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.

1, 51, 381, 1451, 3951, 8801, 17151, 30381, 50101, 78151, 116601, 167751, 234131, 318501, 423851, 553401, 710601, 899131, 1122901, 1386051, 1692951, 2048201, 2456631, 2923301, 3453501, 4052751, 4726801, 5481631, 6323451, 7258701

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 1 + 5*n*(n+1)*(23*n^2 + 23*n + 14)/12.

a(n) = (1/Pi)*Integral_{x=0..Pi} (sin((n+1/2)*x)/sin(x/2))^5. - Yalcin Aktar, Dec 03 2011

G.f.: ( -1 - 46*x - 136*x^2 - 46*x^3 - x^4 ) / (x-1)^5. - R. J. Mathar, Dec 17 2011

a(n) = [x^(5*n)] (Sum_{k=0..2*n} x^k)^5. - Seiichi Manyama, Dec 14 2018

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1}, {1, 51, 381, 1451, 3951}, 30] (* Vincenzo Librandi, Dec 15 2018 *)

Prog

(PARI) a(n)=115/12*n^4+115/6*n^3+185/12*n^2+35/6*n+1
(PARI) {a(n) = polcoeff((sum(k=0, 2*n, x^k))^5, 5*n, x)} \\ Seiichi Manyama, Dec 14 2018
(Magma) [1+5*n*(n+1)*(23*n^2+23*n+14)/12: n in [0..30]]; // Vincenzo Librandi, Dec 15 2018

Crossrefs

Row 5 of A201552.

Cf. A003215, A063496.

Sequence in context: A165087 A359026 A152579 * A273189 A222910 A259692

Adjacent sequences: A083666 A083667 A083668 * A083670 A083671 A083672

Keyword

nonn,easy

Author

Benoit Cloitre, Jun 14 2003

Status

approved