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%I #29 Sep 08 2022 08:45:11
%S 1,26,251,1476,6376,22252,66352,175252,420277,931502,1933503,3796728,
%T 7109128,12773528,22137128,37160504,60634529,96454754,149963979,
%U 228375004,341286880,501309380,724811880,1032814380,1452040005,2016150006,2767184031,3757230256
%N a(n) = Sum_{i=1..n} C(i+3,4)^2.
%H G. C. Greubel, <a href="/A086023/b086023.txt">Table of n, a(n) for n = 1..5000</a>
%H John Engbers and Christopher Stocker, <a href="http://epublications.marquette.edu/mscs_fac/456/">Two Combinatorial Proofs of Identities Involving Sums of Powers of Binomial Coefficients</a>, Integers 16 (2016), #A58.
%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 a(n) = ( C(n+4,5)/126 )*( 126 +420*C(n-1,1) +540*C(n-1,2) +315*C(n-1,3) +70*C(n-1,4) ).
%F a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(35*n^4 +280*n^3 +685*n^2 +500*n +12 )/181440. - _Bruno Berselli_, Sep 05 2013
%F G.f.: x*(x^4+16*x^3+36*x^2+16*x+1) / (x-1)^10. - _Colin Barker_, May 02 2014
%t Table[n*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(35*n^4 + 280*n^3 + 685*n^2 + 500*n + 12)/181440, {n, 1, 50}] (* _G. C. Greubel_, Nov 22 2017 *)
%t Accumulate[Binomial[Range[30]+3,4]^2] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,26,251,1476,6376,22252,66352,175252,420277,931502},30] (* _Harvey P. Dale_, May 06 2018 *)
%o (PARI) a(n) = sum(i=1, n, binomial(i+3, 4)^2); \\ _Michel Marcus_, Sep 05 2013
%o (PARI) Vec(x*(x^4+16*x^3+36*x^2+16*x+1)/(x-1)^10 + O(x^100)) \\ _Colin Barker_, May 02 2014
%o (Magma) [n*(n+1)*(n+2)*(n+3)*(n+4)*(35*n^4 +280*n^3 +685*n^2 +500*n +12 )/181440: n in [1..30]]; // _G. C. Greubel_, Nov 22 2017
%Y Cf. A087127, A024166, A085438, A085439, A085440, A085441, A085442, A086020, A086021, A086022, A086024, A086025, A086026, A086027, A086028, A086029, A086030.
%K nonn,easy
%O 1,2
%A _André F. Labossière_, Jul 11 2003
%E More terms from _Michel Marcus_, Sep 05 2013
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