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 A213639 G.f. satisfies: x = A( x - A(x)^3/x ). 1
 1, 1, 5, 38, 357, 3832, 45189, 572378, 7676653, 107971691, 1581714400, 24012849880, 376361077578, 6071985730614, 100602798234000, 1708558136679750, 29698002444820760, 527661478169200755, 9573199146196780335, 177192815265794698364, 3343432166097650920872 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Table of n, a(n) for n=1..21. FORMULA G.f. satisfies: (1) A(x) = x + A(A(x))^3 / A(x). (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/x^n / n!. (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/x^(n+1) / n! ). EXAMPLE G.f.: A(x) = x + x^2 + 5*x^3 + 38*x^4 + 357*x^5 + 3832*x^6 + 45189*x^7 +... Related series: A(x)^3/x = x^2 + 3*x^3 + 18*x^4 + 145*x^5 + 1389*x^6 + 14967*x^7 +... A(A(x)) = x + 2*x^2 + 12*x^3 + 102*x^4 + 1042*x^5 + 11977*x^6 + 149776*x^7 +... A(A(x))^3/A(x) = x^2 + 5*x^3 + 38*x^4 + 357*x^5 + 3832*x^6 + 45189*x^7 + ... The g.f. satisfies: A(x) = x + A(x)^3/x + [d/dx A(x)^6/x^2]/2! + [d^2/dx^2 A(x)^9/x^3]/3! + [d^3/dx^3 A(x)^12/x^4]/4! +... Logarithmic series: log(A(x)/x) = A(x)^3/x^2 + [d/dx A(x)^6/x^3]/2! + [d^2/dx^2 A(x)^9/x^4]/3! + [d^3/dx^3 A(x)^12/x^5]/4! +... PROG (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^3/x+x*O(x^n))); polcoeff(A, n))} (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(3*m)/x^m)/m!)+x*O(x^n)); polcoeff(A, n)} (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(3*m)/x^(m+1))/m!)+x*O(x^n))); polcoeff(A, n)} for(n=1, 21, print1(a(n), ", ")) CROSSREFS Cf. A213591. Sequence in context: A113207 A158266 A355380 * A243690 A335530 A308877 Adjacent sequences: A213636 A213637 A213638 * A213640 A213641 A213642 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 16 2012 STATUS approved

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Last modified May 26 08:47 EDT 2024. Contains 372815 sequences. (Running on oeis4.)