# Sequences A120588 - A120607 : Formulas?

Paul D. Hanna pauldhanna at juno.com
Mon Nov 12 12:06:28 CET 2007

```Seqfans,
Sequences  A120588 - A120607  are examples of
solutions to functional equations of the form:
r*A(x) = c + b*x + A(x)^n
I list the examples at the bottom of this message.

Can anyone find nice formulas for any of these sequences?
A120588 is simple, but what about A120590 - A120607 ?

In A120588 I give a comment describing the conditions necessary
to satisfy such an equation, but the comment needs to be revised
and made more precise.  Would someone care to help?

Thanks,
Paul
--------------------------------------------------------------
A120588 - COMMENT :

"In order for the g.f. of an integer sequence to satisfy
a functional equation of the form:
r*A(x) = c + b*x + A(x)^n, where n > 1,
where d divides m*n*(n-1)/2 (m>0), and that
the coefficients are given by r = n + d^2/m, c = r-1, and b = d^3/m.

The remaining terms may then be integer and still satisfy:
a_n(k) = r*a(k),
where a_n(k) is the k-th term of the n-th self-convolution of the
sequence."

--------------------------------------------------------------
EXAMPLES: A-numbers, g.f. equation, initial 3 terms.

A120588 : 3*A(x) = 2 + x + A(x)^2       : [1,1,1].
A120590 : 4*A(x) = 3 + x + A(x)^3       : [1,1,3].
A120592 : 5*A(x) = 4 + 4*x + A(x)^3     : [1,2,6].
A120593 : 5*A(x) = 4 + x + A(x)^4       : [1,1,6].
A120594 : 8*A(x) = 7 + 8*x + A(x)^4     : [1,2,6].
A120595 : 13*A(x) = 12 + 27*x + A(x)^4  : [1,3,6].
A120596 : 6*A(x) = 5 + x + A(x)^5       : [1,1,10].
A120597 : 9*A(x) = 8 + 8*x + A(x)^5     : [1,2,10].
A120598 : 30*A(x) = 29 + 125*x + A(x)^5 : [1,5,10].
A120599 : 13*A(x) = 12 + 32*x + A(x)^5  : [1,4,20].
A120600 : 7*A(x) = 6 + x + A(x)^6       : [1,1,15].
A120601 : 15*A(x) = 14 + 27*x + A(x)^6  : [1,3,15].
A120602 : 31*A(x) = 30 + 125*x + A(x)^6 : [1,5,15].
A120603 : 16*A(x) = 15 + 27*x + A(x)^7  : [1,3,21].
A120604 : 24*A(x) = 23 + 64*x + A(x)^8  : [1,4,28].
A120605 : 25*A(x) = 24 + 64*x + A(x)^9  : [1,4,36].
A120606 : 36*A(x) = 35 + 81*x + A(x)^9  : [1,3,12].
A120607 : 37*A(x) = 36 + 81*x + A(x)^10 : [1,3,15].

END.

```