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A382126 G.f. satisfies A(x) = A(x^2)*A(x^3) / (1-x).

%I A382126 #13 Apr 15 2025 08:56:26
%S A382126 1,1,2,3,5,6,11,13,20,26,36,44,66,78,106,132,174,208,282,332,430,520,
%T A382126 656,774,1000,1166,1456,1731,2131,2486,3097,3585,4374,5125,6177,7144,
%U A382126 8700,9994,11966,13874,16482,18908,22598,25800,30472,35014,41062,46802,55178,62624,73094,83384,96834
%N A382126 G.f. satisfies A(x) = A(x^2)*A(x^3) / (1-x).
%H A382126 Paul D. Hanna, <a href="/A382126/b382126.txt">Table of n, a(n) for n = 0..1030</a>
%F A382126 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A382126 (1) A(x) = A(x^2)*A(x^3) / (1-x).
%F A382126 (2) A(x) = A(x^4)*A(x^6)^2*A(x^9) / ((1-x)*(1-x^2)*(1-x^3)).
%F A382126 (3) A(x) = A(x^8)*A(x^12)^3*A(x^18)^3*A(x^27) / ((1-x) * (1-x^2)*(1-x^3) * (1-x^4)*(1-x^6)^2*(1-x^9)).
%F A382126 (4) A(x) = A(x^16)*A(x^24)^4*A(x^36)^6*A(x^54)^4*A(x^81) / ((1-x) * (1-x^2)*(1-x^3) * (1-x^4)*(1-x^6)^2*(1-x^9) * (1-x^8)*(1-x^12)^3*(1-x^18)^3*(1-x^27)).
%F A382126 (5) A(x) = [ Product_{k=0..n} A( x^(2^(n-k)*3^k) )^binomial(n,k) ] / [ Product_{k=0..n-1} Product_{j=0..k} (1 - x^(2^(k-j)*3^j))^binomial(k,j) ] for n >= 1.
%F A382126 (6) A(x) = 1 / Product_{n>=0} Product_{k=0..n} (1 - x^(2^(n-k)*3^k))^binomial(n,k).
%F A382126 (7) A(x) = 1 / Product_{n>=1} (1 - x^A003586(n))^B(n) where B(n) = binomial(F2(n)+F3(n),F3(n)), F2(n) = A007814(A003586(n)), and F3(n) = A007949(A003586(n)).
%e A382126 G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 11*x^6 + 13*x^7 + 20*x^8 + 26*x^9 + 36*x^10 + 44*x^11 + 66*x^12 + 78*x^13 + 106*x^14 + 132*x^15 + ...
%e A382126 where A(x) = A(x^2)*A(x^3) / (1-x).
%e A382126 Also,
%e A382126 A(x) = 1/((1-x) * (1-x^2)*(1-x^3) * (1-x^4)*(1-x^6)^2*(1-x^9) * (1-x^8)*(1-x^12)^3*(1-x^18)^3*(1-x^27) * (1-x^16)*(1-x^24)^4*(1-x^36)^6*(1-x^54)^4*(1-x^81) * ...).
%e A382126 SPECIFIC VALUES.
%e A382126 A(t) = 20 at t = 0.7014984799558170594415639675177795335825631758657...
%e A382126 A(t) = 10 at t = 0.6459007989745013137507136047616010853643546686427...
%e A382126 A(t) = 5 at t = 0.56503953863462028848309645371720743210876751158208...
%e A382126 A(t) = 4 at t = 0.53037049685077322277423751856235866956835682007859...
%e A382126 A(t) = 3 at t = 0.47618735249468901057949356008055501793020059303831...
%e A382126 A(t) = 2 at t = 0.37230216384761004902154570388934366091900945011160...
%e A382126   where 2 = A(t^2)*A(t^3)/(1-t).
%e A382126 A(1/2) = 3.3771233774655473104234437722173818776421879254402816141...
%e A382126   where A(1/2) = 2*A(1/4)*A(1/8).
%e A382126 A(1/3) = 1.77955844576437383134389852350881569628236816392632...
%e A382126   where A(1/3) = (3/2)*A(1/9)*A(1/27).
%e A382126 A(1/4) = 1.45119948201558688211119223245819303991968906141565...
%e A382126   where A(1/4) = (4/3)*A(1/16)*A(1/64).
%e A382126 A(1/8) = 1.16356276973549618417716772166153717349571394213815...
%e A382126 A(1/9) = 1.14080534508777319257895810730361732261052126179176...
%e A382126 A(1/16) = 1.0711276470165363298314165146675228964034666341004...
%e A382126 A(1/27) = 1.0399427932948281565326692726054140704900715611747...
%e A382126 A(1/32) = 1.0332996355617515877322601695093331684320148037290...
%e A382126 A(1/64) = 1.0161250291160556543378749784871318759186544762111...
%e A382126 A(1/81) = 1.0126562735353427211848339688834832435682137255938...
%o A382126 (PARI) {a(n) = my(A=1+x+x*O(x^n)); for(i=1,#binary(n), A = (subst(A, x, x^2)*subst(A, x, x^3)/(1 - x +x*O(x^n))); ); polcoef(A,n)}
%o A382126 for(n=0,55,print1(a(n),", "))
%Y A382126 Cf. A003586, A007814, A007949.
%K A382126 nonn
%O A382126 0,3
%A A382126 _Paul D. Hanna_, Apr 14 2025