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A083669 Number of ordered quintuples (a,b,c,d,e), -n <= a,b,c,d,e <= n, such that a+b+c+d+e = 0.

%I A083669 #30 Sep 08 2022 08:45:10
%S A083669 1,51,381,1451,3951,8801,17151,30381,50101,78151,116601,167751,234131,
%T A083669 318501,423851,553401,710601,899131,1122901,1386051,1692951,2048201,
%U A083669 2456631,2923301,3453501,4052751,4726801,5481631,6323451,7258701
%N A083669 Number of ordered quintuples (a,b,c,d,e), -n <= a,b,c,d,e <= n, such that a+b+c+d+e = 0.
%H A083669 Seiichi Manyama, <a href="/A083669/b083669.txt">Table of n, a(n) for n = 0..10000</a>
%H A083669 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A083669 a(n) = 1 + 5*n*(n+1)*(23*n^2 + 23*n + 14)/12.
%F A083669 a(n) = (1/Pi)*Integral_{x=0..Pi} (sin((n+1/2)*x)/sin(x/2))^5. - _Yalcin Aktar_, Dec 03 2011
%F A083669 G.f.: ( -1 - 46*x - 136*x^2 - 46*x^3 - x^4 ) / (x-1)^5. - _R. J. Mathar_, Dec 17 2011
%F A083669 a(n) = [x^(5*n)] (Sum_{k=0..2*n} x^k)^5. - _Seiichi Manyama_, Dec 14 2018
%t A083669 LinearRecurrence[{5, -10, 10, -5, 1}, {1, 51, 381, 1451, 3951}, 30] (* _Vincenzo Librandi_, Dec 15 2018 *)
%o A083669 (PARI) a(n)=115/12*n^4+115/6*n^3+185/12*n^2+35/6*n+1
%o A083669 (PARI) {a(n) = polcoeff((sum(k=0, 2*n, x^k))^5, 5*n, x)} \\ _Seiichi Manyama_, Dec 14 2018
%o A083669 (Magma) [1+5*n*(n+1)*(23*n^2+23*n+14)/12: n in [0..30]]; // _Vincenzo Librandi_, Dec 15 2018
%Y A083669 Row 5 of A201552.
%Y A083669 Cf. A003215, A063496.
%K A083669 nonn,easy
%O A083669 0,2
%A A083669 _Benoit Cloitre_, Jun 14 2003