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

1,3

Comments

Not the same as A003237.

Links

Table of n, a(n) for n=1..43.

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

Cf. A001190, A003237.

Sequence in context: A282583 A028495 A136752 * A003237 A191519 A165920

Adjacent sequences: A093123 A093124 A093125 * A093127 A093128 A093129

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