A247615 - OEIS

%I #26 Sep 08 2022 08:46:09

%S 1,21,231,1771,10626,53130,230230,888030,3108105,10015005,30045015,

%T 84504355,223651350,558350430,1318250890,2952624906,6296642121,

%U 12834146941,25098124271,47262174531,85990654178,151631858378,259857912678,433877085278,707369215553

%N a(n) = Sum_{k=0..10} binomial(20,k)*binomial(n,k).

%H Vincenzo Librandi, <a href="/A247615/b247615.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_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

%F G.f.: (1 + 10*x + 55*x^2 + 220*x^3 + 715*x^4 + 2002*x^5 + 5005*x^6 + 11440*x^7 + 24310*x^8 + 48620*x^9 + 92378*x^10) / (1-x)^11.

%F a(n) = 11*a(n-1) -55*a(n-2) +165*a(n-3) -330*a(n-4) +462*a(n-5) -462*a(n-6)+330*a(n-7) -165*a(n-8) +55*a(n-9) -11*a(n-10) +a(n-11).

%F a(n) = 1 - 5512999*n/630 + 212329883*n^2/8400 - 134689309*n^3/4536 + 3453077689*n^4/181440 - 64212077*n^5/8640 + 80300707*n^6/43200 - 1817521*n^7/6048 + 1860157*n^8/60480 - 331721*n^9/181440 + 46189*n^10/907200.

%t CoefficientList[Series[(1 + 10 x + 55 x^2 + 220 x^3 + 715 x^4 + 2002 x^5 + 5005 x^6 + 11440 x^7 + 24310 x^8 + 48620 x^9 + 92378 x^10)/(1 - x)^11, {x, 0, 40}], x]

%t Table[Sum[Binomial[20,k]Binomial[n,k],{k,0,10}],{n,0,30}] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,21,231,1771,10626,53130,230230,888030,3108105,10015005,30045015},30] (* _Harvey P. Dale_, May 19 2015 *)

%o (Magma) m:=10; [&+[Binomial(2*m,k)*Binomial(n,k): k in [0..m]]: n in [0..40]]; /* or */ I:=[1, 21, 231, 1771, 10626, 53130, 230230, 888030, 3108105, 10015005, 30045015]; [n le 11 select I[n] else 11*Self(n-1) -55*Self(n-2) +165*Self(n-3) -330*Self(n-4) +462*Self(n-5) -462*Self(n-6) +330*Self(n-7) -165*Self(n-8) +55*Self(n-9) -11*Self(n-10) +Self(n-11): n in [1..40]];

%o (Sage) m=10; [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, A247609, A247610, A247611, A247612.

%K nonn,easy

%O 0,2

%A _Vincenzo Librandi_, Sep 23 2014