A295810 G.f. A(x) satisfies: A(x)^3 = 1+x + x*(A(x)^5 + A(x)^7).

1, 1, 3, 16, 97, 645, 4539, 33242, 250715, 1934131, 15190377, 121050779, 976334857, 7954909796, 65378035310, 541346845867, 4511820592102, 37819912868231, 318639423484669, 2696819879180630, 22918021992024063, 195480539889732302, 1672951468057552136, 14361120982757852887, 123625070638172667688, 1066939084577136857174

Offset: 0

Author

Paul D. Hanna, Nov 27 2017

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 97*x^4 + 645*x^5 + 4539*x^6 + 33242*x^7 + 250715*x^8 + 1934131*x^9 + 15190377*x^10 + 121050779*x^11 + 976334857*x^12 +...
such that A(x)^3 = 1+x + x*(A(x)^5 + A(x)^7).
RELATED SERIES.
A(x)^3 = 1 + 3*x + 12*x^2 + 67*x^3 + 423*x^4 + 2880*x^5 + 20607*x^6 + 152763*x^7 + 1162908*x^8 + 9037195*x^9 + 71398917*x^10 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 150*x^3 + 990*x^4 + 6936*x^5 + 50640*x^6 + 381070*x^7 + 2934665*x^8 + 23016905*x^9 + 183216323*x^10 +...
A(x)^7 = 1 + 7*x + 42*x^2 + 273*x^3 + 1890*x^4 + 13671*x^5 + 102123*x^6 + 781838*x^7 + 6102530*x^8 + 48382012*x^9 + 388548244*x^10 +...
Series_Reversion(A(x) - 1) = x - 3*x^2 + 2*x^3 + 8*x^4 - 21*x^5 + 4*x^6 + 74*x^7 - 137*x^8 - 59*x^9 + 623*x^10 - 797*x^11 - 1083*x^12 + 4840*x^13 - 3793*x^14 - 12355*x^15 +...
which equals x/(1 + 3*x + 7*x^2 + 7*x^3 + 4*x^4 + x^5).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..520

Crossrefs

Cf. A295808. A259757, A295537, A295538.

Programs

PARI

{a(n) = my(A=1+x); for(i=1, n, A = ((1+x) + x*A^5 + x*A^7 +x*O(x^n))^(1/3) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n) = my(A=1+x); A = 1 + serreverse( x/(1 + 3*x + 7*x^2 + 7*x^3 + 4*x^4 + x^5 +x*O(x^n)) ); polcoeff(A,n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:

(1) A(x) = 1 + Series_Reversion( x/(1 + 3*x + 7*x^2 + 7*x^3 + 4*x^4 + x^5) ).

(2) F(A(x)) = x such that F(x) = -(1-x)/(1 - x + x^3 - x^4 + x^5).