This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A157136 G.f. satisfies: A(x) = Sum_{n>=0} x^(n^2) * A(x)^(n^2). 5
 1, 1, 1, 1, 2, 6, 16, 36, 75, 164, 401, 1046, 2718, 6878, 17200, 43486, 112202, 293540, 770535, 2019891, 5296670, 13942944, 36902130, 98097968, 261456388, 697970447, 1866383507, 5001333169, 13432923544, 36154294520, 97475330092, 263188299372 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Apparently: Number of Dyck n-paths with each ascent length being a square number. [David Scambler, May 09 2012] LINKS 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 Cf. A157133, A157134, A157135. Sequence in context: A038503 A079990 A127902 * A178523 A270810 A227035 Adjacent sequences:  A157133 A157134 A157135 * A157137 A157138 A157139 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 24 2009 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 15 20:04 EDT 2019. Contains 328037 sequences. (Running on oeis4.)