cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A157134 G.f. satisfies: A(x) = Sum_{n>=0} x^(n^2) * A(x)^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 7, 11, 18, 33, 63, 117, 211, 383, 713, 1348, 2547, 4793, 9039, 17165, 32785, 62761, 120243, 230768, 444119, 857015, 1656931, 3207990, 6219994, 12079544, 23496417, 45767352, 89256038, 174269488, 340646238, 666604642
Offset: 0

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Author

Paul D. Hanna, Feb 24 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 11*x^7 +...
A(x)^2 = 1 + 2*x + 3*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 27*x^6 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 10*x^3 + 18*x^4 + 36*x^5 + 73*x^6 +...
A(x)^4 = 1 + 4*x + 10*x^2 + 20*x^3 + 39*x^4 + 80*x^5 + 168*x^6 +...
where
A(x) = 1 + x*A(x) + x^4*A(x)^2 + x^9*A(x)^3 + x^16*A(x)^4 +...

Crossrefs

Cf. A157135, A157133, A157136.
Cf. A107595. [From Paul D. Hanna, Apr 25 2010]

Programs

  • 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)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = B(x/A(x)) where B(x) = A(x*B(x)) = g.f. of A157135,
where A157135(n) = [x^n] A(x)^(n+1)/(n+1) for n>=0,
and a(n) = [x^n] -1/B(x)^(n-1)/(n-1) for n>1.
From Paul D. Hanna, Apr 25 2010: (Start)
G.f. A(x) satisfies the continued fraction:
A(x) = 1/(1- x*A(x)/(1- (x^3-x)*A(x)/(1- x^5*A(x)/(1- (x^7-x^3)*A(x)/(1- x^9*A(x)/(1- (x^11-x^5)*A(x)/(1- x^13*A(x)/(1- (x^15-x^7)*A(x)/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
(End)
From Paul D. Hanna, May 05 2010: (Start)
Let A = g.f. A(x) at x=q, then A satisfies the q-series:
A = Sum_{n>=0} q^n*A^n*Product_{k=1..n} (1-q^(4k-3)*A)/(1-q^(4k-1)*A).
(End)

Extensions

Typo in data corrected by D. S. McNeil, Aug 17 2010

A157136 G.f. satisfies: A(x) = Sum_{n>=0} x^(n^2) * A(x)^(n^2).

Original entry on oeis.org

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
Offset: 0

Views

Author

Paul D. Hanna, Feb 24 2009

Keywords

Comments

Apparently: Number of Dyck n-paths with each ascent length being a square number. [David Scambler, May 09 2012]

Examples

			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 +...)).

Crossrefs

Cf. A157133, A157134, A157135.

Programs

  • 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 *)
  • 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

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)

A157133 G.f. satisfies: A(x) = Sum_{n>=0} x^(n(n+1)/2) * A(x)^n.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 14, 30, 62, 129, 278, 604, 1313, 2883, 6386, 14203, 31733, 71272, 160725, 363670, 825653, 1880351, 4293985, 9830499, 22558939, 51880565, 119552907, 276012657, 638348123, 1478749229, 3430799333, 7971134523
Offset: 0

Views

Author

Paul D. Hanna, Feb 24 2009

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 14*x^6 + 30*x^7 +...
A(x)^2 = 1 + 2*x + 3*x^2 + 6*x^3 + 13*x^4 + 26*x^5 + 54*x^6 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 13*x^3 + 30*x^4 + 66*x^5 + 145*x^6 +...
A(x)^4 = 1 + 4*x + 10*x^2 + 24*x^3 + 59*x^4 + 140*x^5 + 326*x^6 +...
where
A(x) = 1 + x*A(x) + x^3*A(x)^2 + x^6*A(x)^3 + x^10*A(x)^4 +...

Crossrefs

Cf. A157134, A157135, A157136.
Cf. A121690. [From Paul D. Hanna, Apr 25 2010]

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,(A=sum(m=0,sqrtint(2*n+1),x^(m*(m+1)/2)*A^m)));polcoeff(A,n)}

Formula

Contribution from Paul D. Hanna, Apr 25 2010: (Start)
G.f. A(x) satisfies the continued fraction:
A(x) = 1/(1- x*A(x)/(1- (x^2-x)*A(x)/(1- x^3*A(x)/(1- (x^4-x^2)*A(x)/(1- x^5*A(x)/(1- (x^6-x^3)*A(x)/(1- x^7*A(x)/(1- (x^8-x^4)*A(x)/(1- ...)))))))))
due to an identity of a partial elliptic theta function.
(End)
Showing 1-3 of 3 results.

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