A267691 a(n) = (n + 1)*(6*n^4 - 21*n^3 + 31*n^2 - 31*n + 30)/30.

1, 1, 2, 18, 99, 355, 980, 2276, 4677, 8773, 15334, 25334, 39975, 60711, 89272, 127688, 178313, 243849, 327370, 432346, 562667, 722667, 917148, 1151404, 1431245, 1763021, 2153646, 2610622, 3142063, 3756719, 4464000, 5274000, 6197521, 7246097, 8432018, 9768354

Offset

0,3

Links

Table of n, a(n) for n=0..35.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Formula

G.f.: (1 - 5*x + 11*x^2 + x^3 + 16*x^4)/(1 - x)^6.

a(n + 1) = a(n) + n^4.

a(n + 1) = A000538(n) + 1.

a(n + 2) - a(n) = A008514(n).

Sum_{n>=0} 1/a(n) = 2.570450909491318975...

Sum_{n>=1} 1/(a(n + 1) - a(n)) = zeta(4) = Pi^4/90.

Example

a(0) = 1,
a(1) = 1 + 0^4 = 1,
a(2) = 1 + 1^4 = 2,
a(3) = 2 + 2^4 = 18,
a(4) = 18+ 3^4 = 99, etc.

Mathematica

Table[(n + 1) (6 n^4 - 21 n^3 + 31 n^2 - 31 n + 30)/30, {n, 0, 30}]
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 1, 2, 18, 99, 355}, 40] (* Vincenzo Librandi, Jan 20 2016 *)

Prog

(PARI) a(n)=(n+1)*(6*n^4-21*n^3+31*n^2-31*n+30)/30 \\ Charles R Greathouse IV, Jan 19 2016
(PARI) Vec((1-5*x+11*x^2+x^3+16*x^4)/(x-1)^6 + O(x^100)) \\ Altug Alkan, Jan 19 2016
(Magma) [(n+1)*(6*n^4-21*n^3+31*n^2-31*n+30)/30: n in [0..35]]; // Vincenzo Librandi, Jan 20 2016

Crossrefs

Essentially the same as A000538.

Cf. A000124, A000583, A008514, A056520, A154323, A263689.

Cf. A013662 (zeta(4)).

Sequence in context: A127553 A055357 A087291 * A219758 A005969 A094251

Adjacent sequences: A267688 A267689 A267690 * A267692 A267693 A267694

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Jan 19 2016

Status

approved