cp's OEIS Frontend

A382314 G.f. satisfies A(x) = 1/(1-x) + 2*x*A(x^2) + 3*x^2*A(x^3).

%I A382314 #14 Apr 15 2025 18:46:03
%S A382314 1,3,4,7,1,18,1,15,13,3,1,58,1,3,4,31,1,81,1,7,4,3,1,162,1,3,40,7,1,
%T A382314 18,1,63,4,3,1,337,1,3,4,15,1,18,1,7,13,3,1,418,1,3,4,7,1,324,1,15,4,
%U A382314 3,1,58,1,3,13,127,1,18,1,7,4,3,1,1161,1,3,4,7,1,18,1,31,121,3,1,58,1,3,4,15,1,81,1,7,4,3,1,1026
%N A382314 G.f. satisfies A(x) = 1/(1-x) + 2*x*A(x^2) + 3*x^2*A(x^3).
%C A382314 Logarithmic derivative of A382126.
%H A382314 Paul D. Hanna, <a href="/A382314/b382314.txt">Table of n, a(n) for n = 0..1030</a>
%F A382314 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A382314 (1) A(x) = 1/(1-x) + 2*x*A(x^2) + 3*x^2*A(x^3).
%F A382314 (2) A(x) = (1-x)^2*(1+2*x)*(1+2*x+3*x^2)/((1-x)*(1-x^2)*(1-x^3)) + 4*x^3*A(x^4) + 12*x^5*A(x^6) + 9*x^8*A(x^9).
%F A382314 (3) A(x) = (1/x)*Sum_{n>=0} Sum_{k=0..n} binomial(n,k) * 2^(n-k)*3^k * x^(2^(n-k)*3^k) / (1 - x^(2^(n-k)*3^k)).
%F A382314 (4) A(x) = (1/x)*Sum_{n>=1} B(n) * A003586(n) * x^A003586(n)/(1 - x^A003586(n)) where B(n) = binomial(F2(n)+F3(n),F3(n)), with F2(n) = A007814(A003586(n)) and F3(n) = A007949(A003586(n)).
%F A382314 (5) A(x) = B'(x)/B(x) where B(x) = B(x^2)*B(x^3)/(1-x) is the g.f. of A382126.
%e A382314 G.f.: A(x) = 1 + 3*x + 4*x^2 + 7*x^3 + x^4 + 18*x^5 + x^6 + 15*x^7 + 13*x^8 + 3*x^9 + x^10 + 58*x^11 + x^12 + 3*x^13 + 4*x^14 + 31*x^15 + ...
%e A382314 where A(x) = 1/(1-x) + 2*x*A(x^2) + 3*x^2*A(x^3).
%e A382314 RELATED SERIES.
%e A382314 The logarithm of the g.f. B(x) for A382126 yields the series
%e A382314 log(B(x)) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + x^5/5 + 18*x^6/6 + x^7/7 + 15*x^8/8 + 13*x^9/9 + 3*x^10/10 + x^11/11 + 58*x^12/12 + ... + a(n-1)*x^n/n + ...
%e A382314 where B(x) = B(x^2)*B(x^3)/(1-x) begins
%e A382314 B(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 + ... + A382126(n)*x^n + ...
%e A382314 so that A(x) = B'(x)/B(x).
%o A382314 (PARI) {a(n) = my(A=1+x +x*O(x^n)); for(i=1,#binary(n), A = 1/(1-x) + 2*x*subst(A,x,x^2) + 3*x^2*subst(A,x,x^3) + x*O(x^n)  ); polcoef(A,n)}
%o A382314 for(n=0,100,print1(a(n),", "))
%Y A382314 Cf. A382126, A072079, A003586, A007814, A007949.
%K A382314 nonn
%O A382314 0,2
%A A382314 _Paul D. Hanna_, Apr 14 2025