A266732 a(n) = 10*binomial(n+4, 5).

0, 10, 60, 210, 560, 1260, 2520, 4620, 7920, 12870, 20020, 30030, 43680, 61880, 85680, 116280, 155040, 203490, 263340, 336490, 425040, 531300, 657800, 807300, 982800, 1187550, 1425060, 1699110, 2013760, 2373360, 2782560, 3246320, 3769920, 4358970, 5019420

Offset: 0

Author

Alan Shore and N. J. A. Sloane, Jan 06 2016

Keywords

• nonn
• easy

Comments

Total number of pips on a set of tetrominoes (4-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. (2010) Vol. 13, Issue 4, Art. No. 10.4.4. See p=4 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 (6,-15,20,-15,6,-1).

Crossrefs

Row 4 of array in A129533. Column k=3 in A253283.

Programs

Magma

[10*Binomial(n+4,5): n in [0..30]]; // G. C. Greubel, Nov 24 2017

Mathematica

Join[{0},10*Binomial[Range[0,40]+5,5]] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{0,10,60,210,560,1260},40] (* Harvey P. Dale, Jun 10 2016 *)

PARI

a(n) = (n*(1+n)*(2+n)*(3+n)*(4+n))/12 \\ Colin Barker, Jan 08 2016

PARI

concat(0, Vec(10*x/(1-x)^6 + O(x^50))) \\ Colin Barker, Jan 08 2016

Formula

From Colin Barker, Jan 08 2016: (Start)

a(n) = n*(1+n)*(2+n)*(3+n)*(4+n)/12.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 5.

G.f.: 10*x / (1-x)^6.

(End)

a(n) = 10*A000389(n+4). - R. J. Mathar, Dec 18 2016

E.g.f.: x*(120 + 240*x + 120*x^2 + 20*x^3 + x^4)*exp(x)/12. - G. C. Greubel, Nov 24 2017