OFFSET
0,2
COMMENTS
Total number of pips on a set of hexominoes (6-celled linear dominoes) with up to n pips in each cell.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq., Vol. 13, Issue 4 (2010), Article 10.4.4. See p=6 in the last equation on page 3.
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
FORMULA
a(n) = 21*A000580(n+6).
From Colin Barker, Jan 08 2016: (Start)
a(n) = n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)/240.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>7.
G.f.: 21*x / (1-x)^8. (End)
From Amiram Eldar, Sep 28 2025: (Start)
Sum_{n>=1} 1/a(n) = 1/18.
Sum_{n>=1} (-1)^(n+1)/a(n) = 64*log(2)/3 - 1327/90. (End)
MATHEMATICA
Table[21 Binomial[n+6, 7], {n, 0, 40}] (* Harvey P. Dale, Jan 13 2021 *)
PROG
(PARI) a(n) = (n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n))/240 \\ Colin Barker, Jan 08 2016
(PARI) concat(0, Vec(21*x/(1-x)^8 + O(x^40))) \\ Colin Barker, Jan 08 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Alan Shore and N. J. A. Sloane, Jan 06 2016
STATUS
approved