A190790 - OEIS

%I #17 Sep 28 2023 08:57:25

%S 1,1,2,6,18,58,198,696,2506,9205,34344,129792,495834,1911640,7428444,

%T 29064650,114404410,452719183,1799994588,7187148262,28807364008,

%U 115865980972,467497031164,1891710323324,7675031497682,31215088847239

%N G.f. satisfies: A(x) = 1 + Sum_{n>=1} q^(2n-1)/(1 - q^(2n-1)) where q = x*A(x).

%H Vaclav Kotesovec, <a href="/A190790/b190790.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f. A(x) satisfies:

%F * A(x) = 1 + Sum_{n>=1} q^(n*(n+1)/2)/(1 - q^n), where q = x*A(x);

%F * A(x) = 1 + Sum_{n>=1} q^n/(1 - q^(2n)), where q = x*A(x);

%F * A(x) = 1 + Sum_{n>=1} A001227(n)*x^n*A(x)^n, where A001227(n) = number of odd divisors of n.

%F Let D(x) = 1 + Sum_{n>=1} A001227(n)*x^n, then

%F * A(x) = D(x*A(x)) and D(x) = A(x/D(x));

%F * A(x) = (1/x)*Series_Reversion(x/D(x));

%F * a(n) = [x^n] D(x)^(n+1)/(n+1), the coefficient of x^n in D(x)^(n+1)/(n+1) for n>=0.

%F a(n) ~ c * d^n / n^(3/2), where d = 4.3154117906555438598489327064282723007551580672340735654761205324876... and c = 0.545252538971019249263783268322061859441589544238489362753993274... - _Vaclav Kotesovec_, Sep 28 2023

%e G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 18*x^4 + 58*x^5 + 198*x^6 +...

%e Let q = x*A(x), then the g.f. A(x) satisfies the following series:

%e * A(x) = 1 + q/(1 - q) + q^3/(1 - q^3) + q^5/(1 - q^5) + q^7/(1 - q^7) +...

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

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

%e * A(x) = 1 + q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 +...+ A001227(n)*q^n +...

%t nmax = 30; A[_] = 0; Do[A[x_] = 1 + Sum[(x*A[x])^(2*k-1)/(1 - (x*A[x])^(2*k-1)), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* _Vaclav Kotesovec_, Sep 28 2023 *)

%t (* Calculation of constant d: *) val = r /. FindRoot[{(Log[1 - r^2*s^2] + QPolyGamma[0, -1/2, r^2*s^2]) / Log[r^2*s^2] == s - 1 - 1/(1 - r*s), r*(1/(-1 + r*s)^2 + (2*r* s*(1/(-1 + r^2*s^2) + Derivative[0, 0, 1][QPolyGamma][0, -1/2, r^2*s^2]))/ Log[r^2*s^2]) == 1 + (2*(s - 1 - 1/(1 - r*s)))/(s*Log[r^2*s^2])}, {r, 1/4}, {s, 2}, WorkingPrecision -> 200]; N[ 1/Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3] (* _Vaclav Kotesovec_, Sep 28 2023 *)

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,(x*A)^(2*m-1)/(1-(x*A)^(2*m-1)+x*O(x^n))));polcoeff(A,n)}

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,(x*A)^(m*(m+1)/2)/(1-(x*A)^m+x*O(x^n))));polcoeff(A,n)}

%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,(x*A)^m/(1-(x*A)^(2*m)+x*O(x^n))));polcoeff(A,n)}

%o (PARI) {a(n)=local(D=1+sum(m=1,n,sumdiv(m, d, d%2)*x^m)+x*O(x^n));polcoeff(1/x*serreverse(x/D),n)}

%o (PARI) {a(n)=local(D=1+sum(m=1,n,sumdiv(m, d, d%2)*x^m)+x*O(x^n));polcoeff(D^(n+1)/(n+1),n)}

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 20 2011