A120603 G.f. satisfies: 16*A(x) = 15 + 27*x + A(x)^7, starting with [1,3,21].

1, 3, 21, 399, 9135, 233709, 6400947, 183585897, 5443737390, 165536020650, 5133935821014, 161768728483362, 5164132704296202, 166660621950110526, 5428573285691233650, 178234125351736454070, 5892439158797172244515

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..641

Formula

G.f.: A(x) = 1 + Series_Reversion((1+16*x - (1+x)^7)/27). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(7*n,n)/(6*n+1) * (15+27*x)^(6*n+1)/16^(7*n+1). - Paul D. Hanna, Jan 24 2008

a(n) ~ 7^(-13/12 + 2*n) * 9^n * (-245 + 32*2^(2/3)*7^(5/6))^(1/2 - n) / (2^(8/3) * n^(3/2) * sqrt(Pi)). - Vaclav Kotesovec, Nov 28 2017

D-finite with recurrence: 137781*(7*n + 11)*(7*n + 23)*(7*n + 5)*(7*n + 17)*(7*n - 1)*(7*n + 29)*a(n) + 26254935*(2*n + 5)*(n + 1)*(1029*n^4 + 10290*n^3 + 35035*n^2 + 46550*n + 19951)*a(n + 1) + 72930375*(n + 1)*(n + 2)*(1029*n^4 + 12348*n^3 + 53949*n^2 + 101430*n + 69023)*a(n + 2) + 3970653750*(2*n + 7)*(n + 3)*(n + 2)*(n + 1)*(7*n^2 + 49*n + 82)*a(n + 3) + 1102959375*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(21*n^2 + 168*n + 334)*a(n + 4) + 2573571875*(2*n + 9)*(n + 5)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 5) - 159704067*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)*a(n + 6) = 0. - Robert Israel, Mar 22 2026

Example

A(x) = 1 + 3*x + 21*x^2 + 399*x^3 + 9135*x^4 + 233709*x^5 +...
A(x)^7 = 1 + 21*x + 336*x^2 + 6384*x^3 + 146160*x^4 + 3739344*x^5 +...

Maple

f:= gfun:-rectoproc({137781*(7*n + 11)*(7*n + 23)*(7*n + 5)*(7*n + 17)*(7*n - 1)*(7*n + 29)*a(n) + 26254935*(2*n + 5)*(n + 1)*(1029*n^4 + 10290*n^3 + 35035*n^2 + 46550*n + 19951)*a(n + 1) + 72930375*(n + 1)*(n + 2)*(1029*n^4 + 12348*n^3 + 53949*n^2 + 101430*n + 69023)*a(n + 2) + 3970653750*(2*n + 7)*(n + 3)*(n + 2)*(n + 1)*(7*n^2 + 49*n + 82)*a(n + 3) + 1102959375*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(21*n^2 + 168*n + 334)*a(n + 4) + 2573571875*(2*n + 9)*(n + 5)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 5) - 159704067*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)*a(n + 6), a(0) = 1, a(1) = 3, a(2) = 21, a(3) = 399, a(4) = 9135, a(5) = 233709}, a(n), remember):
map(f, [$0..20]); # Robert Israel, Mar 22 2026

Mathematica

CoefficientList[1 + InverseSeries[Series[(1+16*x - (1+x)^7)/27, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)

Prog

(PARI) {a(n)=local(A=1+3*x+21*x^2+x*O(x^n)); for(i=0, n, A=A+(-16*A+15+27*x+A^7)/9); polcoeff(A, n)}

Crossrefs

Cf. A120588 - A120602, A120604 - A120607.

Sequence in context: A358950 A271570 A084620 * A386649 A336578 A285380

Adjacent sequences: A120600 A120601 A120602 * A120604 A120605 A120606

Keyword

nonn

Author

Paul D. Hanna, Jun 16 2006

Status

approved