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 1-4 of 4 results.

A106335 Decimal expansion of the radius of convergence of the g.f. of A106336; equals constant A106333 divided by constant A106334.

Original entry on oeis.org

3, 2, 2, 6, 2, 7, 6, 3, 2, 6, 9, 2, 1, 9, 1, 1, 3, 3, 0, 9, 6, 9, 8, 7, 1, 3, 8, 6, 7, 3, 9, 8, 3, 0, 2, 3, 3, 2, 2, 9, 0, 4, 2, 4, 3, 7, 4, 6, 7, 1, 7, 4, 5, 2, 1, 6, 0, 5, 6, 2, 0, 9, 1, 2, 4, 5, 5, 4, 8, 6, 2, 6, 7, 4, 1, 1, 1, 5, 0, 6, 4, 9, 7, 4, 7, 1, 2, 3, 7, 3, 9, 9, 1, 2, 2, 1, 4, 7, 8, 5, 3, 7, 1, 9, 0
Offset: 0

Views

Author

Paul D. Hanna, Apr 29 2005

Keywords

Comments

The g.f. of A106336 equals (1/x)*Series_Reversion( x*eta(x)/eta(x^2)^2 ).
This constant is very close to 2^(3/2) / (3*sqrt(e*Pi)) = 0.3226276326921911330637735905807475397715626276499133673167401123748... - Vaclav Kotesovec, Aug 02 2017

Examples

			x/F(x)=0.322627632692191133096987138673983023322904243746717452160562...
where F(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + ...
so F(x) = 1.9873697211846841452692897833444126... (A106334)
at x = 0.6411803884299545796456448886283011... (A106333).

Crossrefs

Cf. A106333, A106334, A106336.

Programs

  • Mathematica
    digits = 105; x0 = x /. FindRoot[ Sum[(1 - n*(n+1)/2)*x^(n*(n+1)/2), {n, 0, digits}], {x, 1/2}, WorkingPrecision -> digits+5]; f[x_] := EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8)); x0/f[x0] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 05 2013 *)
  • PARI
    A106333=solve(x=.6,.7,sum(n=0,100,(1-n*(n+1)/2)*x^(n*(n+1)/2))); A106334=sum(n=0,100, A106333^(n*(n+1)/2)); A106335=A106333/A106334

Formula

Constant equals the ratio x/F(x) evaluated at the constant x that satisfies: F(x) - x*F'(x) = 0, where F(x) = Sum_{n>=0} x^(n*(n+1)/2).

A106336 Number of ways of writing n as the sum of n+1 triangular numbers, divided by n+1.

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 25, 64, 169, 442, 1172, 3180, 8730, 24116, 67159, 188568, 532741, 1512695, 4315996, 12369324, 35587923, 102747636, 297601382, 864525312, 2518185362, 7353088206, 21520084301, 63115752910, 185474840912, 546042990300, 1610314638958
Offset: 0

Views

Author

Paul D. Hanna, Apr 29 2005

Keywords

Comments

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

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 25*x^6 + 64*x^7 +...
A(x) = F(x*A(x)) where F(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + ...
The radius of convergence equals r = 0.322627632692191133... (A106335)
at which the g.f. converges to A(r) = 1.987369721184684145... (A106334).

Links

Crossrefs

Cf. A106333, A106334, A106335, A106337; related: A109085.

Programs

  • Maple
    b:= proc(n) option remember; expand(`if`(n=0, 1,
      add(`if`(issqr(8*j+1), x*b(n-j), 0), j=1..n)))
    end:
a:= n-> (p-> add(coeff(p, x, i)*binomial(1+n, i),
             i=0..n)/(n+1))(b(n)):
seq(a(n), n=0..35);  # Alois P. Heinz, Jul 31 2017
  • 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, Ceiling[Sqrt[2*d]]],If[d == 0, 1, Ceiling[Sqrt[2*d]] + d]]]]; Table[f[n, n, 0], {n, 0, 30}] (* David Scambler, May 09 2012 *)
  • PARI
    {a(n) = my(X); if(n<0,0,X=x+x*O(x^n); polcoef(eta(X^2)^(2*n+2)/eta(X)^(n+1)/(n+1),n))}
  • PARI
    {a(n) = if(n<0,0,polcoef( sum(k=1,(sqrtint(8*n+1)+1)\2,x^((k^2-k)/2),x*O(x^n))^(n+1)/(n+1),n))}
  • PARI
    {a(n) = my(A=1+x+x*O(x^n)); for(i=1,n, A=prod(m=1,n,(1+(x*A)^m)*(1-(x*A)^(2*m))));polcoef(A,n)} \\ Paul D. Hanna, Oct 23 2010
  • PARI
    {a(n) = my(A=1+x); for(i=1,n, A=exp(sum(m=1,n,(x*A)^m/(1+(x*A)^m+x*O(x^n))/m)));polcoef(A,n)} \\ Paul D. Hanna, Jun 01 2011

Formula

G.f.: A(x) = (1/x) * Series_Reversion( x*eta(x)/eta(x^2)^2 ).
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = F(x*A(x)) where F(x) = Sum_{n>=0} x^(n*(n+1)/2).
(2) log(A(x)) = Sum_{n>=1} A106337(n)/n*x^n.
(3) A(x) = Product_{n>=1} (1 + (x*A(x))^n)*(1 - (x*A(x))^(2*n)). - Paul D. Hanna, Oct 23 2010
(4) A(x) = exp( Sum_{n>=1} (x^n*A(x)^n/(1 + x^n*A(x)^n))/n ). - Paul D. Hanna, Jun 01 2011
From Paul D. Hanna, Jun 11 2025: (Start)
(5) A(x)^4 = Sum_{n>=0} (2*n+1) * (x*A(x))^n / (1 - (x*A(x))^(2*n+1)).
(6) A(x^2)^2 = Sum_{n>=0} (x*A(x^2)^(1/2))^n / (1 + (x*A(x^2)^(1/2))^(2*n+1)).
(End)
a(n) ~ c / (n^(3/2) * A106335^n), where c = A366174 = 0.49833479793360342260635926402850016443069428233051290201996853498... - Vaclav Kotesovec, Oct 07 2020

Extensions

Edited by Paul D. Hanna, Jun 01 2011

A106337 Number of ways of writing n as the sum of n triangular numbers.

Original entry on oeis.org

1, 1, 1, 4, 13, 31, 82, 253, 757, 2173, 6341, 18888, 56266, 167324, 499773, 1499059, 4503557, 13546893, 40824379, 123233868, 372472353, 1127080252, 3414310032, 10353722919, 31425764410, 95463814056, 290222666436, 882954212908, 2688037654049, 8188468874808
Offset: 0

Views

Author

Paul D. Hanna, Apr 29 2005

Keywords

Comments

Number of compositions of n into n triangular numbers with 0's allowed. a(3) = 4: [1,1,1], [0,0,3], [0,3,0], [3,0,0]. - Alois P. Heinz, Jul 31 2017
The radius of convergence is equal to A106335. - Vaclav Kotesovec, Nov 15 2017

Examples

			G106336(x) = exp(x + 1/2*x^2 + 4/3*x^3 + 13/4*x^4 + 31/5*x^5 +...).
G106336(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 +...+ A106336(n)*x^n +...
G106336(x) = 1 + x*G106336(x) + (x*G106336(x))^3 + (x*G106336(x))^6 +...

Links

Crossrefs

Cf. A000217, A007294, A023361, A106333, A106334, A106335, A106336, A296045.
Main diagonal of A286180.

Programs

  • Maple
    b:= proc(n) option remember; expand(`if`(n=0, 1,
      add(`if`(issqr(8*j+1), x*b(n-j), 0), j=1..n)))
    end:
a:= n-> (p-> add(coeff(p, x, i)*binomial(n, i), i=0..n))(b(n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 31 2017
  • Mathematica
    QP = QPochhammer; a[0] = 1; a[n_] := SeriesCoefficient[(QP[-1, x]*QP[x^2]/2 )^n, {x, 0, n}]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jun 04 2017 *)
  • PARI
    {a(n)=local(X); if(n<1,1,X=x+x*O(x^n); polcoeff(eta(X^2)^(2*n)/eta(X)^n,n))}

Formula

Log.g.f.: Sum_{n>=1} a(n)/n*x^n = log(G106336(x)), where G106336(x) is the g.f. of A106336 and satisfies: Sum_{n>=0} (x*G106336(x))^(n*(n+1)/2) = G106336(x).
a(n) = [x^n] Product_{j=1..n} (1+x^j-x^(2*j)-x^(3*j))^n. - Alois P. Heinz, Aug 01 2017

Extensions

a(0) changed to 1 by Alois P. Heinz, Jul 31 2017

A106334 Decimal expansion of the function F(x) evaluated at the constant x that satisfies: F(x) - x*F'(x) = 0, where F(x) = Sum_{n>=0} x^(n*(n+1)/2).

Original entry on oeis.org

1, 9, 8, 7, 3, 6, 9, 7, 2, 1, 1, 8, 4, 6, 8, 4, 1, 4, 5, 2, 6, 9, 2, 8, 9, 7, 8, 3, 3, 4, 4, 4, 1, 2, 6, 1, 8, 3, 4, 2, 7, 1, 7, 7, 2, 9, 8, 5, 5, 4, 5, 7, 4, 7, 0, 3, 5, 6, 2, 2, 3, 1, 0, 3, 8, 2, 6, 9, 5, 8, 9, 3, 8, 8, 6, 6, 2, 5, 5, 4, 7, 7, 6, 2, 0, 9, 7, 6, 2, 9, 9, 6, 3, 3, 6, 5, 7, 2, 7, 4, 6, 8, 1, 3, 5
Offset: 1

Views

Author

Paul D. Hanna, Apr 29 2005

Keywords

Comments

Constant A106333 divided by this constant equals constant A106335, the radius of convergence of the g.f. of A106336.

Examples

			F(x)=1.9873697211846841452692897833444126183427177298554574703562231
where F(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + ...
at x = 0.6411803884299545796456448886283011... (A106333).

Crossrefs

Cf. A106333, A106335, A106332, A106336.

Programs

  • Mathematica
    digits = 105; x0 = x /. FindRoot[ Sum[(1 - n*(n+1)/2)*x^(n*(n+1)/2), {n, 0, digits}], {x, 1/2}, WorkingPrecision -> digits+5]; f[x_] := EllipticTheta[2, 0, Sqrt[x]]/(2*x^(1/8)); f[x0] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Mar 05 2013 *)
  • PARI
    A106333=solve(x=.6,.7,sum(n=0,100,(1-n*(n+1)/2)*x^(n*(n+1)/2))); A106334=sum(n=0,100, A106333^(n*(n+1)/2))
Showing 1-4 of 4 results.

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