A060801 Invert transform of odd numbers: a(n) = Sum_{k=1..n} (2*k+1)*a(n-k), a(0)=1.

1, 3, 14, 64, 292, 1332, 6076, 27716, 126428, 576708, 2630684, 12000004, 54738652, 249693252, 1138988956, 5195558276, 23699813468, 108107950788, 493140127004, 2249484733444, 10261143413212, 46806747599172, 213511451169436

Offset: 0

Author

Vladeta Jovovic, Apr 27 2001

Keywords

• nonn
• easy

Comments

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

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017.
• Stéphane Ouvry and Alexios P. Polychronakos, Signed area enumeration for lattice walks, Séminaire Lotharingien de Combinatoire (2023) Vol. 87B.
• N. J. A. Sloane, Transforms
• Index entries for linear recurrences with constant coefficients, signature (5,-2).

Crossrefs

Cf. A001906, A052530, A033453, A030017, A052913 (partial sums).

Programs

Mathematica

Join[{1}, LinearRecurrence[{5, -2}, {3, 14}, 22]] (* Jean-François Alcover, Aug 07 2018 *)

PARI

Vec((1 - x)^2 / (1 - 5*x + 2*x^2) + O(x^25)) \\ Colin Barker, Mar 19 2019

Formula

G.f.: (x^2-2*x+1)/(2*x^2-5*x+1).

G.f.: 1 / (1 - 3*x - 5*x^2 - 7*x^3 - 9*x^4 - 11*x^5 - ...). - Gary W. Adamson, Jul 27 2009

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

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

a(n) = A052913(n)-A052913(n-1). - R. J. Mathar, Sep 20 2020