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

0,2

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).

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A026243 A058139 A101476 * A292744 A151239 A151240

Adjacent sequences: A060798 A060799 A060800 * A060802 A060803 A060804

Keyword

nonn,easy

Author

Vladeta Jovovic, Apr 27 2001

Status

approved