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A341374
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: [Sum_{n>=0} x^n/(1 - x^(n+1))]^3 = Sum_{n>=0} a(n)*x^n/(1 - x^(n+1))^3.
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2
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1, 3, 12, 22, 63, 57, 181, 174, 318, 302, 714, 444, 1177, 852, 1239, 1349, 2598, 1440, 3586, 2226, 3381, 3282, 6246, 3174, 6980, 5343, 7434, 6031, 12111, 5076, 14638, 9636, 12381, 11513, 16125, 9441, 24115, 15765, 19743, 14982, 32076, 13317, 36726, 21783, 25062
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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) = Sum_{n>=0} a(n)*x^n satisfies:
(1) [ Sum_{n>=0} x^n/(1 - x^(n+1)) ]^3 = Sum_{n>=0} a(n) * x^n / (1 - x^(n+1))^3.
(2) [ Sum_{n>=0} x^n/(1 - x^(n+1)) ]^3 = Sum_{n>=0} (n+1)*(n+2)/2 * x^n * A( x^(n+1) ).
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EXAMPLE
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A(x) = 1 + 3*x + 12*x^2 + 22*x^3 + 63*x^4 + 57*x^5 + 181*x^6 + 174*x^7 + 318*x^8 + 302*x^9 + 714*x^10 + 444*x^11 + 1177*x^12 + ...
such that
D(x)^3 = 1/(1-x)^3 + 3*x/(1-x^2)^3 + 12*x^2/(1-x^3)^3 + 22*x^3/(1-x^4)^3 + 63*x^4/(1-x^5)^3 + 57*x^5/(1-x^6)^3 + ... + a(n)*x^n/(1-x^(n+1))^3 + ...
and
D(x)^3 = A(x) + 3*x*A(x^2) + 6*x^2*A(x^3) + 10*x^3*A(x^4) + 15*x^4*A(x^5) + 21*x^5*A(x^6) + 28*x^6*A(x^7) + ... + (n+1)*(n+2)/2*x^n*A(x^(n+1)) + ...
where
D(x)^3 = 1 + 6*x + 18*x^2 + 41*x^3 + 78*x^4 + 132*x^5 + 209*x^6 + 306*x^7 + 435*x^8 + 591*x^9 + 780*x^10 + 1008*x^11 + ... + A191829(n+1)*x^n + ...
D(x) = 1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 4*x^7 + 3*x^8 + 4*x^9 + 2*x^10 + 6*x^11 + 2*x^12 + ... + A000005(n+1)*x^n + ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(n=0, #A, x^n/(1 - x^(n+1) +x*O(x^#A)) )^3 - sum(n=0, #A-1, A[n+1]*x^n/(1 - x^(n+1) + x*O(x^#A))^3 ), #A-1) ); A[n+1]}
for(n=0, 100, 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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