%I #28 Sep 08 2022 08:45:11
%S 1,2188,282124,10282124,181141499,1982230040,15475158552,93839322648,
%T 467508775773,1989944010148,7445104711204,25010673566116,
%U 76686775501847,217396817767472,575714897767472,1436257466526768,3398894618986905,7674255436599996,16612972826599996
%N a(n) = Sum_{i=1..n} binomial(i+1,2)^7.
%H T. D. Noe, <a href="/A085442/b085442.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).
%F a(n) = (1/823680) *n*(n+1)*(n+2)*(429*n^12 +5148*n^11 +24123*n^10 +52470*n^9 +43047*n^8 -8856*n^7 +4109*n^6 +50430*n^5 -18796*n^4 -44472*n^3 +26864*n^2 +8352*n -5568). - _Vladeta Jovovic_, Jul 07 2003
%F G.f.: x*(x^12 +2172*x^11 +247236*x^10 +6030140*x^9 +49258935*x^8 +163809288*x^7 +242384856*x^6 +163809288*x^5 +49258935*x^4 +6030140*x^3 +247236*x^2 +2172*x+ 1) / (x -1)^16. - _Colin Barker_, May 02 2014
%t Table[Sum[Binomial[k+1,2]^7, {k,1,n}], {n,1,30}] (* _G. C. Greubel_, Nov 22 2017 *)
%t LinearRecurrence[{16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1},{1,2188,282124,10282124,181141499,1982230040,15475158552,93839322648,467508775773,1989944010148,7445104711204,25010673566116,76686775501847,217396817767472,575714897767472,1436257466526768},20] (* _Harvey P. Dale_, May 11 2022 *)
%o (PARI) for(n=1,30, print1(sum(k=1,n, binomial(k+1,2)^7), ", ")) \\ _G. C. Greubel_, Nov 22 2017
%o (Magma) [(1/823680) *n*(n+1)*(n+2)*(429*n^12 +5148*n^11 +24123*n^10 +52470*n^9 +43047*n^8 -8856*n^7 +4109*n^6 +50430*n^5 -18796*n^4 -44472*n^3 +26864*n^2 +8352*n -5568): n in [1..30]]; // _G. C. Greubel_, Nov 22 2017
%Y Column k=7 of A334781.
%Y Cf. A000292, A087127, A024166, A024166, A085438, A085439, A085440, A085441, A000332, A086020, A086021, A086022, A000389, A086023, A086024, A000579, A086025, A086026, A000580, A086027, A086028, A027555, A086029, A086030.
%K easy,nonn
%O 1,2
%A _André F. Labossière_, Jul 07 2003