610b700e-0cf1-4a6d-ada1-d5642f4f5c0f 75 4 5 false false 0 decimal

&[DATE] &[TIME] - &[FILENAME]

The genfunc package contains commands for manipulating rational generating functions.

function finds the rational generating function of the sequence defined by Fn.•      The expression Fn must be a valid closed form expression for a sequence with a rational generating function. genfunc[rgf_encode](2^n,n,z) parse maple 112 z * - /

genfunc[rgf_encode](2^k, k, y, k=2) parse maple 4 y 2^* 12 y * - /

genfunc[rgf_encode](2^n, n, z, [0=2, 1=0]) parse maple 12 z * - 1 + 2 z * 12 z * - (* - 12 z * - /

'loosely' Verify

is same as:

z g 12 z * - 1 + 2 z * 12 z * - (* - 12 z * - / :

z f 112 z * - / 1 + 2 z * - (:

105 i * + g 19.0412 - 9.9783 i * -

105 i * + f 19.0412 - 9.9783 i * -

They don't look the same but using a complex number, get the same result...

maple function must be the full name 'rgf_encode', 'encode' does not work.

Gy (1+y^2)/(1-y-y^2) parse maple : 1 y 2^+ 1 y 1 y + (* - /

• This command determines information about the sequence encoded by the generating function Fz. All queries assume that the first nonzero term is the first term in the sequence.
• The possible values for query are: 'boundary' , 'coeffs' ,'delta' , 'first' ,'firstcf' , 'firstrecur' , 'order' , 'recur'.

genfunc[rgf_sequence]('boundary',(1+y^2)/(1-y-y^2),y) parse maple 01 = 11 = 23 = 31 mat

genfunc[rgf_sequence]('boundary',(1+y^2)/(1-y-y^2),y,t) parse maple 0 t 1 = 1 t 1 = 2 t 3 = 31 mat

genfunc[rgf_sequence]('coeffs',(1+y^2)/(1-y-y^2),y) parse maple 1121 mat

genfunc[rgf_sequence]('coeffs',(1+y^2)/(1-y-y^2),y,t) parse maple 1 t 1 = 2 t 1 = 21 mat

genfunc[rgf_sequence]('coeffs',(1+y^2)/(1-y-y^2),y,a) parse maple 1 a 1 = 2 a 1 = 21 mat

genfunc[rgf_sequence]('delta',(1+y^2)/(1-y-y^2),y) parse maple 01 - =

genfunc[rgf_sequence]('first',(1+y^2)/(1-y-y^2),y) parse maple 0

genfunc[rgf_sequence]('firstcf',(1+y^2)/(1-y-y^2),y) parse maple 1

genfunc[rgf_sequence]('firstrecur',(1+y^2)/(1-y-y^2),y) parse maple 3

genfunc[rgf_sequence]('order',(1+y^2)/(1-y-y^2),y) parse maple 2

genfunc[rgf_sequence]('recur',(1+y^2)/(1-y-y^2), y, t, k) parse maple k t 1 - k + t 2 - k + t + =

Fz z/(1-z-z^2) parse maple : z 1 z 1 z + (* - /

ex n^2*F(n+1)+(n^2+1)*F(n)+(n^2-2*n)*F(n-1)+(n+3)^2*F(n-2) parse maple : n n 1 n + F * 2 - n + (1 - n + F * + (* 1 n 2^+ (n F * + 3 n + (2^2 - n + F * +

genfunc[rgf\005F\simp]\0028\ex\002C\\0020\Fz\002C\\0020\z\002C\\0020\F\002C\\0020\n\0029\ (maple FAIL

Don't know how to pass variables into maple functions yet...

function simplifies occurrences of sequence F in expr. If F is defined by an order k recurrence then all occurrences of F whose index differs from n by an integer amount are simplified to expressions involving F(n),...,F(n−k+1). genfunc[rgf_simp](n^2*F(n+1)+(n^2+1)*F(n)+(n^2-2*n)*F(n-1)+(n+3)^2*F(n-2), z/(1-z-z^2), z, F, n) parse maple 253 n * + (* 3 n 2^* + (n F * 9 - n 8 - n + (* + (1 - n + F * +

genfunc[rgf_simp](n^2*F(n+1)+(n^2+1)*F(n)+(n^2-2*n)*F(n-1)+(n+3)^2*F(n-2), z/(1-z-z^2), z, F, n,n-1) parse maple 12 n * 1 - 2 n * + (* + (1 - n + F * 253 n * + (* 3 n 2^* + (2 - n + F * +

genfunc[rgf_simp](n^2*F(n+1)+(n^2+1)*F(n)+(n^2-2*n)*F(n-1)+(n+3)^2*F(n-2), z/(1-z-z^2), z, F, n,n+1) parse maple 9 - n 8 - n + (* + (1 n + F * 192 n * 7 n + (* + (n F * +

genfunc[rgf_simp](n^2*F(n+1)+(n^2+1)*F(n)+(n^2-2*n)*F(n-1)+(n+3)^2*F(n-2), z/(1-z-z^2), z, F, n,n+1,factor) parse maple 1 n + (9 - n + (* 1 n + F * 192 n * 7 n + (* + (n F * +

genfunc[rgf_simp](F(3*n-2)+F(3*n-11),(1-z+z^2)/(1-4*z-z^2),z,F(3*n-2),n) parse maple 32 - 3 n * + F * - 175 - 3 n * + F * +

genfunc[rgf_simp](F(3*n-2)+F(3*n-11),(1-z+z^2)/(1-4*z-z^2),z,F(3*n-2),n,n+1) parse maple 712 - 3 n * + F * - 1713 n * + F * +

genfunc[rgf_simp](F(3*km-11),(1-z+z^2)/(1-4*z-z^2),z,F(3*n-2),n,km) parse maple 175 - 3 km * + F * 42 - 3 km * + F * -

function shifts and scales fn to determine the sequence encoded by the product of pz and the generating function of fn.termscale(pz, z, fn, n)   pz: polynomial in zz:  name, polynomial variablefn: expression for the nth term in a sequencen:  name, index variable for fn genfunc[termscale](1+a*z, z, n*2^n*t(n) + n^2*t(n-1), n) parse maple n 2 n^n t * n 1 - n + t * + (* a 1 - n + (* 21 - n +^1 - n + t * 1 - n + (2 - n + t * + (* +

procedure finds the generating function of the hybrid terms that results from the term wise multiplication of the sequences encoded by the generating functions gf1, gf2, .... genfunc[rgf_hybrid](z, 1/(1-2*z), 2*z/(1-3*z)) parse maple 4 z * 1 - 6 z * + / -

genfunc[rgf_hybrid](z, z/(1-z-z^2), 1/(1-2*z)) parse maple 2 z * 1 - 2 z * 12 z * + (* + / -

Function finds the homogeneous linear recurrence with constant coefficients of order K that is satisfied by seq.•      If seq satisfies a recurrence of order less than K, that recurrence is found. genfunc[rgf_findrecur](2, [1, 2, 3, 4], t, n) parse maple n t 21 - n + t * 2 - n + t - =

genfunc[rgf_findrecur](2, [1, 2, 4, 8], s, j) parse maple j s 21 - j + s * =

rgf_norm(Fz, z) puts the generating function in normal form, expands the numerator and denominator, and factors the low order coefficient from the denominator. genfunc[rgf_norm](1/(2-3*z+z^2), z) parse maple 123 z * - z 2^+ /

genfunc[rgf_norm](1/(2-3*z+z^2), z, 'factored') parse maple 11 z - (2 z - (* /

genfunc[rgf_norm](1/(5-z^2), z, 'factored'(sqrt(5))) parse maple 15 sqrt z - (5 sqrt z + (* /

Procedure finds the value of the nth term in the sequence encoded by the generating function Fz.•      The value FAIL is returned if Fz is a trivial rational generating function. genfunc[rgf_term](1/(1-2*z), z, 100) parse maple 1267650600228230000000000000000

<less precision than the latest Maple algorithms

genfunc[rgf_term](z/(1-z)^2, z, 1001) parse maple 1001

genfunc[rgf_term](1/(1-a*z), z, 26) parse maple a 26^

genfunc[rgf_term](1/(1-y-y*z), z, 58) parse maple y 58^1 - y + (59^/ -

rgf_charseq: find characteristic sequence of a rational generating function

command returns the characteristic sequence of Fz as a function of Fn. genfunc[rgf_charseq](z/(1-z-z^2), z, F(n), n) parse maple 1 n + F

genfunc[rgf_charseq]((1+2*z)/(1-3*z-4*z^2), z, t(n), n) parse maple 41 - n + t * n t + 3 /