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A264228 G.f. A(x) satisfies: A(x)^3 = A( x^3/(1-3*x) ), with A(0) = 0. 2
1, 1, 2, 5, 13, 35, 97, 274, 785, 2275, 6656, 19630, 58295, 174175, 523238, 1579584, 4789919, 14584723, 44577799, 136732988, 420784888, 1298937282, 4021383654, 12483820395, 38853994422, 121220646116, 379062880051, 1187912517953, 3730305167438, 11736596024002, 36994041916973, 116807229667919, 369415244627269, 1170113816365089 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Radius of convergence is r = (sqrt(13) - 3)/2, where r = r^3/(1-3*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

Table of n, a(n) for n=1..34.

EXAMPLE

G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 35*x^6 + 97*x^7 + 274*x^8 + 785*x^9 + 2275*x^10 + 6656*x^11 + 19630*x^12 + 58295*x^13 + 174175*x^14 +...

where A(x)^3 = A( x^3/(1-3*x) ).

RELATED SERIES.

A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 28*x^6 + 87*x^7 + 270*x^8 + 839*x^9 + 2610*x^10 + 8127*x^11 + 25331*x^12 + 79035*x^13 + 246852*x^14 + 771808*x^15 +...

A( x/(1+x+x^2) ) = x + x^4 + 2*x^7 + 6*x^10 + 22*x^13 + 88*x^16 + 367*x^19 + 1570*x^22 + 6843*x^25 + 30271*x^28 + 135530*x^31 + 612852*x^34 + 2794412*x^37 + 12832472*x^40 +...

Let B(x) = x/Series_Reversion(A(x)), then A(x) = x*B(A(x)), where

B(x) = 1 + x + x^2 + x^3 - x^5 - x^6 + 2*x^8 + 3*x^9 - 6*x^11 - 9*x^12 + 20*x^14 + 30*x^15 - 71*x^17 - 110*x^18 + 267*x^20 + 419*x^21 - 1041*x^23 +...

Let C0(x) and C2(x) be series trisections of B(x), B(x) = C0(x) + x + C2(x):

C0(x) = 1 + x^3 - x^6 + 3*x^9 - 9*x^12 + 30*x^15 - 110*x^18 + 419*x^21 - 1648*x^24 + 6652*x^27 - 27369*x^30 + 114384*x^33 - 484276*x^36 +...

C2(x) =  x^2 - x^5 + 2*x^8 - 6*x^11 + 20*x^14 - 71*x^17 + 267*x^20 - 1041*x^23 + 4168*x^26 - 17047*x^29 + 70902*x^32 - 298967*x^35 + 1275141*x^38 +...

then C2(x) = x^2/C0(x);

further, C2(A(x)) / A(x) = A(x) / C0(A(x)) = M(x), where M(x) is a g.f. of Motzkin numbers: M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x).

PROG

(PARI)  {a(n) = my(A=x); for(i=1, n, A = ( subst(A, x, x^3/(1-3*x +x*O(x^n))) )^(1/3) ); polcoeff(A, n)}

for(n=1, 40, print1(a(n), ", "))

CROSSREFS

Cf. A264229, A264230.

Sequence in context: A005773 A022855 A091190 * A007689 A085281 A082582

Adjacent sequences:  A264225 A264226 A264227 * A264229 A264230 A264231

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 08 2015

STATUS

approved

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Last modified February 25 10:54 EST 2018. Contains 299653 sequences. (Running on oeis4.)