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 1-2 of 2 results.

A228411 G.f.: ( (1 - sqrt(1-32*x)) / (16*x) )^(1/4).

Original entry on oeis.org

1, 2, 26, 476, 10150, 236060, 5807076, 148581048, 3913759878, 105424703020, 2890693930124, 80413849328904, 2263896023453532, 64381391412987672, 1846729385267277960, 53367451809002583408, 1552274439636853988550, 45408989873571191613900, 1335107241077282661195900
Offset: 0

Views

Author

Paul D. Hanna, Nov 09 2013

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 26*x^2 + 476*x^3 + 10150*x^4 + 236060*x^5 +...
where
A(x)^4 = 1 + 8*x + 128*x^2 + 2560*x^3 + 57344*x^4 + 1376256*x^5 +...+ A000108(n)*8^n*x^n +...

Links

Crossrefs

Cf. A000108, A159318.

Programs

  • Mathematica
    CoefficientList[Series[((1-Sqrt[1-32*x])/(16*x))^(1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 10 2013 *)
Table[8^n Binomial[2 n + 1/4, n]/(8 n + 1), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 12 2016 *)
  • PARI
    /* G.f.: ( (1 - sqrt(1-32*x)) / (16*x) )^(1/4): */
{a(n)=polcoeff(( (1 - sqrt(1-32*x +x^2*O(x^n))) / (16*x) )^(1/4),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* G.f.: A(x) = C(8*x)^(1/4), C(x) is Catalan function: */
{a(n)=polcoeff((serreverse(x-8*x^2 +x^2*O(x^n))/x)^(1/4),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    /* G.f.: A(x) = exp( x*A(x)^8 + Integral(A(x)^8 dx) ): */
{a(n)=local(A=1+x);for(i=1,n,A=exp(x*A^8+intformal(A^8+x*O(x^n))));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = exp( x*A(x)^8 + Integral(A(x)^8 dx) ).
(2) A(x)^4 = 1 + 8*x*A(x)^8, thus A(x) = C(8*x)^(1/4) where C(x) is the Catalan function (A000108).
a(n) ~ 2^(5*n-3+1/4)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 10 2013
D-finite with recurrence: n*(4*n+1)*a(n) -2*(8*n-3)*(8*n-7)*a(n-1)=0. - R. J. Mathar, Oct 08 2016
a(n) = 8^n*binomial(2*n + 1/4, n)/(8*n + 1). - Vladimir Reshetnikov, Oct 12 2016

A159319 a(n) = 3^(n^2+n) * C(2*n-1 + 1/3^n, n) / (n*3^n + 1).

Original entry on oeis.org

1, 3, 126, 66708, 379033074, 21399656315607, 11566324342205917416, 58678275719834357303044728, 2762222169999029718435709903699050, 1197781369953334546750963984948238943438411
Offset: 0

Views

Author

Paul D. Hanna, Apr 23 2009

Keywords

Examples

			G.f.: A(x) = 1 + 3*x/3^2 + 126*x^2/3^6 + 66708*x^3/3^12 + 379033074*x^4/3^20 +...
A(x) = Sum_{n>=0} log( (1-sqrt(1-4*x/3^n))/(2*x/3^n) )^n/n!.
A(x) = 1 + log(F(x/3)) + log(F(x/9))^2/2! + log(F(x/27))^3/3! +... where F(x) = (1-sqrt(1-4*x))/(2*x).
Special values.
A(3/4) = 1 + log(2) + log(6-6*sqrt(2/3))^2/2! + log(18-18*sqrt(8/9))^3/3! + log(54-54*sqrt(26/27))^4/4! +...
A(3/4) = 1.6977820781412737038286578011417848301231627494589650...
A(-3/4) = 1 + log(2*sqrt(2)-2) + log(6*sqrt(4/3)-6)^2/2! + log(18*sqrt(10/9)-18)^3/3! + log(54*sqrt(28/27)-54)^4/4! +...
A(-3/4) = 0.8145458917316632938137444904602229430460096517471900...
Illustrate (3^n)-th root formula:
a(n)/3^(n^2+n) = [x^n] F(x)^(1/3^n) or, equivalently,
a(n) = [x^n] F(3^(n+1)*x)^(1/3^n) where F(x)=Catalan(x):
F(3*x) = (1) + 3*x + 18*x^2 + 135*x^3 + 1134*x^4 + 10206*x^5 +...
F(9*x)^(1/3) = 1 + (3)*x + 45*x^2 + 936*x^3 + 22572*x^4 +...
F(27*x)^(1/9) = 1 + 3*x + (126)*x^2 + 7659*x^3 + 546480*x^4 +...
F(81*x)^(1/27) = 1 + 3*x + 369*x^2 + (66708)*x^3 + 14215230*x^4 +...
F(243*x)^(1/81) = 1 + 3*x + 1098*x^2 + 593775*x^3 + (379033074)*x^4 +...
coefficients in parenthesis form the initial terms of this sequence.

Links

Crossrefs

Cf. A159318, A158093, A159558, A159478, A000108.

Programs

  • Magma
    [3^(n^2 +n)*Binomial(2*n -1 +1/3^n, n)/(n*3^n +1): n in [0..40]]; // G. C. Greubel, Jun 26 2018
  • Mathematica
    Table[3^(n^2 +n)*Binomial[2*n -1 +1/3^n, n]/(n*3^n +1), {n, 0, 50}] (* G. C. Greubel, Jun 26 2018 *)
  • PARI
    {a(n)=3^(n^2+n)*binomial(2*n-1+1/3^n, n)/(n*3^n + 1)}
  • PARI
    {a(n)=3^(n^2+n)*polcoeff(1/(1-x+x*O(x^n))^(n+1/3^n)/(n*3^n + 1),n)}
  • PARI
    {a(n)=3^(n^2+n)*polcoeff(((1-sqrt(1-4*x+x^2*O(x^n)))/(2*x))^(1/3^n),n)}
  • PARI
    {a(n)=3^(n^2+n)*polcoeff(sum(k=0,n,log((1-sqrt(1-4*x/3^k+x^2*O(x^n)))/(2*x/3^k))^k/k!),n)}

Formula

G.f.: A(x) = Sum_{n>=0} a(n)*x^n/3^(n^2+n).
G.f.: A(x) = Sum_{n>=0} log(F(x/3^n))^n/n! and
a(n)/3^(n^2+n) = [x^n] F(x)^(1/3^n) where
F(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).
a(n)/3^(n^2+n) = [x^n] 1/(1-x)^(n + 1/3^n)/(n*3^n + 1).
Radius of convergence of series A(x) is |x| <= 3/4.
Showing 1-2 of 2 results.

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