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 A302704 O.g.f. A(x) satisfies:  A(x) = 1 + Integral (x*A(x)^8)' / (x*A(x)^5)' dx. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..400 FORMULA 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. EXAMPLE 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). PROG (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), ", ")) CROSSREFS Cf. A302701, A302705, A303064. Sequence in context: A317261 A024464 A127277 * A235490 A041437 A302279 Adjacent sequences:  A302701 A302702 A302703 * A302705 A302706 A302707 KEYWORD sign AUTHOR Paul D. Hanna, Apr 19 2018 STATUS approved

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Last modified May 16 11:21 EDT 2022. Contains 353704 sequences. (Running on oeis4.)