A128504 Row sums of array A128503 (second convolution of Chebyshev's S(n,x) = U(n,x/2) polynomials).

1, 3, 3, -2, -9, -9, 3, 18, 18, -4, -30, -30, 5, 45, 45, -6, -63, -63, 7, 84, 84, -8, -108, -108, 9, 135, 135, -10, -165, -165, 11, 198, 198, -12, -234, -234, 13, 273, 273, -14, -315, -315, 15, 360, 360, -16, -408, -408, 17, 459, 459

Offset: 0

Author

Wolfdieter Lang Apr 04 2007

Keywords

• sign
• easy

Comments

Second convolution of A010892.

Convolution of A099254 with A010892.

a(n) equals the coefficient of x^2 of the characteristic polynomial of the (n+2)X(n+2) tridiagonal matrix with 1's along the main diagonal, the superdiagonal, and the subdiagonal (see Mathematica code below). [John M. Campbell, Jul 10 2011]

Links

• Tewodros Amdeberhan, George E. Andrews, and Roberto Tauraso, Further study on MacMahon-type sums of divisors, arXiv:2409.20400 [math.NT], 2024. See p. 18.

Crossrefs

Cf. A010892, A099254, A128503.

Programs

Mathematica

Table[Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1, #2] + KroneckerDelta[#1, #2 - 1] + KroneckerDelta[#1, #2 + 1] &, {n + 2, n + 2}], x], x^2], {n, 0, 70}] (* John M. Campbell, Jul 10 2011 *)

PARI

Vec(1/(1-x+x^2)^3+O(x^66)) \\ Joerg Arndt, Jul 02 2013

Formula

a(n) = Sum_{m=0..floor(n/2)} A128503(n,m).

G.f.: 1/(1-x+x^2)^3.

a(n) = (floor(n/3)+1)*(floor(n/3)-floor((n-1)/3)+(3/2)*(floor(n/3)+2)*(3*floor((n+1)/3)-n))*(-1)^n. - Tani Akinari, Jul 03 2013