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%I M1051 N0394 #70 Apr 13 2022 13:25:16
%S 1,2,4,7,12,18,27,38,53,71,94,121,155,194,241,295,359,431,515,609,717,
%T 837,973,1123,1292,1477,1683,1908,2157,2427,2724,3045,3396,3774,4185,
%U 4626,5104,5615,6166,6754,7386,8058,8778,9542,10358,11222,12142,13114
%N Expansion of 1 / ((1-x)^2*(1-x^2)*(1-x^3)*(1-x^4)).
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A002621/b002621.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H P. J. Cameron, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H E. Fix and J. L. Hodges, Jr., <a href="http://dx.doi.org/10.1214/aoms/1177728547">Significance probabilities of the Wilcoxon test</a>, Annals Math. Stat., 26 (1955), 301-312.
%H E. Fix and J. L. Hodges, <a href="/A000601/a000601.pdf">Significance probabilities of the Wilcoxon test</a>, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=199">Encyclopedia of Combinatorial Structures 199</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Thomas Wieder, The number of certain k-combinations of an n-set, <a href="http://www.math.nthu.edu.tw/~amen/">Applied Mathematics Electronic Notes</a>, vol. 8 (2008).
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (2, 0, -1, 0, -2, 2, 0, 1, 0, -2, 1).
%F a(n) = +2*a(n-1) - a(n-3) - 2*a(n-5) + 2*a(n-6) + a(n-8) - 2*a(n-10) + a(n-11).
%F a(n) = 83*n^2/288 +55*n/64 +2815/3456 +11*n^3/288 +n^4/576 +11*(-1)^n/128 +(-1)^n*n/64 + A057077(n)/16 +A061347(n)/27. - _R. J. Mathar_, Mar 15 2011
%F a(n)=floor((n+1)*(9*(-1)^n + n^3 + 21*n^2 + 145*n + 350)/576 + 1/2). - _Tani Akinari_, Nov 10 2012
%p A002621:=-1/(z**2+1)/(z**2+z+1)/(z+1)**2/(z-1)**5; # _Simon Plouffe_ in his 1992 dissertation
%p with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+2), right=Set(U, card<r), U=Sequence(Z, card>=1)}, unlabeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=4..51) ; # _Zerinvary Lajos_, Feb 07 2008
%p A057077 := proc(n) (-1)^floor(n/2) ; end proc:
%p A061347 := proc(n) op(1+(n mod 3),[1,1,-2]) ; end proc:
%p A002621 := proc(n) 83/288*n^2+55/64*n+2815/3456+11/288*n^3+1/576*n^4+11/128*(-1)^n+1/64*(-1)^n*n; %+ A057077(n)/16 +A061347(n)/27; end proc:
%p seq(A002621(n),n=0..10) ; # _R. J. Mathar_, Mar 15 2011
%t CoefficientList[Series[1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^4)),{x,0,60}],x] (* _Stefan Steinerberger_, Jun 10 2007 *)
%t LinearRecurrence[{2, 0, -1, 0, -2, 2, 0, 1, 0, -2, 1}, {1, 2, 4, 7, 12, 18, 27, 38, 53, 71, 94}, 80] (* _Vladimir Joseph Stephan Orlovsky_, Feb 23 2012 *)
%o (PARI) a(n)=(n+1)*(9*(-1)^n+n^3+21*n^2+145*n+350)\/576 \\ _Charles R Greathouse IV_, May 23 2013
%Y Partial sums of A001400. Column 4 of A092905.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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