A247614 - OEIS

%I #23 Jan 14 2025 11:05:33

%S 1,19,190,1330,7315,33649,134596,480700,1562275,4686825,13079352,

%T 34084128,83204745,191006115,414237570,852920310,1675575165,

%U 3155247975,5719519850,10018268150,17013571223,28096825757,45238870040,71179679480,109665022415

%N a(n) = Sum_{k=0..9} binomial(18,k)*binomial(n,k).

%H Vincenzo Librandi, <a href="/A247614/b247614.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_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

%F G.f.: (1 +9*x +45*x^2 +165*x^3 +495*x^4 +1287*x^5 +3003*x^6 + 6435*x^7 +12870*x^8 +24310*x^9) / (1-x)^10.

%F a(n) = 10*a(n-1) -45*a(n-2) +120*a(n-3) -210*a(n-4) +252*a(n-5) -210*a(n-6) +120*a(n-7) -45*a(n-8) +10*a(n-9) -a(n-10).

%F a(n) = (181440 + 462101904*n - 1283316876*n^2 + 1433031524*n^3 - 853620201*n^4 + 303063726*n^5 - 66245634*n^6+8905416*n^7 - 678249*n^8 + 24310*n^9) / 181440.

%t Table[(181440 + 462101904 n - 1283316876 n^2 + 1433031524 n^3 - 853620201 n^4 + 303063726 n^5 - 66245634 n^6 + 8905416 n^7 - 678249 n^8 + 24310 n^9)/181440, {n, 0, 40}] (* or *) CoefficientList[Series[(1 + 9 x + 45 x^2 + 165 x^3 + 495 x^4 + 1287 x^5 + 3003 x^6 + 6435 x^7 + 12870 x^8 + 24310 x^9)/(1 - x)^10, {x, 0, 40}], x]

%t LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,19,190,1330,7315,33649,134596,480700,1562275,4686825},30] (* _Harvey P. Dale_, Jul 19 2019 *)

%o (Magma) m:=9; [&+[Binomial(2*m,k)*Binomial(n,k): k in [0..m]]: n in [0..40]];

%o (Magma) [(181440+462101904*n-1283316876*n^2+1433031524*n^3 -853620201*n^4+303063726*n^5-66245634*n^6 +8905416*n^7-678249*n^8+24310*n^9)/181440: n in [0..40]];

%o (Sage) m=9; [sum((binomial(2*m,k)*binomial(n,k)) for k in (0..m)) for n in (0..40)] # _Bruno Berselli_, Sep 23 2014

%Y Cf. A005408, A056108, A247608 - A247612.

%K nonn,easy

%O 0,2

%A _Vincenzo Librandi_, Sep 23 2014