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A363135
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Expansion of g.f. A(x) satisfying A(x)^2 = Sum_{n=-oo..+oo} (-x)^n * (A(x)^3 + x^(n-1))^(n+1).
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5
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1, 3, 17, 133, 1201, 11796, 122192, 1314266, 14536760, 164299909, 1889209112, 22030014333, 259903138431, 3096573445278, 37205465163246, 450292741602439, 5484622483791007, 67179073566189256, 826958737112517885, 10225081367973756189, 126936946636845618425
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OFFSET
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0,2
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COMMENTS
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Given g.f. G(x,y) of triangle A359670, then A(x) = G(x,y=A(x)^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) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} A359670(n,k) * A(x)^(2*k).
(2) A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^3 + x^(n-1))^(n+1).
(3) A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (A(x)^3 + x^n)^n.
(4) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^3*x^(n+1))^(n-1).
(5) x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + A(x)^3*x^(n+1))^(n+1).
(6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^3 + x^(n-1))^n ].
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (A(x)^3 + x^n)^n ].
(8) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)^3*x^(n+1))^n ].
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x)^3 + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^3*x^n)^n.
(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^3*x^(n+1))^n.
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EXAMPLE
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G.f.: A(x) = 1 + 3*x + 17*x^2 + 133*x^3 + 1201*x^4 + 11796*x^5 + 122192*x^6 + 1314266*x^7 + 14536760*x^8 + 164299909*x^9 + 1889209112*x^10 + ...
where A = A(x) may be generated from triangle A359670 as follows:
A(x) = 1 + x*(2 + A^2) + x^2*(4 + 6*A^2 + A^4) + x^3*(8 + 21*A^2 + 12*A^4 + A^6) + x^4*(14 + 62*A^2 + 68*A^4 + 20*A^6 + A^8) + x^5*(24 + 162*A^2 + 284*A^4 + 170*A^6 + 30*A^8 + A^10) + x^6*(40 + 384*A^2 + 998*A^4 + 970*A^6 + 360*A^8 + 42*A^10 + A^12) + x^7*(64 + 855*A^2 + 3092*A^4 + 4410*A^6 + 2720*A^8 + 679*A^10 + 56*A^12 + A^14) + x^8*(100 + 1806*A^2 + 8724*A^4 + 17172*A^6 + 15627*A^8 + 6608*A^10 + 1176*A^12 + 72*A^14 + A^16) + ... + x^n*(Sum_{k=0..n} A359670(n,k) * A(x)^(2*k)) + ...
RELATED SERIES.
A(x)^2 = 1 + 6*x + 43*x^2 + 368*x^3 + 3489*x^4 + 35320*x^5 + 373683*x^6 + 4082214*x^7 + 45693781*x^8 + 521342486*x^9 + 6040713074*x^10 + ...
A(x)^3 = 1 + 9*x + 78*x^2 + 732*x^3 + 7323*x^4 + 76764*x^5 + 832511*x^6 + 9264354*x^7 + 105198423*x^8 + 1214097433*x^9 + 14199964296*x^10 + ...
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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(1 - sum(n=-#A, #A, (-1)^n * x^n * (Ser(A)^3 + x^(n-1))^(n+1) )/Ser(A)^2, #A-1, x) ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*A^3 + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
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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