OFFSET
0,5
COMMENTS
Apparently: Number of Dyck n-paths with each ascent length being a square number. [David Scambler, May 09 2012]
FORMULA
G.f. satisfies: A(x) = B(x*A(x)) where B(x) = A(x/B(x)) = Sum_{n>=0} x^(n^2),
where a(n) = [x^n] B(x)^(n+1)/(n+1) for n>=0.
G.f.: A(x) = (1/x)*Series_Reversion( x / Sum_{n>=0} x^(n^2) ).
From Paul D. Hanna, Apr 24 2010: (Start)
SPECIAL VALUES:
. at x = 2*exp(-Pi)/(1+Pi^(1/4)/gamma(3/4)) = 0.04142369369176926261...
. A(x) = B(exp(-Pi)) = (1+Pi^(1/4)/gamma(3/4))/2 = 1.043217405606654...
RADIUS OF CONVERGENCE r:
. at r = 0.3529672118496605771445592553666318566205464502456806...,
. A(r) = 1.9530374869760035836323161721583051467541841357702661...,
where r and A(r) are given by:
. r = z/B(z) and
. A(r) = B(z) = Sum_{n>=0} z^(n^2)
such that z is the real root nearest the origin that satisfies:
. B(z) - z*B'(z) = 0, which has solution:
. z = 0.689358196415787767209694723600383373645983284157633311584643...
Here, B(z) = Sum_{n>=0} z^(n^2), the partial Jacobi theta_3 function.
(End)
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 6*x^5 + 16*x^6 + 36*x^7 +...
A(x)^4 = 1 + 4*x + 10*x^2 + 20*x^3 + 39*x^4 + 88*x^5 + 228*x^6 +...
A(x)^9 = 1 + 9*x + 45*x^2 + 165*x^3 + 504*x^4 + 1404*x^5 +...
A(x)^16 = 1 + 16*x + 136*x^2 + 816*x^3 + 3892*x^4 + 15824*x^5 +...
where
A(x) = 1 + x*A(x) + x^4*A(x)^4 + x^9*A(x)^9 + x^16*A(x)^16 +...
A(x) = (1/x)*Series_Reversion(x/(1 + x + x^4 + x^9 + x^16 +...)).
MATHEMATICA
f[x_, y_, d_] := f[x, y, d] = If[x < 0 || y < x, 0, If[x == 0 && y == 0, 1, f[x-1, y, 0] + f[x, y - If[d == 0, 1, Sqrt[d]*2 + 1], If[d == 0, 1, Sqrt[d]*2 + 1 + d]]]]; Table[f[n, n, 0], {n, 0, 31}] (* David Scambler, May 09 2012 *)
PROG
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sum(m=0, sqrtint(n), x^(m^2)*A^(m^2))); polcoeff(A, n)}
(PARI) seq(n)={Vec(serreverse(x/(1 + sum(i=1, sqrtint(n), x^(i^2))) + O(x*x^n)))} \\ Andrew Howroyd, Apr 28 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 24 2009
STATUS
approved