login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A318643 G.f. D(x) satisfies: Sum_{n>=0} n * (x + (-1)^n*A(x))^n = 0, where A(x) = D^8(x), the 8th iteration of D(x), and A(x) is the g.f. of A318640. 4
1, 1, 1, 25, 73, 1025, 4913, 48985, 311305, 2393953, 17903761, 140986201, 1096160649, 7777051265, 61667165361, 597402170649, 4836234935497, 4245154618465, -25145215353455, 12982383457107609, 139920724294631369, -5479397854898810111, -68618853272591110863, 3588130738987950942681, 48514725864891831998601, -2781644195772240632990623 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

a(n) (mod 8) = 1.

a(n) (mod 16) has period 4 after initial term: [1, 1,1,9,9, 1,1,9,9, ...].

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..400

FORMULA

G.f. D(x) satisfies:

(1) D(-D(-x)) = x.

(2) 0 = Sum_{n>=0} (-1)^n * n * ( D(D(D(D(x)))) - (-1)^n*D(D(D(D(-x)) )^n.

(3) 0 = (A-x)*(1 + (A-x)^2)/(1 - (A-x)^2)^2  -  2*(A+x)^2/(1 - (A+x)^2)^2, where A = D^8(x), i.e., A(x) = D(D(D(D(D(D(D(D(x)))))))).

EXAMPLE

G.f.: D(x) = x + x^2 + x^3 + 25*x^4 + 73*x^5 + 1025*x^6 + 4913*x^7 + 48985*x^8 + 311305*x^9 + 2393953*x^10 + 17903761*x^11 + 140986201*x^12 + ...

where D(-D(-x)) = x.

RELATED SERIES.

(a) If D(D(D(D( D(D(D(D(x)))) )))) = A(x) then

A(x) = x + 8*x^2 + 64*x^3 + 704*x^4 + 8704*x^5 + 113536*x^6 + 1544192*x^7 + 21671936*x^8 + 311468032*x^9 + 4560963584*x^10 + ... + A318640(n)*x^n + ...

such that

0 = (x - A(x)) + 2*(x + A(x))^2 + 3*(x - A(x))^3 + 4*(x + A(x))^4 + 5*(x - A(x))^5 + 6*(x + A(x))^6 + 7*(x - A(x))^7 + 8*(x + A(x))^8 + 9*(x - A(x))^9 + 10*(x + A(x))^10 + ...

(b) If D(D(D(D(x)))) = B(x) then

B(x) = x + 4*x^2 + 16*x^3 + 160*x^4 + 1408*x^5 + 13760*x^6 + 140288*x^7 + 1459200*x^8 + 15595520*x^9 + 168584192*x^10 + 1847791616*x^11 + ... + A318641(n)*x^n + ...

such that

0 = (B(x) + B(-x)) - 2*(B(x) - B(-x))^2 + 3*(B(x) + B(-x))^3 - 4*(B(x) - B(-x))^4 + 5*(B(x) + B(-x))^5 - 6*(B(x) - B(-x))^6 + 7*(B(x) + B(-x))^7 - 8*(B(x) - B(-x))^8 + 9*(B(x) + B(-x))^9 - 10*(B(x) - B(-x))^10 +- ...

(c) If D(D(x)) = C(x), then

C(x) = x + 2*x^2 + 4*x^3 + 56*x^4 + 304*x^5 + 2944*x^6 + 22592*x^7 + 196864*x^8 + 1700352*x^9 + 14416896*x^10 + 127798272*x^11 + 1141090304*x^12 + ... + A318642(n)*x^n + ...

where D(-D(-x)) = x.

PROG

(PARI) {HALF(F) = my(H=x); for(i=1, #F, H = (H + subst(F, x, serreverse(H +x*O(x^#F))))/2); H}

{a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff(sum(m=1, #A, m*(x + (-1)^m*x*Ser(A))^m), #A)); polcoeff( HALF(HALF(HALF(x*Ser(A)))), n)}

for(n=1, 30, print1(a(n), ", "))

CROSSREFS

Cf. A318640, A318641, A318642.

Sequence in context: A126379 A114553 A098439 * A044163 A044544 A045180

Adjacent sequences:  A318640 A318641 A318642 * A318644 A318645 A318646

KEYWORD

sign

AUTHOR

Paul D. Hanna, Aug 31 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 10:45 EST 2019. Contains 329751 sequences. (Running on oeis4.)