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A317668 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n) )^n  =  1. 5
1, 4, 26, 356, 8871, 320672, 14811200, 820185072, 52546341422, 3808527303300, 307523461730866, 27352330591164308, 2656394433081980649, 279696497208771609120, 31739466678890197201328, 3862114024795578127697248, 501700135604304149492422266, 69305144023051764776753873168, 10145743117833906529065611237208, 1569100081969097895595627120200512 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..200

FORMULA

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n) )^n.

(2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n.

(3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(4*n+4).

(4) Let B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+1) )^n,

then B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(3*n+3).

(5) Let C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+2) )^n,

then C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(2*n+2).

(6) Let D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+3) )^n,

then D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(n+1).

a(n) ~ 2^(2*n + log(2)/8 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 13 2018

EXAMPLE

G.f.: A(x) = 1 + 4*x + 26*x^2 + 356*x^3 + 8871*x^4 + 320672*x^5 + 14811200*x^6 + 820185072*x^7 + 52546341422*x^8 + 3808527303300*x^9 + 307523461730866*x^10 + ...

such that

1 = 1  +  (1/A(x) - (1-x)^4)  +  (1/A(x) - (1-x)^8)^2  +  (1/A(x) - (1-x)^12)^3  +  (1/A(x) - (1-x)^16)^4  +  (1/A(x) - (1-x)^20)^5  +  (1/A(x) - (1-x)^24)^6  +  (1/A(x) - (1-x)^28)^7  +  (1/A(x) - (1-x)^32)^8  + ...

Also,

A(x) = 1  +  (1/A(x) - (1-x)^8)  +  (1/A(x) - (1-x)^12)^2  +  (1/A(x) - (1-x)^16)^3  +  (1/A(x) - (1-x)^20)^4  +  (1/A(x) - (1-x)^24)^5  +  (1/A(x) - (1-x)^28)^6  +  (1/A(x) - (1-x)^32)^7  +  (1/A(x) - (1-x)^36)^8  + ...

RELATED SERIES.

(1) The series B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+1) )^n begins

B(x) = 1 + x + 5*x^2 + 67*x^3 + 1669*x^4 + 60246*x^5 + 2781335*x^6 + 154062232*x^7 + 9875799121*x^8 + 716231200582*x^9 + 57865799711347*x^10 + ...

also given by B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(3*n+3).

(2) The series C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+2) )^n begins

C(x) = 1 + 2*x + 11*x^2 + 148*x^3 + 3683*x^4 + 132888*x^5 + 6131332*x^6 + 339397944*x^7 + 21742672693*x^8 + 1575995237188*x^9 + 127268039660042*x^10 + ...

also given by C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(2*n+2).

(3) The series D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+3) )^n begins

D(x) = 1 + 3*x + 18*x^2 + 244*x^3 + 6073*x^4 + 219238*x^5 + 10117351*x^6 + 560000464*x^7 + 35868610134*x^8 + 2599382401532*x^9 + 209871544727484*x^10 + ...

also given by D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(n+1).

PROG

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( 1/Ser(A) - (1-x)^(4*m+4) )^m ) )[#A]/2 ); A[n+1]}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A317349, A317666, A317667, A317803.

Sequence in context: A114125 A063182 A293954 * A328419 A194926 A167147

Adjacent sequences:  A317665 A317666 A317667 * A317669 A317670 A317671

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 12 2018

STATUS

approved

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Last modified September 21 20:10 EDT 2021. Contains 347598 sequences. (Running on oeis4.)