A275757 G.f. satisfies: A(x) = x + A( A(x)^3 - A(x)^7 ), an odd function.

1, 1, 3, 11, 46, 207, 977, 4767, 23835, 121424, 627747, 3284055, 17348254, 92387544, 495371637, 2671588333, 14480158111, 78822638280, 430685654483, 2361012092488, 12980509646385, 71547277918984, 395252428706918, 2187886348193235, 12132382884810469, 67383306100049693, 374771558921409855, 2086989709106321626, 11634599273439782284, 64923785744439199536, 362598744217074249165, 2026617482659866472677

Offset: 1

Author

Paul D. Hanna, Aug 20 2016

Keywords

• sign

Comments

Compare to: F(x) = x + F( F(x)^3 - F(x)^9 ) holds when F(x) = x + F(x)^3 is a g.f. of the ternary tree numbers (A001764).

Compare to: C(x) = x + C( C(x)^2 - C(x)^4 ) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

First negative term is a(83). Georg Fischer, Feb 15 2019

Examples

			G.f.: A(x) = x + x^3 + 3*x^5 + 11*x^7 + 46*x^9 + 207*x^11 + 977*x^13 + 4767*x^15 + 23835*x^17 + 121424*x^19 + 627747*x^21 + 3284055*x^23 + 17348254*x^25 +...
such that A(x) = x + A( A(x)^3 - A(x)^7 ).
RELATED SERIES.
A(x)^3 = x^3 + 3*x^5 + 12*x^7 + 52*x^9 + 240*x^11 + 1155*x^13 + 5727*x^15 + 29034*x^17 + 149727*x^19 + 782627*x^21 + 4135668*x^23 + 22051158*x^25 +...
A(x)^7 = x^7 + 7*x^9 + 42*x^11 + 238*x^13 + 1323*x^15 + 7308*x^17 + 40327*x^19 + 222804*x^21 + 1233624*x^23 + 6847281*x^25 + 38102099*x^27 +...
A(x^3 - x^7) = x^3 - x^7 + x^9 - 3*x^13 + 3*x^15 + 3*x^17 - 15*x^19 + 10*x^21 + 30*x^23 - 77*x^25 + 16*x^27 + 231*x^29 - 399*x^31 - 178*x^33 + 1653*x^35 - 1892*x^37 - 2887*x^39 +...
where Series_Reversion(A(x)) = x - A(x^3 - x^7).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A275755, A275756, A275758, A179486.

Programs

PARI

{a(n) = my(A=x); for(i=1, 2*n, A = x + subst(A, x, A^3 - A^7 +x*O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. satisfies:

(1) A(x - A(x^3 - x^7)) = x.

(2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^3-x^7)^(n+1) / (n+1)!.

(3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^3-x^7)^(n+1)/x / (n+1)! ).