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A273162 G.f. A(x) satisfies: A(x*B(x)) = x^2 - x^3, where A(B(x)) = x. 3
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, 67241645122541334, 294268407263148650, 1289830017807048396, 5661906416526228762, 24888336664147079838 (list; graph; refs; listen; history; text; internal format)
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.

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.

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), ", "))

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

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Last modified March 26 23:02 EDT 2019. Contains 321565 sequences. (Running on oeis4.)