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A247610 a(n) = Sum_{k=0..5} binomial(10,k)*binomial(n,k).

%I A247610 #28 Jan 26 2025 19:54:53
%S A247610 1,11,66,286,1001,3003,7798,17858,36873,70003,124130,208110,333025,
%T A247610 512435,762630,1102882,1555697,2147067,2906722,3868382,5070009,
%U A247610 6554059,8367734,10563234,13198009,16335011,20042946,24396526,29476721,35371011,42173638,49985858
%N A247610 a(n) = Sum_{k=0..5} binomial(10,k)*binomial(n,k).
%H A247610 Vincenzo Librandi, <a href="/A247610/b247610.txt">Table of n, a(n) for n = 0..1000</a>
%H A247610 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 A247610 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A247610 G.f.: (1 + 5*x + 15*x^2 + 35*x^3 + 70*x^4 + 126*x^5) / (1-x)^6.
%F A247610 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
%F A247610 a(n) = (20 + 508*n - 925*n^2 + 820*n^3 - 245*n^4 + 42*n^5)/20.
%t A247610 Table[(20 + 508 n - 925 n^2 + 820 n^3 - 245 n^4 + 42 n^5)/20, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 5 x + 15 x^2 + 35 x^3 + 70 x^4 + 126 x^5)/(1 - x)^6, {x, 0, 40}], x]
%t A247610 LinearRecurrence[{6,-15,20,-15,6,-1},{1,11,66,286,1001,3003},40] (* _Harvey P. Dale_, Apr 20 2022 *)
%o A247610 (Magma) [(20+508*n-925*n^2+820*n^3-245*n^4+42*n^5)/20: n in [0..40]];
%o A247610 (Magma) I:=[1,11,66,286,1001,3003]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..40]];
%o A247610 (Sage) m=5; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # _Bruno Berselli_, Sep 22 2014
%Y A247610 Cf. A005408, A056108, A247608.
%K A247610 nonn,easy
%O A247610 0,2
%A A247610 _Vincenzo Librandi_, Sep 22 2014