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A064059 Seventh column of Catalan triangle A009766.

%I A064059 #32 Jan 08 2025 18:40:53
%S A064059 132,429,1001,2002,3640,6188,9996,15504,23256,33915,48279,67298,92092,
%T A064059 123970,164450,215280,278460,356265,451269,566370,704816,870232,
%U A064059 1066648,1298528,1570800,1888887,2258739,2686866,3180372,3746990,4395118,5133856,5973044
%N A064059 Seventh column of Catalan triangle A009766.
%H A064059 G. C. Greubel, <a href="/A064059/b064059.txt">Table of n, a(n) for n = 0..1000</a>
%H A064059 Richard K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
%H A064059 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A064059 G.f.: (132-495*x+770*x^2-616*x^3+252*x^4-42*x^5)/(1-x)^7; numerator polynomial is N(2;5, x) from A062991.
%F A064059 a(n) = A009766(n+6, 6) = (n+1)*binomial(n+12,5)/6.
%F A064059 a(n) = binomial(n+13,6) - 2*binomial(n+12,5). - _Zerinvary Lajos_, Jul 19 2006
%F A064059 a(n) = A214292(n+11,5). - _Reinhard Zumkeller_, Jul 12 2012
%F A064059 From _Amiram Eldar_, Sep 06 2022: (Start)
%F A064059 Sum_{n>=0} 1/a(n) = 25961/2134440.
%F A064059 Sum_{n>=0} (-1)^n/a(n) = 4160*log(2)/77 - 79917773/2134440. (End)
%p A064059 [seq(binomial(n+1,6)-2*binomial(n,5),n=12..55)]; # _Zerinvary Lajos_, Jul 19 2006
%t A064059 CoefficientList[Series[(42 z^5-252 z^4+616 z^3-770 z^2+495 z-132)/(z-1)^7, {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 22 2011 *)
%t A064059 LinearRecurrence[{7,-21,35,-35,21,-7,1},{132,429,1001,2002,3640,6188,9996},40] (* _Harvey P. Dale_, Jan 08 2025 *)
%o A064059 (Magma)
%o A064059 A064059:= func< n | (n+1)*Binomial(n+12,5)/6 >;
%o A064059 [A064059(n): n in [0..40]]; // _G. C. Greubel_, Sep 27 2024
%o A064059 (SageMath)
%o A064059 def A064059(n): return (n+1)*binomial(n+12,5)//6
%o A064059 [A064059(n) for n in range(41)] # _G. C. Greubel_, Sep 27 2024
%Y A064059 Cf. A000096, A005586, A005587, A005557 (third to sixth column).
%K A064059 nonn,easy
%O A064059 0,1
%A A064059 _Wolfdieter Lang_, Sep 13 2001