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A317667 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n) )^n  =  1. 5
1, 3, 15, 154, 2865, 77532, 2684504, 111490839, 5357828286, 291299582266, 17643988446921, 1177175235308976, 85754781272021397, 6772714984220704506, 576470959628636447748, 52613628461306161087953, 5126338275850981999654524, 531146069930403178373329794, 58319563977901655667747310206, 6764879932357508722274792757285 (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)^(3*n) )^n.

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

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

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

then B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+3) )^n * (1-x)^(2*n+2).

(5) Let C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+2) )^n,

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

a(n) ~ 2^(log(2)/6 - 5/2) * 3^n * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 13 2018

EXAMPLE

G.f.: A(x) = 1 + 3*x + 15*x^2 + 154*x^3 + 2865*x^4 + 77532*x^5 + 2684504*x^6 + 111490839*x^7 + 5357828286*x^8 + 291299582266*x^9 + 17643988446921*x^10 + ...

such that

1 = 1  +  (1/A(x) - (1-x)^3)  +  (1/A(x) - (1-x)^6)^2  +  (1/A(x) - (1-x)^9)^3  +  (1/A(x) - (1-x)^12)^4  +  (1/A(x) - (1-x)^15)^5  +  (1/A(x) - (1-x)^18)^6  +  (1/A(x) - (1-x)^21)^7  +  (1/A(x) - (1-x)^24)^8  + ...

Also,

A(x) = 1  +  (1/A(x) - (1-x)^6)  +  (1/A(x) - (1-x)^9)^2  +  (1/A(x) - (1-x)^12)^3  +  (1/A(x) - (1-x)^15)^4  +  (1/A(x) - (1-x)^18)^5  +  (1/A(x) - (1-x)^21)^6  +  (1/A(x) - (1-x)^24)^7  +  (1/A(x) - (1-x)^27)^8  + ...

RELATED SERIES.

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

B(x) = 1 + x + 4*x^2 + 40*x^3 + 743*x^4 + 20073*x^5 + 694477*x^6 + 28841790*x^7 + 1386441234*x^8 + 75408643207*x^9 + 4569235921823*x^10 + ...

restated,

B(x) = 1  +  (1/A(x) - (1-x)^4)  +  (1/A(x) - (1-x)^7)^2  +  (1/A(x) - (1-x)^10)^3  +  (1/A(x) - (1-x)^13)^4  +  (1/A(x) - (1-x)^16)^5  +  (1/A(x) - (1-x)^19)^6  +  (1/A(x) - (1-x)^22)^7  +  (1/A(x) - (1-x)^25)^8  + ...

which can also be written

B(x) = (1-x)^2  +  (1/A(x) - (1-x)^6)*(1-x)^4  +  (1/A(x) - (1-x)^9)^2*(1-x)^6  +  (1/A(x) - (1-x)^12)^3*(1-x)^8  +  (1/A(x) - (1-x)^15)^4*(1-x)^10  +  (1/A(x) - (1-x)^18)^5*(1-x)^12  +  (1/A(x) - (1-x)^21)^6*(1-x)^14  +  (1/A(x) - (1-x)^24)^7*(1-x)^16  +  (1/A(x) - (1-x)^27)^8*(1-x)^18  + ...

...

(2) The series C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n+2) )^n begins

C(x) = 1 + 2*x + 9*x^2 + 91*x^3 + 1690*x^4 + 45661*x^5 + 1579367*x^6 + 65559850*x^7 + 3149821447*x^8 + 171233732325*x^9 + 10371022987322*x^10 + ...

restated,

C(x) = 1  +  (1/A(x) - (1-x)^5)  +  (1/A(x) - (1-x)^8)^2  +  (1/A(x) - (1-x)^11)^3  +  (1/A(x) - (1-x)^14)^4  +  (1/A(x) - (1-x)^17)^5  +  (1/A(x) - (1-x)^20)^6  +  (1/A(x) - (1-x)^23)^7  +  (1/A(x) - (1-x)^26)^8  + ...

which can also be written

C(x) = (1-x)  +  (1/A(x) - (1-x)^6)*(1-x)^2  +  (1/A(x) - (1-x)^9)^2*(1-x)^3  +  (1/A(x) - (1-x)^12)^3*(1-x)^4  +  (1/A(x) - (1-x)^15)^4*(1-x)^5  +  (1/A(x) - (1-x)^18)^5*(1-x)^6  +  (1/A(x) - (1-x)^21)^6*(1-x)^7  +  (1/A(x) - (1-x)^24)^7*(1-x)^8  +  (1/A(x) - (1-x)^27)^8*(1-x)^9  + ...

...

Compare the above series to

1 = (1-x)^3  +  (1/A(x) - (1-x)^6)*(1-x)^6  +  (1/A(x) - (1-x)^9)^2*(1-x)^9  +  (1/A(x) - (1-x)^12)^3*(1-x)^12  +  (1/A(x) - (1-x)^15)^4*(1-x)^15  +  (1/A(x) - (1-x)^18)^5*(1-x)^18  +  (1/A(x) - (1-x)^21)^6*(1-x)^21  +  (1/A(x) - (1-x)^24)^7*(1-x)^24  +  (1/A(x) - (1-x)^27)^8*(1-x)^27  + ...

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)^(3*m+3) )^m ) )[#A]/2 ); A[n+1]}

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

CROSSREFS

Cf. A317349, A317666, A317668, A317802.

Sequence in context: A181535 A162078 A245069 * A228901 A195226 A264558

Adjacent sequences:  A317664 A317665 A317666 * A317668 A317669 A317670

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 12 2018

STATUS

approved

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Last modified June 13 11:36 EDT 2021. Contains 344990 sequences. (Running on oeis4.)