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A064061 Eighth column of Catalan triangle A009766.

%I A064061 #35 Sep 28 2024 02:25:11
%S A064061 429,1430,3432,7072,13260,23256,38760,62016,95931,144210,211508,
%T A064061 303600,427570,592020,807300,1085760,1442025,1893294,2459664,3164480,
%U A064061 4034712,5101360,6399888,7970688,9859575,12118314,14805180,17985552,21732542,26127660,31261516
%N A064061 Eighth column of Catalan triangle A009766.
%H A064061 G. C. Greubel, <a href="/A064061/b064061.txt">Table of n, a(n) for n = 0..1000</a>
%H A064061 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 A064061 <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 A064061 a(n) = A009766(n+7, 7) = (n+1)*binomial(n+14, 6)/7.
%F A064061 G.f.: (429-2002*x+4004*x^2-4368*x^3+2730* x^4-924*x^5+132*x^6)/(1-x)^8; numerator polynomial is N(2;6, x) from A062991.
%F A064061 a(n) = C(n+13,7) - C(n+13,5). - _Zerinvary Lajos_, Nov 25 2006
%F A064061 a(n) = A214292(n+13,6). - _Reinhard Zumkeller_, Jul 12 2012
%F A064061 From _Amiram Eldar_, Sep 06 2022: (Start)
%F A064061 Sum_{n>=0} 1/a(n) = 323171/88339680.
%F A064061 Sum_{n>=0} (-1)^n/a(n) = 7929257917/88339680 - 55552*log(2)/429. (End)
%p A064061 [seq(binomial(n,7)-binomial(n,5),n=13..37)]; # _Zerinvary Lajos_, Nov 25 2006
%t A064061 CoefficientList[Series[(132*z^6 - 924*z^5 + 2730*z^4 - 4368*z^3 + 4004*z^2 - 2002*z + 429)/(z - 1)^8, {z, 0, 100}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jul 16 2011 *)
%t A064061 Table[Binomial[n,7]-Binomial[n,5],{n,13,50}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{429,1430,3432,7072,13260,23256,38760,62016},40] (* _Harvey P. Dale_, Sep 03 2015 *)
%o A064061 (Magma)
%o A064061 A064061:= func< n | (n+1)*Binomial(n+14, 6)/7 >;
%o A064061 [A064061(n): n in [0..40]]; // _G. C. Greubel_, Sep 28 2024
%o A064061 (SageMath)
%o A064061 def A064061(n): return (n+1)*binomial(n+14,6)//7
%o A064061 [A064061(n) for n in range(41)] # _G. C. Greubel_, Sep 28 2024
%Y A064061 Cf. A009766, A064059 (seventh column), A062991, A214292.
%K A064061 nonn,easy
%O A064061 0,1
%A A064061 _Wolfdieter Lang_, Sep 13 2001