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 A203266 G.f.: A(x) = exp( Sum_{n>=1} G_n(x^n)^2 * x^n/n ) such that G_n(x^n) = Product_{k=0..n-1} A(u^k*x) where u is an n-th root of unity. 2
 1, 1, 3, 10, 43, 172, 852, 3719, 19290, 90469, 481825, 2295973, 12812880, 62122518, 346770241, 1744884177, 9830723932, 49268101457, 285020577850, 1438021808125, 8300553742452, 42955378420263, 247456097485363, 1271978597532857, 7520668963944205, 38700952324615078 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Table of n, a(n) for n=0..25. FORMULA G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * exp( Sum_{k>=1} 2*A203265(n*k)*x^(n*k)/k ) ) where A(x) = exp( Sum_{n>=1} A203265(n)*x^n/n ). The logarithmic derivative yields A203265. EXAMPLE G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 43*x^4 + 172*x^5 + 852*x^6 + 3719*x^7 +... G.f.: A(x) = exp( Sum_{n>=1} A203265(n) * x^n/n ), where A(x) = exp( Sum_{n>=1} G_n(x^n)^2 * x^n/n ) and G_n(x) = exp( Sum_{k>=1} A203265(n*k)*x^k/k ), which begin: G_1(x) = A(x); G_2(x) = 1 + 5*x + 75*x^2 + 1518*x^3 + 34663*x^4 + 867760*x^5 +...; G_3(x) = 1 + 22*x + 2019*x^2 + 214648*x^3 + 31221037*x^4 +...; G_4(x) = 1 + 125*x + 59771*x^2 + 40659310*x^3 + 31438395303*x^4 +...; G_5(x) = 1 + 576*x + 1760688*x^2 + 6380121685*x^3 +...; G_6(x) = 1 + 3554*x + 57073923*x^2 + 1295238092004*x^3 +...; G_7(x) = 1 + 16843*x + 1719312892*x^2 + 212162358939394*x^3 +...; G_8(x) = 1 + 103917*x + 56284535547*x^2 + 44125115136389518*x^3 +...; ... Also, G_n(x^n) = Product_{k=0..n-1} A(u^k*x) where u = n-th root of unity: G_2(x^2) = A(x)*A(-x); G_3(x^3) = A(x)*A(u*x)*A(u^2*x) where u = exp(2*Pi*I/3); G_4(x^4) = A(x)*A(u*x)*A(u^2*x)*A(u^3*x) where u = exp(2*Pi*I/4); ... The logarithmic derivative of this sequence yields A203265: A203265 = [1,5,22,125,576,3554,16843,103917,521338,3189600,...]. PROG (PARI) {a(n)=local(L=vector(n, i, 1)); for(i=1, n, L=Vec(deriv(sum(m=1, n, x^m/m*exp(sum(k=1, floor(n/m), 2*L[m*k]*x^(m*k)/k)+x*O(x^n)))))); polcoeff(exp(x*Ser(vector(n, m, L[m]/m))), n)} (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, x^m/m*round(prod(k=0, m-1, subst(A^2, x, exp(2*Pi*I*k/m)*x+x*O(x^n))))))); polcoeff(A, n)} CROSSREFS Cf. A203265 (log), A203254, A203268. Sequence in context: A288813 A306237 A042545 * A151084 A151085 A082936 Adjacent sequences: A203263 A203264 A203265 * A203267 A203268 A203269 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 30 2011 STATUS approved

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Last modified September 18 11:53 EDT 2024. Contains 376000 sequences. (Running on oeis4.)