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A245107
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G.f. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * (A(x)^k - x^k)^k * x^(k*(n-k)).
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2
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1, 2, 3, 15, 71, 444, 3156, 24021, 195445, 1678810, 15109730, 141918887, 1387000940, 14077705892, 148201957507, 1616833069407, 18267949067293, 213646700978736, 2584967961734726, 32337901074348415, 417993768656658465, 5578096978483927773, 76784593772859188463, 1089213101301826525900
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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. satisfies: A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} C(n,k) * (1 - x^(n-k)*A(x)^k)^(n-k) * x^(k*(n-k)) * A(x)^(k^2).
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 3*x^2 + 15*x^3 + 71*x^4 + 444*x^5 + 3156*x^6 +...
Let A = A(x) then
A(x) = 1 + x*(1 + (A-x))
+ x^2*(1 + 2*(A-x)*x + (A^2-x^2)^2)
+ x^3*(1 + 3*(A-x)*x^2 + 3*(A^2-x^2)^2*x^2 + (A^3-x^3)^3)
+ x^4*(1 + 4*(A-x)*x^3 + 6*(A^2-x^2)^2*x^4 + 4*(A^3-x^3)^3*x^3 + (A^4-x^4)^4)
+ x^5*(1 + 5*(A-x)*x^4 + 10*(A^2-x^2)^2*x^6 + 10*(A^3-x^3)^3*x^6 + 5*(A^4-x^4)^4*x^4 + (A^5-x^5)^5)
+ x^6*(1 + 6*(A-x)*x^5 + 15*(A^2-x^2)^2*x^8 + 20*(A^3-x^3)^3*x^9 + 15*(A^4-x^4)^4*x^8 + 6*(A^5-x^5)^5*x^5 + (A^6-x^6)^6) +...
where we have the identity
A(x) = 1 + x*((1-x) + A)
+ x^2*((1-x^2)^2 + 2*(1-x*A)*x*A + A^4)
+ x^3*((1-x^3)^3 + 3*(1-x^2*A)^2*x^2*A + 3*(1-x*A^2)*x^2*A^4 + A^9)
+ x^4*((1-x^4)^4 + 4*(1-x^3*A)^3*x^3*A + 6*(1-x^2*A^2)^2*x^4*A^4 + 4*(1-x*A^3)*x^3*A^9 + A^16)
+ x^5*((1-x^5)^5 + 5*(1-x^4*A)^4*x^4*A + 10*(1-x^3*A^2)^3*x^6*A^4 + 10*(1-x^2*A^3)^2*x^6*A^9 + 5*(1-x*A^4)*x^4*A^16 + A^25)
+ x^6*((1-x^6)^6 + 6*(1-x^5*A)^5*x^5*A + 15*(1-x^4*A^2)^4*x^8*A^4 + 20*(1-x^3*A^3)^3*x^9*A^9 + 15*(1-x^2*A^4)^2*x^8*A^16 + 6*(1-x*A^5)*x^5*A^25 + A^36) +...
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)*(A^k - x^k)^k * x^(k*(m-k)) +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)*(1 - x^(m-k)*A^k)^(m-k) * x^(k*(m-k))*A^(k^2) +x*O(x^n)))); polcoeff(A, n)}
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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