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%I N0523 #58 Apr 21 2023 13:00:10
%S 1,2,5,9,15,23,34,47,64,84,108,136,169,206,249,297,351,411,478,551,
%T 632,720,816,920,1033,1154,1285,1425,1575,1735,1906,2087,2280,2484,
%U 2700,2928,3169,3422,3689,3969,4263,4571,4894,5231,5584,5952,6336,6736,7153,7586
%N Number of partitions of 2*n into at most 4 parts.
%C Bisection of A001400.
%C Molien series for 4-dimensional group of structure S_4 X C_2 and order 48, arising from complete weight enumerators of even trace-Hermitian self-dual additive codes over GF(4) containing the all-ones vector.
%C Partial sums of A156040. - _Bob Selcoe_, Feb 08 2014
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%H Vincenzo Librandi, <a href="/A014126/b014126.txt">Table of n, a(n) for n = 0..1000</a>
%H L. Colmenarejo, <a href="http://arxiv.org/abs/1604.00803">Combinatorics on several families of Kronecker coefficients related to plane partitions</a>, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
%H H. R. Henze and C. M. Blair, <a href="http://dx.doi.org/10.1021/ja01329a033">The number of structurally isomeric Hydrocarbons of the Ethylene Series</a>, J. Amer. Chem. Soc., 55 (2) (1933), 680-686.
%H H. R. Henze and C. M. Blair, <a href="/A000631/a000631.pdf">The number of structurally isomeric Hydrocarbons of the Ethylene Series</a>, J. Amer. Chem. Soc., 55 (2) (1933), 680-685. (Annotated scanned copy)
%H G. Nebe, E. M. Rains, and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,-1,0,2,-1).
%F G.f.: (1+x^2)/((1-x)^2*(1-x^2)*(1-x^3)). - _James A. Sellers_
%F a(n) = (1/72) * (4*n^3 + 30*n^2 + 72*n + 55 + 8*A049347(n) + 9*(-1)^n ). - _Ralf Stephan_, Aug 15 2013
%F E.g.f.: exp(-x)*(27 + 3*exp(2*x)*(55 + 106*x + 42*x^2 + 4*x^3) + 8*exp(x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)))/216. - _Stefano Spezia_, Apr 05 2023
%p with(combstruct): seq(count(Partition((2*n+4)), size=4), n=0..50); # _Zerinvary Lajos_, Mar 28 2008
%t CoefficientList[Series[(1 + x^2) / ((1 - x)^2 (1 - x^2) (1 - x^3)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Aug 15 2013 *)
%t LinearRecurrence[{2,0,-1,-1,0,2,-1},{1,2,5,9,15,23,34},50] (* _Harvey P. Dale_, Aug 31 2015 *)
%o (PARI) a(n)=(4*n^3+30*n^2+72*n+55+8*[1,-1,0][(n%3)+1]+9*(-1)^n)/72
%Y Cf. A000631, A001400, A014125, A049347, A092498, A156040.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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