%I #34 Jul 08 2024 10:37:22

%S 10,45,126,280,540,945,1540,2376,3510,5005,6930,9360,12376,16065,

%T 20520,25840,32130,39501,48070,57960,69300,82225,96876,113400,131950,

%U 152685,175770,201376,229680,260865,295120,332640,373626,418285,466830,519480,576460

%N a(n) = binomial(n+2,2)*binomial(n+5,2).

%H Nathaniel Johnston, <a href="/A105938/b105938.txt">Table of n, a(n) for n = 0..5000</a>

%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_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F G.f.: (x^2-5*x+10)/(1-x)^5. - _Alois P. Heinz_, Oct 16 2008

%F a(0)=10, a(1)=45, a(2)=126, a(3)=280, a(4)=540; for n>4, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Harvey P. Dale_, Sep 05 2013

%F a(n) = A000217(n+1)*A000217(n+4). - _R. J. Mathar_, Nov 29 2015

%F a(n) = A000096(n+1)*A000096(n+2). - _Bruno Berselli_, Sep 21 2016

%F From _Amiram Eldar_, Jan 06 2021: (Start)

%F Sum_{n>=0} 1/a(n) = 5/36.

%F Sum_{n>=0} (-1)^n/a(n) = 1/12. (End)

%e If n=0 then C(0+2,0)*C(0+5,2) = C(2,0)*C(5,2) = 1*10 = 10.

%e If n=9 then C(9+2,9)*C(9+5,2) = C(11,9)*C(14,2) = 55*91 = 5005.

%p a:= n-> binomial(n+2,n)*binomial(n+5,2):

%p seq(a(n), n=0..40); # _Alois P. Heinz_, Oct 16 2008

%t Table[n (n + 1) (n + 3) (n + 4)/4, {n, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 26 2011 *)

%t Table[Binomial[n + 2, n] Binomial[n + 5, 2], {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {10, 45, 126, 280, 540}, 40] (* _Harvey P. Dale_, Sep 05 2013 *)

%Y Subsequence of A085780.

%Y Cf. A000096, A000217, A062145.

%K nonn,easy

%O 0,1

%A _Zerinvary Lajos_, Apr 27 2005