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A112574 G.f. A(x) satisfies: A(x)^2 equals the g.f. of A110649, which consists entirely of numbers 1 through 12.

%I A112574 #8 Mar 10 2015 02:12:20
%S A112574 1,3,0,2,0,3,-8,30,-90,290,-930,3000,-9696,31461,-102420,334467,
%T A112574 -1095510,3598464,-11852026,39136629,-129548493,429817733,-1429178703,
%U A112574 4761992751,-15898024868,53174651133,-178168302693,597971203902,-2010093276240,6767100270918
%N A112574 G.f. A(x) satisfies: A(x)^2 equals the g.f. of A110649, which consists entirely of numbers 1 through 12.
%C A112574 A110649 is formed from every 2nd term of A084067, which also consists entirely of numbers 1 through 12.
%C A112574 Why are so many a(n) divisible by 3, i.e., 22 of the first 28? - _Jonathan Vos Post_, Sep 14 2005
%F A112574 G.f. A(x) satisfies: A(x)^4 (mod 8) = g.f. of A084067.
%e A112574 A(x) = 1 + 3*x + 2*x^3 + 3*x^5 - 8*x^6 + 30*x^7 - 90*x^8 +..
%e A112574 A(x)^2 = 1 + 6*x + 9*x^2 + 4*x^3 + 12*x^4 + 6*x^5 +...
%e A112574 A(x)^4 = 1 + 12*x + 54*x^2 + 116*x^3 + 153*x^4 + 228*x^5 +..
%e A112574 A(x)^4 (mod 8) = 1 + 4*x + 6*x^2 + 4*x^3 + x^4 + 4*x^5 +...
%e A112574 G(x) = 1 + 12*x + 6*x^2 + 4*x^3 + 9*x^4 + 12*x^5 + 4*x^6 +..
%e A112574 where G(x) is the g.f. of A084067.
%o A112574 (PARI) {a(n)=local(d=2,m=12,A=1+m*x); for(j=2,d*n, for(k=1,m,t=polcoeff((A+k*x^j+x*O(x^j))^(1/m),j); if(denominator(t)==1,A=A+k*x^j;break))); polcoeff(Ser(vector(n+1,i,polcoeff(A,d*(i-1))))^(1/2),n)}
%Y A112574 Cf. A110649, A084067.
%K A112574 sign
%O A112574 0,2
%A A112574 _Paul D. Hanna_, Sep 14 2005