A271934 G.f. A(x) satisfies: A(x) = A( x^3 + 6*x*A(x)^3 )^(1/3), with A(0)=0, A'(0)=1.

1, 2, 8, 38, 196, 1064, 5988, 34632, 204672, 1231082, 7514052, 46433088, 289976404, 1827459072, 11608240000, 74249294704, 477826080368, 3091718252320, 20101537759256, 131262924427560, 860504352317040, 5661120688863216, 37363827222888640, 247331149667685440, 1641642515512685408, 10923380539408947456, 72850297774044995328, 486886413558080754198, 3260469757311730139044, 21874082006618739609864

Offset

1,2

Comments

Compare the g.f. to the following identity:

C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),

where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Links

Table of n, a(n) for n=1..30.

Formula

G.f. A(x) satisfies: A( B(x) - 2*x^3/B(x) ) = x, where B(x) is described in the examples.

Example

G.f.: A(x) = x + 2*x^2 + 8*x^3 + 38*x^4 + 196*x^5 + 1064*x^6 + 5988*x^7 + 34632*x^8 + 204672*x^9 + 1231082*x^10 + 7514052*x^11 + 46433088*x^12 +...
where A(x)^3 = A( x^3 + 6*x*A(x)^3 ).
RELATED SERIES.
A(x)^3 = x^3 + 6*x^4 + 36*x^5 + 218*x^6 + 1332*x^7 + 8208*x^8 + 50984*x^9 + 319056*x^10 + 2010528*x^11 + 12750950*x^12 + 81348948*x^13 + 521839944*x^14 + 3364421812*x^15 + 21791976192*x^16 +...
Series reversion of A(x) equals B(x) - 2*x^3/B(x) where
B(x) = x + 2*x^4 - 20*x^7 + 302*x^10 - 5436*x^13 + 108072*x^16 - 2286160*x^19 + 50475256*x^22 - 1149822240*x^25 + 26825146770*x^28 - 637700980612*x^31 + 15391872726072*x^34 - 376193675011356*x^37 + 9291840570002312*x^40 +...

Prog

(PARI) {a(n) = my(A=x+x^2, X=x+x*O(x^n)); for(i=1, n, A = subst(A, x, x^3 + 6*X*A^3)^(1/3) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A271931.

Sequence in context: A026939 A291088 A047098 * A364723 A266797 A369208

Adjacent sequences: A271931 A271932 A271933 * A271935 A271936 A271937

Keyword

nonn

Author

Paul D. Hanna, Apr 16 2016

Status

approved