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%I #40 Oct 08 2022 16:39:33
%S 0,2,12,42,110,240,462,812,1332,2070,3080,4422,6162,8372,11130,14520,
%T 18632,23562,29412,36290,44310,53592,64262,76452,90300,105950,123552,
%U 143262,165242,189660,216690,246512,279312,315282,354620,397530,444222,494912,549822
%N a(n) = n*(n + 1)*(n^2 + n + 2)/4.
%C Number of integer sequences of length n+1 with sum zero and sum of absolute values 4. - _R. H. Hardin_, Feb 22 2009
%C Partial sums of A034262. - _Jeremy Gardiner_, Jun 23 2013
%H T. D. Noe, <a href="/A001621/b001621.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F Equals 2 * A002817 and (A058919(n-1) - 1)/2.
%F G.f.: (-2*x*(x^2+x+1))/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
%F a(n) = A000217(n) * A000124(n). - _Torlach Rush_, Aug 05 2018
%F E.g.f.: exp(x)*x*(8 + 16*x + 8*x^2 + x^3)/4. - _Stefano Spezia_, Oct 08 2022
%t a[n_]:=Sum[i+i^3, {i, 1, n}]; (* _Vladimir Joseph Stephan Orlovsky_, Dec 05 2008 *)
%t Array[# (# + 1) (#^2 + # + 2)/4 &, 39, 0] (* or *)
%t CoefficientList[Series[-2x (x^2 + x + 1)/(x - 1)^5, {x, 0, 38}], x] (* or *)
%t LinearRecurrence[{5, -10, 10, -5, 1}, {0, 2, 12, 42, 110}, 39] (* _Robert G. Wilson v_, Aug 05 2018 *)
%Y Cf. A002817, A058919, A092365.
%Y Cf. A000124, A000217, A034262.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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