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A340895
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G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(3*n)/(1 - x*A(x)^n) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(3*n+2)).
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8
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1, 1, 4, 30, 266, 2578, 26501, 284016, 3139627, 35546887, 410192986, 4807352738, 57070869569, 684923802002, 8296759274813, 101314431713383, 1245926701296828, 15417454366803538, 191837536701986616, 2398860867832185041, 30131118130388007583, 379999736040117142018
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OFFSET
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0,3
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COMMENTS
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Equals row k = 3 of rectangular table A340940.
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LINKS
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FORMULA
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Given g.f. A(x), the following sums are all equal:
(1) B(x) = Sum_{n>=0} x^n*A(x)^(3*n)/(1 - x*A(x)^n),
(2) B(x) = Sum_{n>=0} x^n*A(x)^(2*n)/(1 - x*A(x)^(3*n+1)),
(3) B(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(3*n+2)),
(4) B(x) = Sum_{n>=0} x^n/(1 - x*A(x)^(n+3)),
(5) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2+3*n) * (1 - x^2*A(x)^(2*n+3)) / ((1 - x*A(x)^n)*(1 - x*A(x)^(n+3))),
(6) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(3*n^2+3*n) * (1 - x^2*A(x)^(6*n+3)) / ((1 - x*A(x)^(3*n+1))*(1 - x*A(x)^(3*n+2)));
see the example section for the value of B(x).
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EXAMPLE
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G.f.: A(x) = 1 + x + 4*x^2 + 30*x^3 + 266*x^4 + 2578*x^5 + 26501*x^6 + 284016*x^7 + 3139627*x^8 + 35546887*x^9 + 410192986*x^10 + 4807352738*x^11 + ...
such that
B(x) = 1/(1-x) + x*A(x)^3/(1 - x*A(x)) + x^2*A(x)^6/(1 - x*A(x)^2) + x^3*A(x)^9/(1 - x*A(x)^3) + x^4*A(x)^12/(1 - x*A(x)^4) + ...
and
B(x) = 1/(1-x*A(x)) + x*A(x)^2/(1 - x*A(x)^4) + x^2*A(x)^4/(1 - x*A(x)^7) + x^3*A(x)^6/(1 - x*A(x)^10) + x^4*A(x)^8/(1 - x*A(x)^13) + ...
also
B(x) = 1/(1-x*A(x)^2) + x*A(x)/(1 - x*A(x)^5) + x^2*A(x)^2/(1 - x*A(x)^8) + x^3*A(x)^3/(1 - x*A(x)^11) + x^4*A(x)^4/(1 - x*A(x)^14) + ...
further
B(x) = 1/(1-x*A(x)^3) + x/(1 - x*A(x)^4) + x^2/(1 - x*A(x)^5) + x^3/(1 - x*A(x)^6) + x^4/(1 - x*A(x)^7) + x^5/(1 - x*A(x)^8) + ...
where
B(x) = 1 + 2*x + 6*x^2 + 29*x^3 + 203*x^4 + 1738*x^5 + 16574*x^6 + 168779*x^7 + 1797190*x^8 + 19770290*x^9 + 222969428*x^10 + 2564646302*x^11 + ...
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PROG
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(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); H=A; A=concat(A, 0);
H[#A-1] = -polcoeff( sum(m=0, #A, x^m/(1 - x*Ser(A)^(m+3)) ) - sum(m=0, #A, x^m*Ser(A)^m/(1 - x*Ser(A)^(3*m+2)) ), #A)/2; A=H); 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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