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.

A200312 a(n) = A000108(n)*A006130(n), where A000108 is the Catalan numbers and A006130(n) = A006130(n-1) + 3*A006130(n-2).

Original entry on oeis.org

1, 1, 8, 35, 266, 1680, 12804, 93093, 726440, 5635058, 45063668, 362121760, 2955642508, 24284658100, 201428123040, 1680921310635, 14119413718770, 119205791509200, 1011387051005100, 8617021562542470, 73704123363739440, 632601537174078420
Offset: 0

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Author

Paul D. Hanna, Nov 16 2011

Keywords

Comments

More generally, given {S} such that: S(n) = b*S(n-1) + c*S(n-2), S(0)=1, |b|>0, |c|>0, then Sum_{n>=0} S(n)*Catalan(n)*x^n = sqrt( (1-2*b*x - sqrt(1-4*b*x-16*c*x^2))/(2*b^2+8*c) )/x.

Examples

			G.f.: A(x) = 1 + x + 2*4*x^2 + 5*7*x^3 + 14*19*x^4 + 42*40*x^5 + 132*97*x^6 + 429*217*x^7 + ... + A000108(n)*A006130(n)*x^n + ...
where the g.f. of A006130, F(x) = 1/(1-x-3*x^2), begins:
F(x) = 1 + x + 4*x^2 + 7*x^3 + 19*x^4 + 40*x^5 + 97*x^6 + 217*x^7 + ...

Links

Crossrefs

Cf. A098614, A098616, A200375.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt((1-2*x - Sqrt(1-4*x-48*x^2))/26)/x)); // G. C. Greubel, Jul 27 2018
  • Mathematica
    CoefficientList[Series[Sqrt[(1 - 2*x - Sqrt[1 - 4*x - 48*x^2])/26]/x, {x, 0, 30}], x] (* G. C. Greubel, Jul 27 2018 *)
  • PARI
    {a(n)=binomial(2*n, n)/(n+1)*polcoeff(1/(1-x-3*x^2+x*O(x^n)),n)}
  • PARI
    {a(n)=polcoeff(sqrt((1-2*x - sqrt(1-4*x-48*x^2+x^3*O(x^n)))/26)/x,n)}
  • PARI
    {a(n)=polcoeff(serreverse(x*sqrt(1-12*x^2+x^2*O(x^n)) - x^2)/x,n)}
  • PARI
    {a(n)=polcoeff((1/x)*serreverse(x-x^2 - 6*x^3*sum(m=0,n\2,binomial(2*m,m)/(m+1)*3^m*x^(2*m))+x^3*O(x^n)),n)}

Formula

G.f.: sqrt( (1-2*x - sqrt(1-4*x-48*x^2))/26 )/x.
G.f.: (1/x)*Series_Reversion( x*sqrt(1-12*x^2) - x^2 ).
G.f.: (1/x)*Series_Reversion( x-x^2 - 6*x^3*Sum_{n>=0} A000108(n)*3^n*x^(2*n) ).
G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x)^2 + 13*x^2*A(x)^4).
Conjecture: n*(n+1)*a(n) -2*n*(2*n-1)*a(n-1) -12*(2*n-1)*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 17 2011
a(n) = ( ((1+sqrt(13))/2)^(n+1) - ((1-sqrt(13))/2)^(n+1) )/sqrt(13) * binomial(2*n+1,n)/(2*n+1). - Paul D. Hanna, Sep 25 2012
0 = +a(n)*(+110592*a(n+3) -9216*a(n+4) -7392*a(n+5) +858*a(n+6)) +a(n+1)*(+6912*a(n+3) -1968*a(n+4) -910*a(n+5) +154*a(n+6)) +a(n+2)*(-240*a(n+3) -2*a(n+4) +41*a(n+5) -4*a(n+6)) +a(n+3)*(+6*a(n+3) +5*a(n+4) +3*a(n+5) -a(n+6)) for all n in Z. - Michael Somos, Jul 28 2018

A200376 G.f.: 1/sqrt(1-10*x^2 + x^4/(1-8*x^2)) + x/(1-9*x^2).

Original entry on oeis.org

1, 1, 5, 9, 37, 81, 301, 729, 2549, 6561, 22045, 59049, 193029, 531441, 1703469, 4782969, 15111573, 43046721, 134539837, 387420489, 1200901157, 3486784401, 10739313997, 31381059609, 96172251061, 282429536481, 862142190941, 2541865828329, 7734936371269, 22876792454961, 69439155241581
Offset: 0

Views

Author

Paul D. Hanna, Nov 16 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 9*x^3 + 37*x^4 + 81*x^5 + 301*x^6 + 729*x^7 +...
The g.f. of A200375(n) = A000108(n)*A001045(n) begins:
G(x) = 1 + x + 2*3*x^2 + 5*5*x^3 + 14*11*x^4 + 42*21*x^5 + 132*43*x^6 +...
where A(x) = G(x/A(x)) and G(x) = A(x*G(x)).

Crossrefs

Cf. A200375, A098615, A098617.

Programs

  • Mathematica
    CoefficientList[Series[1/Sqrt[1-10x^2+x^4/(1-8x^2)]+x/(1-9x^2),{x,0,30}], x] (* Harvey P. Dale, Nov 19 2011 *)
  • PARI
    {a(n)=polcoeff(1/sqrt(1-10*x^2 + x^4/(1-8*x^2 +x*O(x^n))) + x/(1-9*x^2 +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=local(G=sum(m=0,n,binomial(2*m, m)/(m+1)*polcoeff(1/(1-x-2*x^2+x*O(x^m)), m)*x^m)+x*O(x^n)); polcoeff(x/serreverse(x*G),n)}
for(n=0,30,print1(a(n),", "))

Formula

D-finite with recurrence: n*a(n) +(n-1)*a(n-1) +(24-17*n)*a(n-2) +(41-17*n)*a(n-3) +72*(n-3)*a(n-4) +72*(n-4)*a(n-5)=0. - R. J. Mathar, Nov 17 2011
G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x) + 9*x^2*A(x)^2). - Paul D. Hanna, Nov 18 2014
Let G(x) = g.f. of A200375, then g.f. A(x) satisfies:
(1) A(x) = x/Series_Reversion(x*G(x)),
(2) A(x) = G(x/A(x)) and G(x) = A(x*G(x)),
where A200375(n) = A000108(n)*A001045(n), the product of Catalan and Jacobsthal numbers.
a(n) ~ 3^(n-1). - Vaclav Kotesovec, Jun 29 2013

A200538 Product of Jacobsthal and Motzkin numbers: a(n) = A001045(n+1)*A001006(n).

Original entry on oeis.org

1, 1, 6, 20, 99, 441, 2193, 10795, 55233, 284735, 1494404, 7914270, 42360541, 228460935, 1241224182, 6784445340, 37288826697, 205937705799, 1142317727466, 6361104740100, 35548154733969, 199295884785459, 1120615326442269, 6318077793648075, 35710056983891367, 202297486497822121
Offset: 0

Views

Author

Paul D. Hanna, Nov 18 2011

Keywords

Comments

The g.f. for the Jacobsthal numbers is 1/(1-x-2*x^2) and the g.f. M(x) for the Motzkin numbers satisfy: M(x) = 1 + x*M(x) + x^2*M(x)^2.

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 20*x^3 + 99*x^4 + 441*x^5 + 2193*x^6 +...
where A(x) = 1*1 + 1*1*x + 3*2*x^2 + 5*4*x^3 + 11*9*x^4 + 21*21*x^5 + 43*51*x^6 + 85*127*x^7 + 171*323*x^8 +...+ A001045(n+1)*A001006(n)*x^n +...

Crossrefs

Cf. A200375, A200539, A200540, A001045, A001006.

Programs

  • PARI
    {A001006(n)=polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2),n)}
{A001045(n)=polcoeff( x/(1-x-2*x^2+x*O(x^n)),n)}
{a(n)=A001045(n+1)*A001006(n)}
Showing 1-3 of 3 results.

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