cp's OEIS Frontend

A247613 a(n) = Sum_{k=0..8} binomial(16,k)*binomial(n,k).

%I A247613 #27 Jan 14 2025 11:05:26
%S A247613 1,17,153,969,4845,20349,74613,245157,735471,2031535,5189327,12316239,
%T A247613 27322191,57029103,112740255,212383935,383358645,666220005,1119362365,
%U A247613 1824861005,2895653673,4484253081,6793194849,10087438257,14708950035,21093714291
%N A247613 a(n) = Sum_{k=0..8} binomial(16,k)*binomial(n,k).
%H A247613 Vincenzo Librandi, <a href="/A247613/b247613.txt">Table of n, a(n) for n = 0..1000</a>
%H A247613 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 A247613 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A247613 G.f.: (1 + 8*x + 36*x^2 + 120*x^3 + 330*x^4 + 792*x^5 + 1716*x^6 + 3432*x^7  + 6435*x^8) / (1-x)^9.
%F A247613 a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9).
%F A247613 a(n) = (20160 - 15076944*n + 40499716*n^2 - 42247940*n^3 + 23174515*n^4 - 7234136*n^5 + 1335334*n^6 - 134420*n^7 + 6435*n^8) / 20160.
%t A247613 Table[(20160 - 15076944 n + 40499716 n^2 - 42247940 n^3 + 23174515 n^4 - 7234136 n^5 + 1335334 n^6 - 134420 n^7 + 6435 n^8)/20160, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 8 x + 36 x^2 + 120 x^3 + 330 x^4 + 792 x^5 + 1716 x^6 + 3432 x^7 + 6435 x^8)/(1 - x)^9, {x, 0, 40}], x]
%t A247613 Table[Sum[Binomial[16,k]Binomial[n,k],{k,0,8}],{n,0,30}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,17,153,969,4845,20349,74613,245157,735471},40] (* _Harvey P. Dale_, Mar 25 2015 *)
%o A247613 (Magma) m:=8; [&+[Binomial(2*m,k)*Binomial(n,k): k in [0..m]]: n in [0..40]];
%o A247613 (Magma) [(20160-15076944*n+40499716*n^2-42247940*n^3 +23174515*n^4-7234136*n^5+1335334*n^6-134420*n^7 +6435*n^8)/20160: n in [0..40]];
%o A247613 (Sage) m=8; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # _Bruno Berselli_, Sep 23 2014
%Y A247613 Cf. A005408, A056108, A247608 - A247612.
%K A247613 nonn,easy
%O A247613 0,2
%A A247613 _Vincenzo Librandi_, Sep 23 2014