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A204218
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G.f. satisfies: A(x) = 1 + Sum_{n>=1} n * (x*A(x))^(n*(n+1)/2) / (1 - x^n*A(x)^n).
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1
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1, 1, 2, 7, 22, 74, 271, 1012, 3858, 15046, 59579, 238825, 967873, 3958517, 16316594, 67716000, 282719162, 1186633647, 5004102122, 21192022233, 90089538788, 384305738731, 1644544501988, 7057705570877, 30368821119351, 130993073168419, 566297366630412, 2453269044761359
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. satisfies: A(x) = 1 + 1/Theta4(x*A(x))^2 * Sum_{n>=1} (-1)^(n-1) * n * (x*A(x))^(n*(n+1)/2) * (1 - x^n*A(x)^n)/(1 + x^n*A(x)^n)^2 where Theta4(x) = 1 + 2*Sum_{n>=1} (-x)^(n^2), due to an identity of Ramanujan.
G.f. satisfies: A(x) = 1 + Sum_{n>=1} A204217(n)*x^n*A(x)^n.
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 22*x^4 + 74*x^5 + 271*x^6 + 1012*x^7 +...
where A(x) = 1 + x*A(x)/(1 - x*A(x)) + 2*x^3*A(x)^3/(1 - x^2*A(x)^2) + 3*x^6*A(x)^6/(1 - x^3*A(x)^3) + 4*x^10*A(x)^10/(1 - x^4*A(x)^4) +...
Also, by a Ramanujan identity:
(A(x) - 1)*Theta4(x*A(x))^2 = x*A(x)*(1 - x*A(x))/(1 + x*A(x))^2 - 2*x^3*A(x)^3*(1 - x^2*A(x)^2)/(1 + x^2*A(x)^2)^2 + 3*x^6*A(x)^6*(1 - x^3*A(x)^3)/(1 + x^3*A(x)^3)^2 - 4*x^10*A(x)^10*(1 - x^4*A(x)^4)/(1 + x^4*A(x)^4)^2 +...
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MATHEMATICA
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a[n_] := (For[A = 1+x; i = 1, i <= n, i++, A = 1+Sum[m*(x*A)^(m*(m+1)/2)/(1 - (x*A + x*O[x]^n)^m), {m, 1, n}]]; Coefficient[A, x, n]); Table[an = a[n]; Print[an]; an, {n, 0, 27}] (* Jean-François Alcover, Jun 04 2017, translated from PARI *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, m*(x*A)^(m*(m+1)/2)/(1-(x*A +x*O(x^n))^m))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x, Theta4=1+2*sum(m=1, sqrtint(n+1), (-x)^(m^2))); for(i=1, n, A=1+1/subst(Theta4^2, x, x*A+x*O(x^n))*sum(m=1, sqrtint(2*n+1), (-1)^(m-1)*m*(x*A)^(m*(m+1)/2)*(1-(x*A)^m)/(1+(x*A)^m+x*O(x^n))^2)); polcoeff(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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