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..621
FORMULA
G.f.: A(x) = 1 + Series_Reversion((1+24*x - (1+x)^8)/64). Lagrange Inversion yields: G.f.: A(x) = Sum_{n>=0} C(8*n,n)/(7*n+1) * (23+64*x)^(7*n+1)/24^(8*n+1). - Paul D. Hanna, Jan 24 2008
a(n) ~ 4^(-1 + 3*n) * (-23 + 21*3^(1/7))^(1/2 - n) / (3^(3/7) * n^(3/2) * sqrt(7*Pi)). - Vaclav Kotesovec, Nov 28 2017
D-finite with recurrence: -2097152*(4*n + 3)*(2*n + 5)*(4*n + 17)*(8*n - 1)*(8*n + 13)*(8*n + 27)*(8*n + 41)*a(n) - 1507328*(n + 1)*(458752*n^6 + 8257536*n^5 + 58992640*n^4 + 212459520*n^3 + 402661168*n^2 + 375323424*n + 131542875)*a(n + 1) - 91004928*(2*n + 7)*(n + 2)*(n + 1)*(4096*n^4 + 57344*n^3 + 286976*n^2 + 603904*n + 449895)*a(n + 2) - 109016320*(n + 1)*(n + 2)*(n + 3)*(4096*n^4 + 65536*n^3 + 386816*n^2 + 997376*n + 947067)*a(n + 3) - 2507375360*(2*n + 9)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*(32*n^2 + 288*n + 631)*a(n + 4) - 1081305624*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(32*n^2 + 320*n + 797)*a(n + 5) - 2072502446*(2*n + 11)*(n + 6)*(n + 5)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 6) + 124902511*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)*(n + 7)*a(n + 7) = 0. - Robert Israel, Mar 22 2026
EXAMPLE
A(x) = 1 + 4*x + 28*x^2 + 616*x^3 + 15820*x^4 + 453208*x^5 +...
A(x)^8 = 1 + 32*x + 672*x^2 + 14784*x^3 + 379680*x^4 + 10876992*x^5 +...
MAPLE
f:= gfun:-rectoproc({-2097152*(4*n + 3)*(2*n + 5)*(4*n + 17)*(8*n - 1)*(8*n + 13)*(8*n + 27)*(8*n + 41)*a(n) - 1507328*(n + 1)*(458752*n^6 + 8257536*n^5 + 58992640*n^4 + 212459520*n^3 + 402661168*n^2 + 375323424*n + 131542875)*a(n + 1) - 91004928*(2*n + 7)*(n + 2)*(n + 1)*(4096*n^4 + 57344*n^3 + 286976*n^2 + 603904*n + 449895)*a(n + 2) - 109016320*(n + 1)*(n + 2)*(n + 3)*(4096*n^4 + 65536*n^3 + 386816*n^2 + 997376*n + 947067)*a(n + 3) - 2507375360*(2*n + 9)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*(32*n^2 + 288*n + 631)*a(n + 4) - 1081305624*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(32*n^2 + 320*n + 797)*a(n + 5) - 2072502446*(2*n + 11)*(n + 6)*(n + 5)*(n + 4)*(n + 3)*(n + 2)*(n + 1)*a(n + 6) + 124902511*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n + 5)*(n + 6)*(n + 7)*a(n + 7), a(0) = 1, a(1) = 4, a(2) = 28, a(3) = 616, a(4) = 15820, a(5) = 453208, a(6) = 13894552}, a(n), remember):
map(f, [$0..20]); # Robert Israel, Mar 22 2026
MATHEMATICA
CoefficientList[1 + InverseSeries[Series[(1+24*x - (1+x)^8)/64, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Nov 28 2017 *)
PROG
(PARI) {a(n)=local(A=1+4*x+28*x^2+x*O(x^n)); for(i=0, n, A=A+(-24*A+23+64*x+A^8)/16); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 16 2006
STATUS
approved