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%I #37 Aug 21 2023 12:44:32
%S 1,3,14,64,292,1332,6076,27716,126428,576708,2630684,12000004,
%T 54738652,249693252,1138988956,5195558276,23699813468,108107950788,
%U 493140127004,2249484733444,10261143413212,46806747599172,213511451169436
%N Invert transform of odd numbers: a(n) = Sum_{k=1..n} (2*k+1)*a(n-k), a(0)=1.
%C a(n) is the number of generalized compositions of n when there are 2*i+1 different types of i, (i=1,2,...). - _Milan Janjic_, Sep 24 2010
%H Colin Barker, <a href="/A060801/b060801.txt">Table of n, a(n) for n = 0..1000</a>
%H Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, <a href="https://arxiv.org/abs/1707.07798">(an + b)-color compositions</a>, arXiv:1707.07798 [math.CO], 2017.
%H Stéphane Ouvry and Alexios P. Polychronakos, <a href="https://par.nsf.gov/servlets/purl/10440181">Signed area enumeration for lattice walks</a>, Séminaire Lotharingien de Combinatoire (2023) Vol. 87B.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-2).
%F G.f.: (x^2-2*x+1)/(2*x^2-5*x+1).
%F G.f.: 1 / (1 - 3*x - 5*x^2 - 7*x^3 - 9*x^4 - 11*x^5 - ...). - _Gary W. Adamson_, Jul 27 2009
%F a(n) = 5*a(n-1) - 2*a(n-2) with a(1) = 3, a(2) = 14, for n >= 3. - _Taras Goy_, Mar 19 2019
%F a(n) = (2^(-2-n)*((5-sqrt(17))^n*(-7+sqrt(17)) + (5+sqrt(17))^n*(7+sqrt(17)))) / sqrt(17) for n > 0. - _Colin Barker_, Mar 19 2019
%F a(n) = A052913(n)-A052913(n-1). - _R. J. Mathar_, Sep 20 2020
%t Join[{1}, LinearRecurrence[{5, -2}, {3, 14}, 22]] (* _Jean-François Alcover_, Aug 07 2018 *)
%o (PARI) Vec((1 - x)^2 / (1 - 5*x + 2*x^2) + O(x^25)) \\ _Colin Barker_, Mar 19 2019
%Y Cf. A001906, A052530, A033453, A030017, A052913 (partial sums).
%K nonn,easy
%O 0,2
%A _Vladeta Jovovic_, Apr 27 2001
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