%I #28 Jul 06 2024 10:30:37

%S 0,21,168,756,2520,6930,16632,36036,72072,135135,240240,408408,668304,

%T 1058148,1627920,2441880,3581424,5148297,7268184,10094700,13813800,

%U 18648630,24864840,32776380,42751800,55221075,70682976,89713008,112971936,141214920,175301280

%N a(n) = 21*binomial(n+6,7).

%C Total number of pips on a set of hexominoes (6-celled linear dominoes) with up to n pips in each cell.

%H Colin Barker, <a href="/A266733/b266733.txt">Table of n, a(n) for n = 0..1000</a>

%H Steve Butler and Pavel Karasik, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Butler/butler7.html">A note on nested sums</a>, J. Int. Seq. (2010) Vol. 13, Issue 4, Art. No. 10.4.4. See p=6 in the last equation on page 3.

%H Sela Fried, <a href="https://arxiv.org/abs/2406.18923">Counting r X s rectangles in nondecreasing and Smirnov words</a>, arXiv:2406.18923 [math.CO], 2024. See p. 9.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

%F a(n) = 21*A000580(n+6).

%F From _Colin Barker_, Jan 08 2016: (Start)

%F a(n) = n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)/240.

%F 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.

%F G.f.: 21*x / (1-x)^8.

%F (End)

%t Table[21 Binomial[n+6,7],{n,0,40}] (* _Harvey P. Dale_, Jan 13 2021 *)

%o (PARI) a(n) = (n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n))/240 \\ _Colin Barker_, Jan 08 2016

%o (PARI) concat(0, Vec(21*x/(1-x)^8 + O(x^40))) \\ _Colin Barker_, Jan 08 2016

%Y Row 6 of array in A129533.

%K nonn,easy

%O 0,2

%A _Alan Shore_ and _N. J. A. Sloane_, Jan 06 2016