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A302704
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O.g.f. A(x) satisfies: A(x) = 1 + Integral (x*A(x)^8)' / (x*A(x)^5)' dx.
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5
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1, 1, 3, 7, 10, 24, 186, 492, -1863, -5240, 79369, 220350, -2492912, -6984296, 90693060, 254955852, -3412605726, -9625060440, 133881917577, 378533393025, -5412043255536, -15332556581976, 224289105628470, 636469447338144, -9487486533101850, -26960087538403992, 408305313050817591, 1161625141535962012, -17832202665017550896, -50783861201670203640, 788741951929695672520
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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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O.g.f. A(x) satisfies:
(1) A(x) = 1 + Integral (x*A(x)^8)' / (x*A(x)^5)' dx.
(2) A(x) = 1 + Integral A(x)^3 * (A(x) + 8*x*A'(x)) / (A(x) + 5*x*A'(x)) dx.
(3) A(x) = 1 + Integral A(x) * (sqrt( 1 + 4*x*A(x)^2 + 64*x^2*A(x)^4 ) - (1 - 8*x*A(x)^2))/(10*x) dx.
(4) 0 = A(x)^4 - A(x)*(1 - 8*x*A(x)^2)*A'(x) - 5*x*A'(x)^2.
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 10*x^4 + 24*x^5 + 186*x^6 + 492*x^7 - 1863*x^8 - 5240*x^9 + 79369*x^10 + 220350*x^11 - 2492912*x^12 + ...
RELATED SERIES.
(x*A(x)^8)' / (x*A(x)^5)' = 1 + 6*x + 21*x^2 + 40*x^3 + 120*x^4 + 1116*x^5 + 3444*x^6 - 14904*x^7 - 47160*x^8 + 793690*x^9 + 2423850*x^10 + ...
which equals A'(x).
The logarithmic derivative of the g.f. begins:
A'(x)/A(x) = 1 + 5*x + 13*x^2 + 5*x^3 + 31*x^4 + 905*x^5 + 1975*x^6 - 21595*x^7 - 41270*x^8 + 883355*x^9 + 1736824*x^10 - 34567735*x^11 + ...
which equals (sqrt(1 + 4*x*A(x)^2 + 64*x^2*A(x)^4) - (1 - 8*x*A(x)^2))/(10*x).
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PROG
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(PARI) {a(n) = my(A=1); for(i=1, n, A = 1 + intformal( (x*A^8)'/(x*A^5 +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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sign
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AUTHOR
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STATUS
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approved
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