This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A105938 #38 Mar 11 2025 04:37:49
%S A105938 10,45,126,280,540,945,1540,2376,3510,5005,6930,9360,12376,16065,
%T A105938 20520,25840,32130,39501,48070,57960,69300,82225,96876,113400,131950,
%U A105938 152685,175770,201376,229680,260865,295120,332640,373626,418285,466830,519480,576460
%N A105938 a(n) = binomial(n+2,2)*binomial(n+5,2).
%H A105938 Nathaniel Johnston, <a href="/A105938/b105938.txt">Table of n, a(n) for n = 0..5000</a>
%H A105938 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 A105938 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A105938 G.f.: (10 - 5*x + x^2)/(1-x)^5. - _Alois P. Heinz_, Oct 16 2008
%F A105938 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 A105938 a(n) = A000217(n+1)*A000217(n+4). - _R. J. Mathar_, Nov 29 2015
%F A105938 a(n) = A000096(n+1)*A000096(n+2). - _Bruno Berselli_, Sep 21 2016
%F A105938 From _Amiram Eldar_, Jan 06 2021: (Start)
%F A105938 Sum_{n>=0} 1/a(n) = 5/36.
%F A105938 Sum_{n>=0} (-1)^n/a(n) = 1/12. (End)
%F A105938 From _G. C. Greubel_, Mar 11 2025: (Start)
%F A105938 a(n) = 30*A000389(n+5)/(n+3).
%F A105938 E.g.f.: (1/4)*(40 + 140*x + 92*x^2 + 18*x^3 + x^4)*exp(x). (End)
%e A105938 If n=0 then C(0+2,0)*C(0+5,2) = C(2,0)*C(5,2) = 1*10 = 10.
%e A105938 If n=9 then C(9+2,9)*C(9+5,2) = C(11,9)*C(14,2) = 55*91 = 5005.
%p A105938 a:= n-> binomial(n+2,n)*binomial(n+5,2):
%p A105938 seq(a(n), n=0..40); # _Alois P. Heinz_, Oct 16 2008
%t A105938 Table[n(n+1)(n+3)(n+4)/4, {n, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 26 2011 *)
%t A105938 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 *)
%o A105938 (Magma)
%o A105938 A105938:= func< n | 30*Binomial(n+5,5)/(n+3) >;
%o A105938 [A105938(n): n in [0..40]]; // _G. C. Greubel_, Mar 11 2025
%o A105938 (SageMath)
%o A105938 def A105938(n): return 30*binomial(n+5,5)//(n+3)
%o A105938 print([A105938(n) for n in range(41)]) # _G. C. Greubel_, Mar 11 2025
%Y A105938 Subsequence of A085780.
%Y A105938 Cf. A000096, A000217, A000389, A062145.
%K A105938 nonn,easy
%O A105938 0,1
%A A105938 _Zerinvary Lajos_, Apr 27 2005