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 A222013 G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^(n*(n+1)/2) / Product_{k=1..n} (1 + k*x*A(x)^k). 1
 1, 1, 2, 7, 32, 172, 1038, 6865, 49098, 376942, 3094812, 27129690, 253821716, 2534600760, 27012498668, 307083883519, 3719224056464, 47898505899624, 654343988611350, 9455986402701388, 144138413744793426, 2311030293590097634, 38871924229882607774 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Compare the g.f. to the identities: (1) 1/(1-x) = Sum_{n>=0} n!*x^n / Product_{k=1..n} (1 + k*x). (2) C(x) = Sum_{n>=0} n!*x^n*C(x)^n / Product_{k=1..n} (1 + k*x*C(x)), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108). LINKS EXAMPLE G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 32*x^4 + 172*x^5 + 1038*x^6 +... where A(x) = 1 + x*A(x)/(1+x*A(x)) + 2!*x^2*A(x)^3/((1+x*A(x))*(1+2*x*A(x)^2)) + 3!*x^3*A(x)^6/((1+x*A(x))*(1+2*x*A(x)^2)*(1+3*x*A(x)^3)) + 4!*x^4*A(x)^10/((1+x*A(x))*(1+2*x*A(x)^2)*(1+3*x*A(x)^3)*(1+4*x*A(x)^4)) +... PROG (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, m!*x^m*A^(m*(m+1)/2)/prod(k=1, m, 1+k*x*(A+x*O(x^n))^k))); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A222014, A221586. Sequence in context: A107593 A125277 A321688 * A179488 A191809 A161392 Adjacent sequences:  A222010 A222011 A222012 * A222014 A222015 A222016 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 04 2013 STATUS approved

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Last modified September 25 12:23 EDT 2020. Contains 337343 sequences. (Running on oeis4.)