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A363105
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Expansion of g.f. A(x) satisfying 5 = Sum_{n=-oo..+oo} (-x)^n * (5*A(x) + x^(n-1))^(n+1).
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6
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1, 7, 59, 538, 5149, 51059, 520035, 5407889, 57181230, 612910369, 6644662132, 72731584789, 802696690614, 8922392225233, 99798739026795, 1122441028044882, 12686176392341722, 144013323190860339, 1641303449002365323, 18772674107796041770, 215413772477355781876
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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 may be described as follows.
(1) 5 = Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(n-1))^(n+1).
(2) 5 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (5*A(x) + x^n)^n.
(3) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^(n-1).
(4) 5*x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^(n+1).
(5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(n-1))^n ].
(6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (5*A(x) + x^n)^n ].
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 5*A(x)*x^(n+1))^n ].
(8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (5*A(x) + x^n)^(n+1).
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^n)^n.
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^n.
a(n) = Sum_{k=0..n} A359670(n,k) * 5^k for n >= 0.
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EXAMPLE
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G.f.: A(x) = 1 + 7*x + 59*x^2 + 538*x^3 + 5149*x^4 + 51059*x^5 + 520035*x^6 + 5407889*x^7 + 57181230*x^8 + 612910369*x^9 + 6644662132*x^10 + ...
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PROG
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(PARI) {a(n) = my(A=1, y=5); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 25, print1( a(n), ", "))
(PARI) {a(n) = my(A=[1], y=5); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-y + sum(n=-#A, #A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y), #A-1, x) ); A[n+1]}
for(n=0, 25, 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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