A299198 a(n) = n^4/6 - 2*n^3/3 - n^2/6 + 5*n/3 + 1.

2, 1, 0, 5, 26, 77, 176, 345, 610, 1001, 1552, 2301, 3290, 4565, 6176, 8177, 10626, 13585, 17120, 21301, 26202, 31901, 38480, 46025, 54626, 64377, 75376, 87725, 101530, 116901, 133952, 152801, 173570, 196385, 221376, 248677, 278426, 310765, 345840, 383801, 424802, 469001

Offset: 1

Author

Juri-Stepan Gerasimov, Feb 04 2018

Keywords

• nonn
• easy

Examples

			For n=2, a(2) = 1^4/6 - 2*1^3/3 - 1^2/6 + 5*1/3 + 1 = 2.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A001263, A067998, A077415, A299120, A299146.

Programs

GAP

List([1..50], n -> n^4/6-2*n^3/3-n^2/6+5*n/3+1); # Muniru A Asiru, Feb 04 2018

Julia

[div((n-3)*(n+1)*(n^2-2*n-2),6) for n in 1:50] |> println # Bruno Berselli, Apr 11 2018

Magma

[n^4/6-2*n^3/3-n^2/6+5*n/3+1: n in [1..50]];

Maple

seq(n^4/6-2*n^3/3-n^2/6+5*n/3+1,n=1..50); # Muniru A Asiru, Feb 04 2018

Mathematica

f[n_] := n^4/6 - 2 n^3/3 - n^2/6 + 5 n/3 + 1; Array[f, 50] (* or *)
CoefficientList[ Series[(-2 + 9 x - 15 x^2 + 5 x^3 - x^4)/(-1 + x)^5, {x, 0, 50}], x] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {2, 1, 0, 5, 26}, 50] (* Robert G. Wilson v, Feb 09 2018 *)

PARI

Vec(x*(2 - 9*x + 15*x^2 - 5*x^3 + x^4) / (1 - x)^5 + O(x^50)) \\ Colin Barker, Feb 05 2018

Formula

a(n) = (n - 3)*(n + 1)*(n^2 - 2*n - 2)/6 = A299120(n-1) + A299120(1-n).

From Colin Barker, Feb 05 2018: (Start)

G.f.: x*(2 - 9*x + 15*x^2 - 5*x^3 + x^4) / (1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

(End)

E.g.f.: exp(x)*(6 + 6*x - 6*x^2 + 2*x^3 + x^4)/6. - Iain Fox, Feb 09 2018

6*a(n) = A067998(n)^2 - 5*A067998(n) + 6. - Bruno Berselli, Apr 11 2018