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A321603
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G.f. A(x) satisfies: 1 = Sum_{n>=0} ((1+x)^(3*n) - A(x))^n.
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6
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1, 3, 12, 235, 7872, 351924, 19340668, 1250971416, 92720438955, 7733929764167, 716488771114410, 72981787493017014, 8107675760704948748, 975749719762368998037, 126491959992115408069503, 17576241581408197850363955, 2606439876885873198662077692, 410925212330248782377865281826, 68641203626673300062880912740755, 12110976733338358608040713750036252
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^(3*n) - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(3*n^2) / (1 + (1+x)^(3*n)*A(x))^(n+1).
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 12*x^2 + 235*x^3 + 7872*x^4 + 351924*x^5 + 19340668*x^6 + 1250971416*x^7 + 92720438955*x^8 + 7733929764167*x^9 + ...
such that
1 = 1 + ((1+x)^3 - A(x)) + ((1+x)^6 - A(x))^2 + ((1+x)^9 - A(x))^3 + ((1+x)^12 - A(x))^4 + ((1+x)^15 - A(x))^5 + ((1+x)^18 - A(x))^6 + ((1+x)^21 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + (1+x)^3/(1 + (1+x)^3*A(x))^2 + (1+x)^12/(1 + (1+x)^6*A(x))^3 + (1+x)^27/(1 + (1+x)^9*A(x))^4 + (1+x)^48/(1 + (1+x)^12*A(x))^5 + (1+x)^75/(1 + (1+x)^15*A(x))^6 + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 3 + 15*x + 624*x^2 + 28731*x^3 + 1638798*x^4 + 109462350*x^5 + 8333782509*x^6 + 710574703107*x^7 + 67015908514587*x^8 + ...
the coefficients of which are all divisible by 3:
(1/3) * A'(x)/A(x) = 1 + 5*x + 208*x^2 + 9577*x^3 + 546266*x^4 + 36487450*x^5 + 2777927503*x^6 + 236858234369*x^7 + 22338636171529*x^8 + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(3*m) - Ser(A))^m ) )[#A] ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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