A273162 G.f. A(x) satisfies: A(x*B(x)) = x^2 - x^3, where A(B(x)) = x.

1, 1, 3, 8, 28, 95, 351, 1309, 5056, 19787, 78847, 317705, 1294673, 5321598, 22047985, 91957296, 385832452, 1627351862, 6896087775, 29345806842, 125353612440, 537303633158, 2310270577888, 9962069922553, 43070357677938, 186663591654655, 810799482641934, 3529141491880136, 15390864728209348

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..515

Formula

G.f. A(x) satisfies: A(x*A(x)) = A(x)^2 - A(x)^3.

G.f.: A(x) = Product_{n>=0} F(n), where F(0) = x, F(1) = 1-x, and F(n+1) = 1 - (1 - F(n))^2 * F(n) for n > 0. - Paul D. Hanna, Dec 30 2021

Example

G.f.: A(x) = x + x^2 + 3*x^3 + 8*x^4 + 28*x^5 + 95*x^6 + 351*x^7 + 1309*x^8 + 5056*x^9 + 19787*x^10 + 78847*x^11 + 317705*x^12 +...
such that A(x*B(x)) = x^2 - x^3, where A(B(x)) = x.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 22*x^5 + 81*x^6 + 294*x^7 + 1124*x^8 + 4338*x^9 + 17140*x^10 + 68476*x^11 + 277229*x^12 + 1132716*x^13 + 4669477*x^14 +...
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 43*x^6 + 168*x^7 + 648*x^8 + 2574*x^9 + 10284*x^10 + 41691*x^11 + 170333*x^12 + 702273*x^13 + 2915100*x^14 +...
A(x*A(x)) = A(x)^2 - A(x)^3 where
A(x)^2 - A(x)^3 = x^2 + x^3 + 4*x^4 + 10*x^5 + 38*x^6 + 126*x^7 + 476*x^8 + 1764*x^9 + 6856*x^10 + 26785*x^11 + 106896*x^12 + 430443*x^13 + 1754377*x^14 +...
Let B(x) be the series reversion of g.f. A(x) so that A(B(x)) = x, then
B(x) = x - x^2 - x^3 + 2*x^4 - 2*x^5 + 3*x^6 - x^7 - 7*x^8 + 10*x^9 + 2*x^10 - 15*x^11 + 2*x^12 + 34*x^13 - 51*x^14 + 17*x^15 + 73*x^16 - 218*x^17 + 323*x^18 - 135*x^19 - 467*x^20 + 1139*x^21 - 1279*x^22 + 430*x^23 + 1587*x^24 +...
such that A(x*B(x)) = x^2 - x^3,
also, B(x) = B(x^2 - x^3)/x.
From Paul D. Hanna, Dec 30 2021: (Start)
GENERATING METHOD.
Define F(n), a polynomial in x of order 3^(n-1), by the following recurrence:
F(0) = x,
F(1) = (1 - x),
F(2) = (1 - x^2 * (1-x)),
F(3) = (1 - x^4 * (1-x)^2 * F(2)),
F(4) = (1 - x^8 * (1-x)^4 * F(2)^2 * F(3)),
F(5) = (1 - x^16 * (1-x)^8 * F(2)^4 * F(3)^2 * F(4)),
...
F(n+1) = 1 - (F(n) - 1)^2 * F(n)
...
Then the series reversion B(x) equals the infinite product:
B(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
that is,
B(x) = x * (1-x) * (1 - x^2*(1-x)) * (1 - x^4*(1-x)^2*(1 - x^2*(1-x))) * (1 - x^8*(1-x)^4*(1 - x^2*(1-x))^2*(1 - x^4*(1-x)^2*(1 - x^2*(1-x)))) * ...
(End)

Prog

(PARI) {a(n) = my(A=[1, 1], F, B); for(i=1, n, A=concat(A, 0); F=x*Ser(A); B=serreverse(F); A[#A] = Vec(subst(F, x, x*B))[#A]); A[n]}
for(n=1, 50, print1(a(n), ", "))
(PARI) /* Using Infinite Product Formula for Series Reversion */
{F(n) = my(G=x); if(n==0, G=x, if(n==1, G=1-x, G = 1 - (F(n-1) - 1)^2 * F(n-1) )); G}
{a(n) = my(A, B = prod(k=0, #binary(n), F(k) +x*O(x^n)));
A = serreverse(B); polcoeff(A, n)}
for(n=1, 50, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 30 2021

Crossrefs

Cf. A273203, A273095.

Sequence in context: A148856 A151482 A241585 * A148857 A148858 A148859

Adjacent sequences: A273159 A273160 A273161 * A273163 A273164 A273165

Keyword

nonn

Author

Paul D. Hanna, May 16 2016

Status

approved