A259117 G.f. A(x) satisfies: A(x)^2 = A( x^2*(1+x)/(1-x) ), with A(0) = 0.

1, 1, 1, 2, 3, 5, 9, 18, 38, 80, 166, 342, 705, 1463, 3065, 6492, 13917, 30217, 66455, 147920, 332700, 754544, 1721596, 3943344, 9051155, 20789895, 47741587, 109543782, 251076670, 574820868, 1314663050, 3004304786, 6861785924, 15668223170, 35778071864, 81723401730, 186773111883, 427182189689, 977960176139

Offset

1,4

Comments

Radius of convergence is r = sqrt(2)-1, where r = r^2*(1+r)/(1-r), with A(r) = 1.

Compare to a g.f. M(x) of Motzkin numbers: M(x)^2 = M(x^2/(1-2*x)) where M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x).

Links

Paul D. Hanna, Table of n, a(n) for n = 1..520

Example

G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 9*x^7 + 18*x^8 + 38*x^9 + 80*x^10 + 166*x^11 + 342*x^12 + 705*x^13 + 1463*x^14 + 3065*x^15 + 6492*x^16 +...
such that A(x^2*(1+x)/(1-x)) = A(x)^2, where
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 11*x^6 + 20*x^7 + 38*x^8 + 76*x^9 + 159*x^10 + 338*x^11 + 719*x^12 + 1526*x^13 + 3235*x^14 + 6868*x^15 + 14638*x^16 +...
RELATED SERIES.
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 + 5*x^5 - 12*x^6 + 29*x^7 - 75*x^8 + 201*x^9 - 542*x^10 + 1480*x^11 - 4112*x^12 + 11562*x^13 - 32770*x^14 + 93593*x^15 +...
where
B(x^2) = B(x)^2*(1 + B(x))/(1 - B(x)).

Prog

(PARI) {a(n) = my(A=x); for(i=1, n, A = sqrt( subst(A, x, x^2*(1+x)/(1-x +x*O(x^n))) ) ); polcoeff(A, n)}
for(n=1, 50, print1(a(n), ", "))

Crossrefs

Sequence in context: A047121 A363307 A182080 * A096753 A022862 A292541

Adjacent sequences: A259114 A259115 A259116 * A259118 A259119 A259120

Keyword

nonn

Author

Paul D. Hanna, Nov 08 2015

Status

approved