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 A258994 E.g.f.: A'(x) = 1 + A(x)^6, with A(0)=1. 6
 1, 2, 12, 192, 4272, 124992, 4531392, 195869952, 9832326912, 562125837312, 36056880110592, 2564230500421632, 200237330428342272, 17032391106795159552, 1567547894591436275712, 155196096043697480466432, 16447362605632117421309952, 1857733260790463501532659712 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS In general, for k>1, if e.g.f. satisfies A'(x) = 1 + A(x)^k, with A(0)=1, then a(n) ~ n! * d^(n + 1/(k-1)) / ((k-1)^(1/(k-1)) * Gamma(1/(k-1)) * n^(1-1/(k-1))), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(k*j-1). LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..50 FORMULA a(n) ~ n! * d^(n+1/5) / (5^(1/5) * Gamma(1/5) * n^(4/5)), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(6*j-1) = 6/(Pi - sqrt(3)*log(2+sqrt(3))) = 6.97224737278326506475991855023425659249063565... E.g.f.: 1 + Series_Reversion( Integral 1/(1 + (1+x)^6) dx ). - Paul D. Hanna, Jun 16 2015 EXAMPLE A(x) = 1 + 2*x + 12*x^2/2! + 192*x^3/3! + 4272*x^4/4! + 124992*x^5/5! + ... A'(x) = 2 + 12*x + 96*x^2 + 712*x^3 + 5208*x^4 + 188808*x^5/5 + ... 1 + A(x)^6 = 2 + 12*x + 96*x^2 + 712*x^3 + 5208*x^4 + 188808*x^5/5 + ... MATHEMATICA nmax=20; Subscript[a, 0]=1; egf=Sum[Subscript[a, k]*x^k, {k, 0, nmax+1}]; Table[Subscript[a, k]*k!, {k, 0, nmax}] /.Solve[Take[CoefficientList[Expand[1+egf^6-D[egf, x]], x], nmax]==ConstantArray[0, nmax]][[1]] PROG (PARI) {a(n) = local(A=1); A = 1 + serreverse( intformal( 1/(1 + (1+x)^6 +x*O(x^n)) )); n!*polcoeff(A, n)} for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 16 2015 CROSSREFS Cf. A000831 (k=2), A258969 (k=3), A258970 (k=4), A258971 (k=5), A258927. Sequence in context: A015195 A051421 A182162 * A110105 A182163 A245358 Adjacent sequences:  A258991 A258992 A258993 * A258995 A258996 A258997 KEYWORD nonn AUTHOR Vaclav Kotesovec, Jun 16 2015 STATUS approved

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Last modified January 25 17:35 EST 2022. Contains 350572 sequences. (Running on oeis4.)