A158416 Expansion of g.f. (1+x-x^3)/(1-x^2)^2.

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 1, 14, 1, 15, 1, 16, 1, 17, 1, 18, 1, 19, 1, 20, 1, 21, 1, 22, 1, 23, 1, 24, 1, 25, 1, 26, 1, 27, 1, 28, 1, 29, 1, 30, 1, 31, 1, 32, 1, 33, 1, 34, 1, 35, 1, 36, 1, 37, 1, 38, 1, 39, 1, 40, 1, 41, 1, 42, 1, 43, 1, 44, 1

Offset: 0

Author

Paul Barry, Mar 18 2009

Keywords

• nonn
• easy

Comments

Binomial transform is A111297. Binomial transform of [1,1,1,2,1,3,1,...] is A109975.

Essentially the same as A152271 and A133622. - R. J. Mathar, Mar 20 2009

Also defined by: a(0)=1; thereafter, a(n) = number of copies of a(n-1) in the list [a(0), a(1), ..., a(n-1)]. - N. J. A. Sloane, Feb 12 2016

Links

• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Crossrefs

Cf. A109975, A111297, A133622, A152271.

Related to A268696.

Programs

Mathematica

CoefficientList[Series[(1+x-x^3)/(1-x^2)^2,{x,0,100}],x] (* or *) LinearRecurrence[{0,2,0,-1},{1,1,2,1},100] (* Harvey P. Dale, Aug 17 2016 *)

PARI

a(n)=1+!(n%2)*n/2 \\ Jaume Oliver Lafont, Mar 21 2009

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(k+1,n-k).

G.f.: Q(0)/x - 1/x, where Q(k)= 1 + (k+1)*x/(1 - x/(x + (k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 23 2013

E.g.f.: cosh(x) + (2 + x)*sinh(x)/2. - Stefano Spezia, Sep 06 2023