A093126 G.f.: A(x) = x/(1 - x - G001190(x^2)), where G001190 is the g.f. of A001190, the Wedderburn-Etherington numbers (binary rooted trees).
1, 1, 2, 3, 6, 10, 19, 33, 62, 110, 204, 366, 675, 1219, 2239, 4059, 7439, 13518, 24737, 45018, 82304, 149924, 273929, 499290, 911902, 1662787, 3036105, 5537577, 10109364, 18441799, 33663239, 61416729, 112099746, 204536183, 373305550, 681166986, 1243173492, 2268490929, 4140035734, 7554756990, 13787320832, 25159612832, 45915363672
Offset: 1
Keywords
Examples
A(x) = x + x^2 + 2x^3 + 3x^4 + 6x^5 + 10x^6 + 19x^7 + 33x^8 + ... = x/(1-x -(x^2 + x^4 + x^6 + 2x^8 + 3x^10 + 11x^12 + 23x^14 + ...)).
Programs
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PARI
{a(n) = my(A=x,u,v); for(k=2,n,u=A+x*O(x^k); v=subst(u,x,x^2); A-=x^k*polcoeff(u^2 -v*(1+2*u+2*u^2),k+1)/2); polcoeff(A,n)} for(n=1,30,print1(a(n),", "))
Formula
G.f. satisfies the following identities:
(1) A(x^2) = A(x)^2 / (1 + 2*A(x) + 2*A(x)^2),
(2) A(-x) = -A(x) / (1 + 2*A(x)),
(3) A(x) + A(-x) = -2*A(x)*A(-x),
(4) A(x)^2 / (1 + 2*A(x)) = A(x^2) / (1 - 2*A(x^2)).
Extensions
Changed offset to 1 and removed leading zero. - Paul D. Hanna, Aug 16 2016
Comments