A391610 Triangle T(n,k) read by rows: T(n,k) is the coefficient of x^k of the monic polynomial (1+x)^n + ((2*(1+x))^n - (2+x)^n) / x.

1, 2, 1, 5, 5, 1, 13, 21, 10, 1, 33, 76, 62, 19, 1, 81, 245, 290, 160, 36, 1, 193, 726, 1135, 920, 387, 69, 1, 449, 2023, 3941, 4235, 2639, 903, 134, 1, 1025, 5384, 12572, 16856, 13958, 7112, 2060, 263, 1, 2305, 13833, 37668, 60564, 62622, 42462, 18372, 4626, 520, 1

Offset

0,2

Comments

These should be the f-vectors of some reflexive polytopes, see also A388426.

The constant terms are given by A005183.

The values at x=1 are given by A083324.

The subleading coefficients are given by A052944.

The associated h-vectors are essentially given by A008949.

Links

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

Formula

T(n,k) = [x^k] (1+x)^n + ((2*(1+x))^n - (2+x)^n) / x.

Example

The first few polynomials are:
 1,
 x + 2,
 x^2 + 5*x + 5,
 x^3 + 10*x^2 + 21*x + 13,
 x^4 + 19*x^3 + 62*x^2 + 76*x + 33,

Maple

T:= (n, k)-> coeff((1+x)^n+((2*(1+x))^n-(2+x)^n)/x, x, k):
seq(seq(T(n, k), k=0..n), n=0..9);  # Alois P. Heinz, Dec 15 2025

Mathematica

row[n_]:=CoefficientList[Series[(1+x)^n + ((2*(1+x))^n - (2+x)^n) / x, {x, 0, n}], x]; Array[row, 10, 0]//Flatten (* Stefano Spezia, Dec 14 2025 *)

Prog

(SageMath)
x = polygen(ZZ, 'x')
def A(n):
    y = x + 1
    return y**n + ((2*y)**n - (1+y)**n) // (y - 1)
[A(n) for n in range(6)]
(PARI) row(n) = Vecrev(Vec((1+x)^n + ((2*(1+x))^n - (2+x)^n) / x)); \\ Michel Marcus, Dec 15 2025

Crossrefs

Cf. A005183, A008949, A052944, A083324, A388426.

Sequence in context: A126124 A123971 A060920 * A107842 A126216 A124733

Adjacent sequences: A391607 A391608 A391609 * A391611 A391612 A391613

Keyword

nonn,tabl

Author

F. Chapoton, Dec 14 2025

Status

approved