OFFSET
0,2
COMMENTS
Self-convolution square root yields the integer sequence A231617.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..800
FORMULA
G.f. A(x) satisfies:
(1) A(x) = exp( x*(4*A(x)^2 - 3*A(x)) + Integral(4*A(x)^2 - 3*A(x) dx) ).
(2) A(x) = (1/x)*Series_Reversion( x*(1-2*x)*(1-6*x)/(1-3*x)^2 ).
(3) A(x) = 1 + x*A(x)*(4*A(x) - 3)*(2 - 3*x*A(x)).
(4) A(x) = 1 + Sum_{n>=2} 3^(n-2) * n * x^(n-1) * A(x)^n.
Recurrence: 16*n*(n+1)*(109*n-135)*a(n) = 8*n*(3706*n^2 - 6443*n + 2403)*a(n-1) - 9*(5341*n^3 - 17297*n^2 + 15870*n - 3744)*a(n-2) + 162*(n-3)*(2*n-3)*(109*n-26)*a(n-3). - Vaclav Kotesovec, Dec 29 2013
a(n) ~ 2^(2*n-7/2) * 3^(5*n-1) / (n^(3/2) * sqrt(Pi*s) * r^n), where r = 63.59903834580526343... is the root of the equation -45349632 + 793152*r - 1323*r^2 + r^3 = 0 and s = 0.002323998891900610500... is the root of the equation -109 + 37278*s + 3860055*s^2 + 120932352*s^3 = 0. - Vaclav Kotesovec, Dec 29 2013
EXAMPLE
G.f.: A(x) = 1 + 2*x + 17*x^2 + 166*x^3 + 1790*x^4 + 20604*x^5 +...
Related expansions.
(1 - 3*x*A(x))^2 = 1 - 6*x - 3*x^2 - 66*x^3 - 654*x^4 - 7140*x^5 -...
4*A(x) - 3 = 1 + 8*x + 68*x^2 + 664*x^3 + 7160*x^4 + 82416*x^5 +...
4*A(x)^2 - 3*A(x) = 1 + 10*x + 101*x^2 + 1102*x^3 + 12762*x^4 +...
log(A(x)) = 2*x + 30*x^2/2 + 404*x^3/3 + 5510*x^4/4 + 76572*x^5/5 +...
The square root of the g.f. yields an integer series (A231617):
sqrt(A(x)) = 1 + x + 8*x^2 + 75*x^3 + 788*x^4 + 8914*x^5 + 106006*x^6 +...
MATHEMATICA
CoefficientList[1/x*InverseSeries[Series[x*(1-2*x)*(1-6*x)/(1-3*x)^2, {x, 0, 20}], x], x] (* Vaclav Kotesovec, Dec 29 2013 *)
PROG
(PARI) {a(n)=polcoeff((serreverse(x*(1-2*x)*(1-6*x)/(1-3*x)^2 +x^2*O(x^n))/x), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); for(i=1, n, A=exp(x*(4*A^2-3*A)+intformal(4*A^2-3*A +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); for(i=1, n, A=1+x*A*(4*A-3)*(2-3*x*A) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 11 2013
STATUS
approved