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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
OFFSET
0,2
LINKS
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 12 2018
STATUS
approved