1, 5, 1],
  [1, 16, 16, 1],
  [1, 43, 121, 43, 1],
  [1, 106, 666, 666, 106, 1]
  ...

Prog

(SageMath)
Bs = PolynomialRing(QQ, 's')
s = Bs.gen()
La = LazyPowerSeriesRing(Bs, 'u')
u = La.gen()
f = (s-1)/(s-exp((s-1)*u))*exp(s*u*(s-1)/(s-exp((s-1)*u)))
[list(f[n] * factorial(n)) for n in range(9)]

Crossrefs

Row sums are A002720. Alternating row sums are A354020.

Similar to A046802.

Cf. A053221.

Sequence in context: A174159 A074060 A157637 * A157181 A347974 A029847

Adjacent sequences: A388423 A388424 A388425 * A388427 A388428 A388429

Keyword

nonn,tabl

Author

F. Chapoton, Oct 15 2025

Status

approved