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Expansion of g.f. A(x) satisfying A(x) = x + x^2 + (2*A(x)^3 + A(x^3))/3.
3

%I #9 Aug 22 2024 04:22:39

%S 1,1,1,2,4,9,20,48,117,290,734,1880,4868,12730,33556,89072,237904,

%T 638873,1723930,4672008,12710904,34703894,95054188,261116816,

%U 719223064,1985934212,5496123033,15242821108,42357113994,117918233704,328833828334,918470764376,2569238134248,7197046596440

%N Expansion of g.f. A(x) satisfying A(x) = x + x^2 + (2*A(x)^3 + A(x^3))/3.

%C Conjecture: a(n) is odd iff n is in A038754, which consists of numbers of the form 3^k and 2*3^k.

%H Paul D. Hanna, <a href="/A375439/b375439.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 2.9308423020191987018531615662206918839933116797613922... and c = 0.186346847470275688362452238277535367815900456286173... - _Vaclav Kotesovec_, Aug 22 2024

%e G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 20*x^7 + 48*x^8 + 117*x^9 + 290*x^10 + 734*x^11 + 1880*x^12 + 4868*x^13 + 12730*x^14 + 33556*x^15 + ...

%e where A(x) = x + x^2 + (2*A(x)^3 + A(x^3))/3.

%e RELATED SERIES.

%e A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 30*x^7 + 72*x^8 + 175*x^9 + 435*x^10 + 1101*x^11 + 2819*x^12 + 7302*x^13 + 19095*x^14 + 50332*x^15 + ...

%e Let B(x) be the series reversion of A(x), B(A(x)) = x, then B(x) begins

%e B(x) = x - x^2 + x^3 - 2*x^4 + 4*x^5 - 9*x^6 + 22*x^7 - 57*x^8 + 152*x^9 - 411*x^10 + 1119*x^11 - 3063*x^12 + 8436*x^13 - 23405*x^14 + 65452*x^15 + ...

%e SPECIFIC VALUES.

%e A(1/3) = 0.6046115975458048490061476622502250915528261368314569825...

%e where A(1/3) = 4/9 + (2*A(1/3)^3 + A(1/27))/3.

%e A(1/4) = 0.345218924086872316546119663994502755734706567000751...

%e A(1/5) = 0.253555647303827972834265469178971877524548605418192...

%e A(1/6) = 0.201444567662949882659512632012060178593075505771758...

%e A(1/7) = 0.167365364255434800795732539120237367470157092655512...

%e A(1/8) = 0.143236474390624253781858259379882809014038308155736...

%e A(1/27) = 0.03846365186207481603806452459437536518999937182129...

%o (PARI) {a(n) = my(A=[0,1],Ax=x); for(i=1,n, A = concat(A,0); Ax=Ser(A);

%o A[#A] = polcoeff( x + x^2 + ( 2*Ax^3 + subst(Ax,x,x^3) )/3 - Ax,#A-1) );A[n+1]}

%o for(n=1,40,print1(a(n),", "))

%Y Cf. A375438, A357538, A000625.

%K nonn

%O 1,4

%A _Paul D. Hanna_, Aug 21 2024