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A247609 a(n) = Sum_{k=0..4} binomial(8,k)*binomial(n,k).

%I A247609 #29 Jan 26 2025 20:27:54
%S A247609 1,9,45,165,495,1231,2639,5055,8885,14605,22761,33969,48915,68355,
%T A247609 93115,124091,162249,208625,264325,330525,408471,499479,604935,726295,
%U A247609 865085,1022901,1201409,1402345,1627515,1878795,2158131,2467539,2809105,3184985,3597405
%N A247609 a(n) = Sum_{k=0..4} binomial(8,k)*binomial(n,k).
%H A247609 Vincenzo Librandi, <a href="/A247609/b247609.txt">Table of n, a(n) for n = 0..1000</a>
%H A247609 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 A247609 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A247609 G.f.: (1 + 4*x + 10*x^2 + 20*x^3 + 35*x^4)/(1 - x)^5.
%F A247609 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F A247609 a(n) = (12 - 58*n + 217*n^2 - 98*n^3 + 35*n^4)/12.
%F A247609 a(n) = 1 + 8*Binomial(n, 1) + 28*Binomial(n, 2) + 56*Binomial(n, 3) + 70*Binomial(n, 4).
%t A247609 Table[(12 - 58 n + 217 n^2 - 98 n^3 + 35 n^4)/12, {n, 0, 50}] (* or *) CoefficientList[Series[(1 + 4 x + 10 x^2 + 20 x^3 + 35 x^4)/(1 - x)^5, {x, 0, 50}], x]
%t A247609 LinearRecurrence[{5,-10,10,-5,1},{1,9,45,165,495},40] (* _Harvey P. Dale_, Oct 19 2024 *)
%o A247609 (Magma) [(12-58*n+217*n^2-98*n^3+35*n^4)/12: n in [0..40]];
%o A247609 (Magma) [1+8*Binomial(n, 1)+28*Binomial(n, 2)+56*Binomial(n, 3)+70*Binomial(n,4): n in [0..40]];
%o A247609 (Magma) I:=[1,9,45,165,495]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]];
%o A247609 (Sage) m=4; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # _Bruno Berselli_, Sep 22 2014
%Y A247609 Cf. A005408, A056108, A247608.
%K A247609 nonn,easy
%O A247609 0,2
%A A247609 _Vincenzo Librandi_, Sep 22 2014