A099193 a(n) = n*(4*n^6 + 70*n^4 + 196*n^2 + 45)/315.

0, 1, 14, 99, 476, 1765, 5418, 14407, 34232, 74313, 149830, 284075, 511380, 880685, 1459810, 2340495, 3644272, 5529233, 8197758, 11905267, 16970060, 23784309, 32826266, 44673751, 60018984, 79684825, 104642486, 136030779, 175176964, 223619261, 283131090, 355747103

Offset: 0

Author

Jonathan Vos Post, Nov 16 2004

Keywords

• easy
• nonn

Comments

Kim asserts that every nonnegative integer can be represented by the sum of no more than 21 of these numbers.

Starting with 1 = binomial transform of [1, 13, 72, 220, 400, 432, 256, 0, 0, 0, ...], where (1, 13, 72, 220, 400, 432, 256) = row 7 of the Chebyshev triangle A081277. Also = row 7 of the array in A142978. - Gary W. Adamson, Jul 19 2008

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
• Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
• Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Crossrefs

Similar sequences: A005900 (m=3), A014820(n-1) (m=4), A069038 (m=5), A069039 (m=6), A099195 (m=8), A099196 (m=9), A099197 (m=10).

Cf. A000332.

Cf. A142978, A081277.

Programs

Mathematica

Table[SeriesCoefficient[x (1 + x)^6/(1 - x)^8, {x, 0, n}], {n, 0, 31}] (* Michael De Vlieger, Dec 14 2015 *)

PARI

concat(0, Vec(x*(1+x)^6/(1-x)^8 + O(x^40))) \\ Michel Marcus, Dec 14 2015

Formula

a(n) = n*(4*n^6 + 70*n^4 + 196*n^2 + 45)/315.

G.f.: x*(1+x)^6/(1-x)^8. - R. J. Mathar, Jul 18 2009

a(n) = 14*a(n-1)/(n-1) + a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018

Extensions

More terms from Michel Marcus, Dec 14 2015