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A325286 G.f. satisfies: 1 = Sum_{n>=0} (1+x)^(n*(n-1)/2) / A(x)^n * 1/2^(n+1).

%I A325286 #14 Jul 30 2023 08:45:32
%S A325286 1,1,3,25,343,6441,150975,4203201,134852079,4886641681,197154406591,
%T A325286 8760602600193,425074860993439,22363792326962881,1268239233311498079,
%U A325286 77129745316500047745,5008173999379887257151,345838251972031108425345,25309861534968595801377279,1956926079593452273940279169,159406563966400881627947865279,13645204581985719926987977747329,1224591755319676016226530026499583,114980206425267526899287638805977857
%N A325286 G.f. satisfies: 1 = Sum_{n>=0} (1+x)^(n*(n-1)/2) / A(x)^n * 1/2^(n+1).
%C A325286 a(n) == 1 (mod 2) for n >= 0.
%C A325286 Conjecture: a(n) == 1 (mod 3) iff n is a number whose base-3 representation contains no 2 (cf. A005836), otherwise a(n) == 0 (mod 3).
%H A325286 Paul D. Hanna, <a href="/A325286/b325286.txt">Table of n, a(n) for n = 0..175</a>
%F A325286 G.f. satisfies:
%F A325286 (1) 1 = Sum_{n>=0} (1+x)^(n*(n-1)/2) / A(x)^n * 1/2^(n+1).
%F A325286 (2) 1 = Sum_{n>=0} (1+x)^(n*(n+1)/2) / A(x)^(n+1) * 1/2^(n+1).
%e A325286 G.f.: A(x) = 1 + x + 3*x^2 + 25*x^3 + 343*x^4 + 6441*x^5 + 150975*x^6 + 4203201*x^7 + 134852079*x^8 + 4886641681*x^9 + 197154406591*x^10 + ...
%e A325286 such that
%e A325286 1 = 1/2 + 1/(2^2*A(x)) + (1+x)/(2^3*A(x)^2) + (1+x)^3/(2^4*A(x)^3) + (1+x)^6/(2^5*A(x)^4) + (1+x)^10/(2^6*A(x)^5) + (1+x)^15/(2^7*A(x)^6) + (1+x)^21/(2^8*A(x)^7) + ...
%e A325286 also,
%e A325286 1 = 1/(2*A(x)) + (1+x)/(2*A(x))^2 + (1+x)^3/(2*A(x))^3 + (1+x)^6/(2*A(x))^4 + (1+x)^10/(2*A(x))^5 + (1+x)^15/(2*A(x))^6 + (1+x)^21/(2*A(x))^7 + (1+x)^28/(2*A(x))^8 + ...
%o A325286 (PARI) /* Requires adequate precision */
%o A325286 {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); A[#A] = round( polcoeff( sum(m=0,10*#A+100, (1+x+x*O(x^#A))^(m*(m-1)/2)/Ser(A)^m/2^(m+1)*1.),#A-1)));A[n+1]}
%o A325286 for(n=0,25,print1(a(n),", "))
%Y A325286 Cf. A303290.
%K A325286 nonn
%O A325286 0,3
%A A325286 _Paul D. Hanna_, Apr 18 2019