A001621 a(n) = n*(n + 1)*(n^2 + n + 2)/4.

0, 2, 12, 42, 110, 240, 462, 812, 1332, 2070, 3080, 4422, 6162, 8372, 11130, 14520, 18632, 23562, 29412, 36290, 44310, 53592, 64262, 76452, 90300, 105950, 123552, 143262, 165242, 189660, 216690, 246512, 279312, 315282, 354620, 397530, 444222, 494912, 549822

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of integer sequences of length n+1 with sum zero and sum of absolute values 4. - R. H. Hardin, Feb 22 2009

Partial sums of A034262. - Jeremy Gardiner, Jun 23 2013

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A002817, A058919, A092365.

Cf. A000124, A000217, A034262.

Programs

Mathematica

a[n_]:=Sum[i+i^3, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)
Array[# (# + 1) (#^2 + # + 2)/4 &, 39, 0] (* or *)
CoefficientList[Series[-2x (x^2 + x + 1)/(x - 1)^5, {x, 0, 38}], x] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 2, 12, 42, 110}, 39] (* Robert G. Wilson v, Aug 05 2018 *)

Formula

Equals 2 * A002817 and (A058919(n-1) - 1)/2.

G.f.: (-2*x*(x^2+x+1))/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

a(n) = A000217(n) * A000124(n). - Torlach Rush, Aug 05 2018

E.g.f.: exp(x)*x*(8 + 16*x + 8*x^2 + x^3)/4. - Stefano Spezia, Oct 08 2022