A378262 - OEIS

%I #9 Dec 06 2024 11:09:49

%S 1,1,3,5,14,25,60,117,257,504,1053,2067,4197,8248,16491,32533,64919,

%T 128893,257923,515770,1036024,2080743,4185781,8411269,16893534,

%U 33867059,67793691,135460708,270330258,538950081,1074174949,2141296967,4271640535,8530158021,17054867115,34138204669,68398842318

%N G.f. satisfies A(x) = Sum_{n>=1} 2^(n-1) * A(x^(2*n))/A(x^n), with A(0) = 0 and A'(0) = 1.

%H Paul D. Hanna, <a href="/A378262/b378262.txt">Table of n, a(n) for n = 1..520</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) A(x) = Sum_{n>=1} 2^(n-1) * A(x^(2*n))/A(x^n).

%F (2) A(x) = Sum_{n>=1} x^n/(1 - 2*x^n) * ([x^n] A(x^2)/A(x)).

%e G.f.: A(x) = x + x^2 + 3*x^3 + 5*x^4 + 14*x^5 + 25*x^6 + 60*x^7 + 117*x^8 + 257*x^9 + 504*x^10 + 1053*x^11 + 2067*x^12 + 4197*x^13 + 8248*x^14 + 16491*x^15 + ...

%e where

%e A(x) = A(x^2)/A(x) + 2*A(x^4)/A(x^2) + 2^2*A(x^6)/A(x^3) + 2^3*A(x^8)/A(x^4) + 2^4*A(x^10)/A(x^5) + ... + 2^(n-1)*A(x^(2*n))/A(x^n) + ...

%e with

%e A(x^2)/A(x) = x - x^2 - x^3 - x^4 - 2*x^5 - x^6 - 4*x^7 - x^8 + 5*x^9 + 12*x^10 + 29*x^11 + 65*x^12 + 101*x^13 + 128*x^14 + 131*x^15 - 97*x^16 + ...

%e From the coefficients of x^n in A(x^2)/A(x) we may form A(x) like so

%e A(x) = x/(1-2*x) - x^2/(1-2*x^2) - x^3/(1-2*x^3) - x^4/(1-2*x^4) - 2*x^5/(1-2*x^5) - x^6/(1-2*x^6) - 4*x^7/(1-2*x^7) - x^8/(1-2*x^8) + 5*x^9/(1-2*x^9) + 12*x^10/(1-2*x^10) + 29*x^11/(1-2*x^11) + 65*x^12/(1-2*x^12) + ... + ([x^n] A(x^2)/A(x)) * x^n/(1 - 2*x^n) + ...

%e SPECIFIC VALUES.

%e A(t) = 3 at t = 0.43733814488282063301680910555646948170999223090757...

%e A(t) = 2 at t = 0.41397004526177482844685461509046496062468756075210...

%e A(t) = 3/2 at t = 0.39330222443727410417944329172595352498212069922925...

%e A(t) = 1 at t = 0.35749402156274824941243704576004713164226979305829...

%e A(t) = 2/3 at t = 0.31333794936725576481707196951511233228426643397267...

%e A(t) = 1/2 at t = 0.27748211306635183931075005365310539042990531754751...

%e A(2/5) = 1.63758629524692945778740862999490236385081146642444...

%e A(1/3) = 0.79365065960676358258532147504412273666409442743886...

%e A(1/4) = 0.40608804950229998821473272051417196450715525962124...

%e A(1/5) = 0.27936643003430423804549184204540603443863382773108...

%o (PARI) {a(n) = my(V=[1],A,B); for(i=1,n, V = concat(V,0); A = x*Ser(V); B = subst(A,x, x^2)/A;

%o V[#V] = (1/2) * polcoef( sum(m=1,#V, 2^(m-1) * subst(B,x, x^m +x*O(x^#V)) ) - A, #V) ); V[n]}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A378259.

%K nonn,new

%O 1,3

%A _Paul D. Hanna_, Dec 05 2024