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A341958
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G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(n+4)) = Sum_{n>=0} x^n*A(x)^(2*n)/(1 - x*A(x)^(5*n+3)).
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1
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1, 2, 24, 460, 10480, 262264, 6968920, 193069480, 5515713624, 161332352440, 4807707617064, 145452700810040, 4455938243565144, 137951634380458936, 4309354120105494920, 135662827893112703864, 4299732712210509636536, 137088360221810475982712
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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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Given g.f. A(x), the following sums are all equal:
(1) B(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(n+4)),
(2) B(x) = Sum_{n>=0} x^n*A(x)^(2*n)/(1 - x*A(x)^(5*n+3)),
(3) B(x) = Sum_{n>=0} x^n*A(x)^(3*n)/(1 - x*A(x)^(5*n+2)),
(4) B(x) = Sum_{n>=0} x^n*A(x)^(4*n)/(1 - x*A(x)^(n+1)),
(5) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2+5*n) * (1 - x^2*A(x)^(2*n+5)) / ((1 - x*A(x)^(n+1))*(1 - x*A(x)^(n+4))),
(6) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(5*n^2+5*n) * (1 - x^2*A(x)^(10*n+5)) / ((1 - x*A(x)^(5*n+2))*(1 - x*A(x)^(5*n+3)));
see the example section for the value of B(x).
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 24*x^2 + 460*x^3 + 10480*x^4 + 262264*x^5 + 6968920*x^6 + 193069480*x^7 + 5515713624*x^8 + 161332352440*x^9 + 4807707617064*x^10 + ...
such that
B(x) = 1/(1-x*A(x)^4) + x*A(x)/(1 - x*A(x)^5) + x^2*A(x)^2/(1 - x*A(x)^6) + x^3*A(x)^3/(1 - x*A(x)^7) + x^4*A(x)^4/(1 - x*A(x)^8) + ...
and
B(x) = 1/(1-x*A(x)^3) + x*A(x)^2/(1 - x*A(x)^8) + x^2*A(x)^4/(1 - x*A(x)^13) + x^3*A(x)^6/(1 - x*A(x)^18) + x^4*A(x)^8/(1 - x*A(x)^23) + ...
also
B(x) = 1/(1-x*A(x)^2) + x*A(x)^3/(1 - x*A(x)^7) + x^2*A(x)^6/(1 - x*A(x)^12) + x^3*A(x)^9/(1 - x*A(x)^17) + x^4*A(x)^12/(1 - x*A(x)^22) + ...
further
B(x) = 1/(1-x*A(x)) + x*A(x)^4/(1 - x*A(x)^2) + x^2*A(x)^8/(1 - x*A(x)^3) + x^3*A(x)^12/(1 - x*A(x)^4) + x^4*A(x)^16/(1 - x*A(x)^5) + ...
where
B(x) = 1 + 2*x + 13*x^2 + 180*x^3 + 3541*x^4 + 81806*x^5 + 2066157*x^6 + 55253320*x^7 + 1537808581*x^8 + 44084046546*x^9 + 1292851214781*x^10 + ...
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PROG
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(PARI) {a(n) = my(A=[1, 2]); for(i=1, n, A=concat(A, 0); H=A; A=concat(A, 0);
H[#A-1] = -polcoeff( sum(m=0, #A, x^m*Ser(A)^m/(1 - x*Ser(A)^(m+4)) ) - sum(m=0, #A, x^m*Ser(A)^(2*m)/(1 - x*Ser(A)^(5*m+3)) ), #A)/4; A=H); A[n+1] }
for(n=0, 20, 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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