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%I #30 Sep 08 2022 08:46:09
%S 1,15,120,680,3060,11628,38760,116280,316767,788161,1805100,3840420,
%T 7660250,14446134,25947612,44668692,74091645,118941555,185495056,
%U 281936688,418766304,609260960,869994720,1221419808,1688512539,2301487461,3096583140,4116923020
%N a(n) = Sum_{k=0..7} binomial(14,k)*binomial(n,k).
%H Vincenzo Librandi, <a href="/A247612/b247612.txt">Table of n, a(n) for n = 0..1000</a>
%H C. Krattenthaler, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s42kratt.html">Advanced determinant calculus</a> Séminaire Lotharingien de Combinatoire, B42q (1999), 67 pp, (see p. 54).
%H <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 G.f.: (1 + 7*x + 28*x^2 + 84*x^3 + 210*x^4 + 462*x^5 + 924*x^6 + 1716*x^7) / (1-x)^8.
%F a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
%F a(n) = (1680 + 386012*n - 958048*n^2 + 943761*n^3 - 455455*n^4 + 123123*n^5 - 17017*n^6 + 1144*n^7)/1680.
%t Table[(1680 + 386012 n - 958048 n^2 + 943761 n^3 - 455455 n^4 + 123123 n^5 - 17017 n^6 + 1144 n^7)/1680, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 7 x + 28 x^2 + 84 x^3 + 210 x^4 + 462 x^5 + 924 x^6 + 1716 x^7)/(1 - x)^8, {x, 0, 40}], x]
%t LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,15,120,680,3060,11628,38760,116280},30] (* _Harvey P. Dale_, May 12 2017 *)
%o (Magma) [(1680+386012*n-958048*n^2+943761*n^3-455455*n^4+123123*n^5- 17017*n^6+1144*n^7)/1680: n in [0..40]]; /* or */ I:=[1,15,120,680,3060,11628,38760, 116280]; [n le 8 select I[n] else 8*Self(n-1)-28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5) -28*Self(n-6)+8*Self(n-7)-Self(n-8): n in [1..40]];
%o (Sage) m=7; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # _Bruno Berselli_, Sep 22 2014
%Y Cf. A005408, A056108, A247608 - A247611
%K nonn,easy
%O 0,2
%A _Vincenzo Librandi_, Sep 22 2014
%E Definition edited by _Robert Israel_, Sep 22 2014
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