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%I M3360 #65 May 22 2024 12:56:13
%S 0,1,4,9,17,28,43,62,86,115,150,191,239,294,357,428,508,597,696,805,
%T 925,1056,1199,1354,1522,1703,1898,2107,2331,2570,2825,3096,3384,3689,
%U 4012,4353,4713,5092,5491,5910,6350,6811,7294,7799,8327,8878,9453,10052
%N Expansion of x*(1+x-x^2)/((1-x)^4*(1+x)).
%C Number of n-covers of a 2-set.
%C Boolean switching functions a(n,s) for s = 2.
%C Without the initial 0, this is row 1 of the convolution array A213778. - _Clark Kimberling_, Jun 21 2012
%C a(n) equals the second column of the triangle A355754. - _Eric W. Weisstein_, Mar 12 2024
%D R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A005744/b005744.txt">Table of n, a(n) for n = 0..1000</a>
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences </a>, J. Int. Seq. 13 (2010) # 10.7.8.
%H A. V. Jayanthan, S. A. Seyed Fakhari, I. Swanson, and S. Yassemi, <a href="https://arxiv.org/abs/2405.06781">Induced matching, ordered matching and Castelnuovo-Mumford regularity of bipartite graphs</a>, arXiv:2405.06781 [math.AC], 2024. See p. 17.
%H Vladeta Jovovic, <a href="/A005748/a005748.pdf">Binary matrices up to row and column permutations</a>.
%H <a href="/index/Bo#Boolean">Index entries for sequences related to Boolean functions</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F a(n) = A002623(n) - (n+1).
%F a(n) = n*(n-1)/2 + Sum_{j=1..floor((n+1)/2)} (n-2*j+1)*(n-2*j)/2. - _N. J. A. Sloane_, Nov 28 2003
%F From _R. J. Mathar_, Apr 01 2010: (Start)
%F a(n) = 5*n/12 - 1/16 + 5*n^2/8 + n^3/12 + (-1)^n/16.
%F a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). (End)
%F a(n) = A181971(n+1, n-1) for n > 0. - _Reinhard Zumkeller_, Jul 09 2012
%F a(n) + a(n+1) = A008778(n). - _R. J. Mathar_, Mar 13 2021
%F E.g.f.: (x*(2*x^2 + 21*x + 27)*cosh(x) + (2*x^3 + 21*x^2 + 27*x - 3)*sinh(x))/24. - _Stefano Spezia_, Jul 27 2022
%t CoefficientList[Series[x (1+x-x^2)/((1-x)^4(1+x)),{x,0,50}],x] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{0,1,4,9,17},50] (* _Harvey P. Dale_, Apr 10 2012 *)
%o (PARI) a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,3,-2,-2,3]^n*[0;1;4;9;17])[1,1] \\ _Charles R Greathouse IV_, Feb 06 2017
%Y _John W. Layman_ observes that A003453 appears to be the alternating sum transform (PSumSIGN) of A005744.
%Y Cf. A002623, A005745, A005746, A005747, A005748, A005771, A003180, A008778, A052265.
%Y Cf. A181971, A213778.
%Y Cf. A355754.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_, _Simon Plouffe_
%E Additional comments from _Alford Arnold_
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