A120602 G.f. satisfies: 31*A(x) = 30 + 125*x + A(x)^6, starting with [1,5,15].

1, 5, 15, 190, 2550, 38070, 609205, 10199640, 176483340, 3130904150, 56641633455, 1040985874470, 19381240377460, 364777461207360, 6929053224018750, 132665646902812800, 2557591625106894075, 49604907701733017850, 967242234362414552850, 18950004748051829487750, 372848735641925952297900

Offset

0,2

Comments

See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.

Links

Robert Israel, Table of n, a(n) for n = 0..754

Formula

G.f.: A(x) = 1 + Series_Reversion((1+31*x - (1+x)^6)/125). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(6*n,n)/(5*n+1) * (30+125*x)^(5*n+1)/31^(6*n+1). - Paul D. Hanna, Jan 24 2008

a(n) ~ 5^(-1/2 + 3*n) * (-30 + 5*(31/6)^(6/5))^(1/2 - n) / (2^(3/5) * 3^(1/10) * 31^(2/5) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

D-finite with recurrence: -5625000*(6*n - 1)*(3*n + 2)*(2*n + 3)*(3*n + 7)*(6*n + 19)*a(n) - 2700000*(n + 1)*(1620*n^4 + 12960*n^3 + 36315*n^2 + 41580*n + 16024)*a(n + 1) - 43740000*(2*n + 5)*(n + 2)*(n + 1)*(24*n^2 + 120*n + 137)*a(n + 2) - 6998400*(n + 1)*(n + 2)*(n + 3)*(72*n^2 + 432*n + 641)*a(n + 3) - 30233088*(2*n + 7)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 4) + 4197653*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*a(n + 5) = 0. - Robert Israel, Mar 22 2026

Example

A(x) = 1 + 5*x + 15*x^2 + 190*x^3 + 2550*x^4 + 38070*x^5 +...
A(x)^6 = 1 + 30*x + 465*x^2 + 5890*x^3 + 79050*x^4 + 1180170*x^5 +...

Maple

f:= gfun:-rectoproc({-5625000*(6*n - 1)*(3*n + 2)*(2*n + 3)*(3*n + 7)*(6*n + 19)*a(n) - 2700000*(n + 1)*(1620*n^4 + 12960*n^3 + 36315*n^2 + 41580*n + 16024)*a(n + 1) - 43740000*(2*n + 5)*(n + 2)*(n + 1)*(24*n^2 + 120*n + 137)*a(n + 2) - 6998400*(n + 1)*(n + 2)*(n + 3)*(72*n^2 + 432*n + 641)*a(n + 3) - 30233088*(2*n + 7)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 4) + 4197653*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*a(n + 5), a(0) = 1, a(1) = 5, a(2) = 15, a(3) = 190, a(4) = 2550}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 22 2026

Mathematica

CoefficientList[1 + InverseSeries[Series[(1+31*x - (1+x)^6)/125, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)

Prog

(PARI) {a(n)=local(A=1+5*x+15*x^2+x*O(x^n)); for(i=0, n, A=A+(-31*A+30+125*x+A^6)/25); polcoeff(A, n)}

Crossrefs

Cf. A120588 - A120601, A120603 - A120607.

Sequence in context: A174850 A143048 A261843 * A004130 A088869 A134715

Adjacent sequences: A120599 A120600 A120601 * A120603 A120604 A120605

Keyword

nonn

Author

Paul D. Hanna, Jun 16 2006

Extensions

More terms from Robert Israel, Mar 22 2026

Status

approved