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URL: https://oeis.org/A093126

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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, 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
1,3
COMMENTS
Not the same as A003237.
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)).
EXAMPLE
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 + ...)).
PROG
(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), ", "))
CROSSREFS
Sequence in context: A282583 A028495 A136752 * A003237 A191519 A165920
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 23 2004
EXTENSIONS
Changed offset to 1 and removed leading zero. - Paul D. Hanna, Aug 16 2016
STATUS
approved