cp's OEIS Frontend

A341373 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: [Sum_{n>=0} x^n/(1 - x^(n+1))]^2 = Sum_{n>=0} a(n)*x^n/(1 - x^(n+1))^2.

%I A341373 #12 Feb 15 2021 22:41:18
%S A341373 1,2,5,6,15,6,30,16,34,14,69,12,95,20,51,46,152,8,179,34,90,34,253,26,
%T A341373 210,46,174,52,371,-30,402,128,179,36,254,40,527,78,225,58,647,-56,
%U A341373 673,140,178,38,813,46,600,32,334,104,963,-24,467,180,381,26,1169,-10,1119,120,318,236,649
%N A341373 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: [Sum_{n>=0} x^n/(1 - x^(n+1))]^2 = Sum_{n>=0} a(n)*x^n/(1 - x^(n+1))^2.
%F A341373 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A341373 (1) [ Sum_{n>=0} x^n/(1 - x^(n+1)) ]^2 = Sum_{n>=0} a(n) * x^n / (1 - x^(n+1))^2.
%F A341373 (2) [ Sum_{n>=0} x^n/(1 - x^(n+1)) ]^2 = Sum_{n>=0} (n+1) * x^n * A( x^(n+1) ).
%e A341373 G.f.: A(x) = 1 + 2*x + 5*x^2 + 6*x^3 + 15*x^4 + 6*x^5 + 30*x^6 + 16*x^7 + 34*x^8 + 14*x^9 + 69*x^10 + 12*x^11 + 95*x^12 + 20*x^13 + 51*x^14 + 46*x^15 + ...
%e A341373 such that
%e A341373 D(x)^2 = 1/(1-x)^2 + 2*x/(1-x^2)^2 + 5*x^2/(1-x^3)^2 + 6*x^3/(1-x^4)^2 + 15*x^4/(1-x^5)^2 + 6*x^5/(1-x^6)^2 + 30*x^6/(1-x^7)^2 + ... + a(n)*x^n/(1-x^(n+1))^2 + ...
%e A341373 and
%e A341373 D(x)^2 = A(x) + 2*x*A(x^2) + 3*x^2*A(x^3) + 4*x^3*A(x^4) + 5*x^4*A(x^5) + 6*x^5*A(x^6) + 7*x^6*A(x^7) + ... + (n+1)*x^n*A(x^(n+1)) + ...
%e A341373 where
%e A341373 D(x)^2 = 1 + 4*x + 8*x^2 + 14*x^3 + 20*x^4 + 28*x^5 + 37*x^6 + 44*x^7 + 58*x^8 + 64*x^9 + 80*x^10 + 86*x^11 + 108*x^12 + ... + A055507(n+1)*x^n + ...
%e A341373 D(x) = 1 + 2*x + 2*x^2 + 3*x^3 + 2*x^4 + 4*x^5 + 2*x^6 + 4*x^7 + 3*x^8 + 4*x^9 + 2*x^10 + 6*x^11 + 2*x^12 + ... + A000005(n+1)*x^n + ...
%o A341373 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A341373 A[#A] = polcoeff( sum(n=0,#A, x^n/(1 - x^(n+1) +x*O(x^#A)) )^2 - sum(n=0,#A-1,A[n+1]*x^n/(1 - x^(n+1) + x*O(x^#A))^2 ), #A-1) );A[n+1]}
%o A341373 for(n=0,100,print1(a(n),", "))
%Y A341373 Cf. A341374, A341375, A055507, A000005.
%K A341373 sign
%O A341373 0,2
%A A341373 _Paul D. Hanna_, Feb 11 2021