OFFSET
0,6
COMMENTS
Compare to Ramanujan's g- and G-functions, which are related by:
G = ( ( g^8 + sqrt(g^16 + 1/g^8) )/2 )^(1/8), and
g = ( ( G^8 + sqrt(G^16 - 1/G^8) )/2 )^(1/8),
where G^8 * g^8 * (G^8 - g^8) = 1/4.
One may explore a similar relation between some functions A and B given by
A = ( ( B^8 + sqrt(B^16 + R/B^8) )/2 )^(1/8), and
B = ( ( A^8 + sqrt(A^16 - R/A^8) )/2 )^(1/8),
where R = 4 * A^8 * B^8 * (A^8 - B^8) is an arbitrary value. If we set A = A(x) and B = A(-x), R is an odd function for which we have chosen R = 64*x as the simplest case that generates integer coefficients in A(x). After choosing this value for R, the resulting relation describes an infinite number of functions A(x); thus, we imposed the additional constraint A(-x) = 2 - A(x) to arrive at the unique solution A(x) specified by this sequence.
LINKS
Eric Weisstein's World of Mathematics, Ramanujan g- and G-Functions.
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) A(x) = ( ( A(-x)^8 + sqrt(A(-x)^16 + 64*x/A(-x)^8) )/2 )^(1/8).
(2) A(-x) = ( ( A(x)^8 + sqrt(A(x)^16 - 64*x/A(x)^8) )/2 )^(1/8).
(3) A(-x) = 2 - A(x).
(4) A(x)^8 * A(-x)^8 * (A(x)^8 - A(-x)^8) = 16*x.
(5) A(x) = 1 + sqrt(1 - A(x)*A(-x)).
(6) A(x) = 1 + Series_Reversion( x*(1 - x^2)^8*(1 + x^2)*(1 + 6*x^2 + x^4) ).
(7) x = A(x)^8 * A(-x)^8 * (A(x) - 1) * (A(x)^2 + A(-x)^2)/2 * (A(x)^4 + A(-x)^4)/2.
(8) x = A(x)^8 * (2 - A(x))^8 * (A(x) - 1) * (2 - 2*A(x) + A(x)^2) * (8 - 16*A(x) + 12*A(x)^2 - 4*A(x)^3 + A(x)^4).
EXAMPLE
G.f.: A(x) = 1 + x + x^3 + 24*x^5 + 95*x^7 + 2699*x^9 + 15803*x^11 + 426524*x^13 + 3152930*x^15 + 78893000*x^17 + 687247207*x^19 + ...
where A(x)^8 = (A(-x)^8 + sqrt(A(-x)^16 + 64*x/A(-x)^8))/2.
Related series.
Series_Reversion(A(x) - 1) = x * (1 - x^2)^8 * (1 + x^2) * (1 + 6*x^2 + x^4) = x - x^3 - 21*x^5 + 85*x^7 - 134*x^9 + 70*x^11 + 70*x^13 - 134*x^15 + 85*x^17 - 21*x^19 - x^21 + x^23.
A(x)^8 = 1 + 8*x + 28*x^2 + 64*x^3 + 126*x^4 + 416*x^5 + 1680*x^6 + 5248*x^7 + 13973*x^8 + 53008*x^9 + 224092*x^10 + ... + A353325(n)*x^n + ...
A(x)^16 = 1 + 16*x + 120*x^2 + 576*x^3 + 2060*x^4 + 6432*x^5 + 21168*x^6 + 76800*x^7 + 275118*x^8 + 943344*x^9 + 3346960*x^10 + ...
sqrt(A(x)^16 - 64*x/A(x)^8) = 1 - 24*x + 28*x^2 - 192*x^3 + 126*x^4 - 1248*x^5 + 1680*x^6 - 15744*x^7 + 13973*x^8 - 159024*x^9 + 224092*x^10 + ...
A(x)^8 - A(-x)^8 = 16*x + 128*x^3 + 832*x^5 + 10496*x^7 + 106016*x^9 + 1536256*x^11 + 18183424*x^13 + 277839360*x^15 + 3583519472*x^17 + ...
A(x) * A(-x) = 1 - x^2 - 2*x^4 - 49*x^6 - 238*x^8 - 6164*x^10 - 41564*x^12 - 1023231*x^14 - 8430262*x^16 - 194852183*x^18 - ...
A(x)^8 * A(-x)^8 = 1 - 8*x^2 + 12*x^4 - 336*x^6 + 686*x^8 - 38896*x^10 + 48064*x^12 - 6011040*x^14 + 132853*x^16 - 1072398368*x^18 + ...
PROG
(PARI) {a(n) = my(A=1+x, B=1-x); for(i=1, n,
A = ((B^8 + sqrt(B^16 + 64*x/B^8 +x*O(x^n) ))/2)^(1/8);
B = ((A^8 + sqrt(A^16 - 64*x/A^8 +x*O(x^n) ))/2)^(1/8);
B = 1 - (A - B)/2; ); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) /* Series Reversion formula */
{a(n) = my(A = 1 + serreverse(x*(1 - x^2)^8*(1 + x^2)*(1 + 6*x^2 + x^4) +x^2*O(x^n) )); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 13 2022
STATUS
approved