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 A187826 G.f. satisfies: A(x) = Sum_{n>=0} ((1 + x*A(x))^n - 1)^n / (1 + x*A(x))^(n^2). 4
 1, 1, 4, 26, 219, 2227, 26438, 359904, 5555201, 96383191, 1864908541, 39929905561, 938897407239, 24069888638463, 668309231078015, 19977542570492051, 639571311256259372, 21828488143257352752, 791044181963746918758, 30331001954496565907536 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Limit n->infinity A220353(n)/A187826(n) = 1. - Vaclav Kotesovec, Nov 08 2014 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..140 FORMULA G.f. satisfies: A(x) = Sum_{n>=1} ((1+x*A(x))^n - 1)^(n-1) / (1+x*A(x))^(n^2). a(n) ~ c * n^n / (exp(n) * (log(2))^(2*n)), where c = 2.341658334181687683758... . - Vaclav Kotesovec, Nov 08 2014 In closed form, c = 1 / (log(2) * sqrt(1-log(2)) * 2^((1-log(2))/2)). - Vaclav Kotesovec, May 03 2015 EXAMPLE G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 219*x^4 + 2227*x^5 + 26438*x^6 +... where the g.f. satisfies the identities: (1) A(x) = 1 + x*A(x)/(1+x*A(x)) + ((1 + x*A(x))^2 - 1)^2/(1+x*A(x))^4 + ((1 + x*A(x))^3 - 1)^3/(1+x*A(x))^9 + ((1 + x*A(x))^4 - 1)^4/(1+x*A(x))^16 +... (2) A(x) = 1/(1+x*A(x)) + ((1 + x*A(x))^2 - 1)/(1+x*A(x))^4 + ((1 + x*A(x))^3 - 1)^2/(1+x*A(x))^9 + ((1 + x*A(x))^4 - 1)^3/(1+x*A(x))^16 +... PROG (PARI) {a(n)=local(q, A=1); for(i=1, n, q=1+x*A+x*O(x^n); A=sum(k=0, n+1, q^(-k^2)*(q^k-1)^k)); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) (PARI) {a(n)=local(q, A=1); for(i=1, n, q=1+x*A+x*O(x^n); A=sum(k=1, n+1, q^(-k^2)*(q^k-1)^(k-1))); polcoeff(A, n)} for(n=0, 30, print1(a(n), ", ")) CROSSREFS Cf. A220352, A187827, A220353. Sequence in context: A143436 A135884 A120971 * A145347 A219780 A259902 Adjacent sequences: A187823 A187824 A187825 * A187827 A187828 A187829 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 27 2012 STATUS approved

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Last modified June 3 03:45 EDT 2023. Contains 363103 sequences. (Running on oeis4.)