A317667 - OEIS

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A317667

G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^(3*n) )^n = 1.

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)^(3n) )^n.` `(2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3n+3) )^n.` `(3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(3n+3) )^n (1-x)^(3n+3).` `(4) Let B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3n+1) )^n,` `then B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3n+3) )^n (1-x)^(2n+2).` `(5) Let C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3n+2) )^n,` `then C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(3n+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 + 3x + 15x^2 + 154x^3 + 2865x^4 + 77532x^5 + 2684504x^6 + 111490839x^7 + 5357828286x^8 + 291299582266x^9 + 17643988446921x^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)^(3n+1) )^n begins` `B(x) = 1 + x + 4x^2 + 40x^3 + 743x^4 + 20073x^5 + 694477x^6 + 28841790x^7 + 1386441234x^8 + 75408643207x^9 + 4569235921823x^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)^(3n+2) )^n begins` `C(x) = 1 + 2x + 9x^2 + 91x^3 + 1690x^4 + 45661x^5 + 1579367x^6 + 65559850x^7 + 3149821447x^8 + 171233732325x^9 + 10371022987322x^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 March 31 03:00 EDT 2023. Contains 361626 sequences. (Running on oeis4.)