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 1-5 of 5 results.

A364161 G.f. satisfies A(x) = 1 + x*A(x)^2/(1 - x^3*A(x)).

Original entry on oeis.org

1, 1, 2, 5, 15, 47, 153, 514, 1769, 6205, 22102, 79733, 290721, 1069688, 3966739, 14810348, 55627778, 210046102, 796864028, 3035912900, 11610468138, 44556451207, 171529074168, 662238211929, 2563524741603, 9947573055828, 38687704042595
Offset: 0

Views

Author

Seiichi Manyama, Aug 28 2023

Keywords

Crossrefs

Cf. A001003, A119370.
Cf. A218251, A364833, A365247.

Programs

  • Maple
    A364161 := proc(n)
    add( binomial(n-2*k-1,k)*binomial(2*n-5*k+1,n-3*k)/(2*n5*k+1),k=0..floor(n/3)) ;
end proc:
seq(A364161(n),n=0..80); # R. J. Mathar, Aug 29 2023
  • PARI
    a(n) = sum(k=0, n\3, binomial(n-2*k-1, k)*binomial(2*n-5*k+1, n-3*k)/(2*n-5*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * binomial(2*n-5*k+1,n-3*k)/(2*n-5*k+1).
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(n+1)*a(n-2) +3*(-2*n+3)*a(n-3) +(-2*n+7)*a(n-5) +(n-8)*a(n-6) +(n-8)*a(n-8)=0. - R. J. Mathar, Aug 29 2023

A364833 G.f. satisfies A(x) = 1 + x*A(x)^2/(1 - x^3*A(x)^3).

Original entry on oeis.org

1, 1, 2, 5, 15, 49, 168, 595, 2160, 7997, 30083, 114660, 441840, 1718531, 6737820, 26600784, 105659970, 421949492, 1693120779, 6823018035, 27602090087, 112053680381, 456343848121, 1863893501065, 7633232165286, 31337360839387, 128944120202510
Offset: 0

Views

Author

Seiichi Manyama, Aug 28 2023

Keywords

Crossrefs

Cf. A112806, A219537.
Cf. A218251, A364161, A365247.
Cf. A001003, A049124.

Programs

  • Maple
    A364833 := proc(n)
    add( binomial(n-2*k-1,k)*binomial(2*n-3*k+1,n-3*k)/ (2*n-3*k+1),k=0..floor(n/3)) ;
end proc:
seq(A364833(n),n=0..80); # R. J. Mathar, Aug 29 2023
  • PARI
    a(n) = sum(k=0, n\3, binomial(n-2*k-1, k)*binomial(2*n-3*k+1, n-3*k)/(2*n-3*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * binomial(2*n-3*k+1,n-3*k)/(2*n-3*k+1).
D-finite with recurrence 31*n*(626109182191*n-1858292669035) *(n-1)*(n+1) *a(n) -n*(n-1) *(244150473843619*n^2 -1454194662255591*n +2175006457069082) *a(n-1) +3*(n-1) *(292927551362415*n^3 -2593205532882651*n^2 +7084566217454162*n -5823331737745632)*a(n-2) +(-843955616916167*n^4 +9932491073296715*n^3 -42016891739306929*n^2 +76184884157722453*n -50166914106142776) *a(n-3) +18*(1509721335071*n^4 -40413442328880*n^3 +330301781039401*n^2 -1078322794857576*n +1231650372542192) *a(n-4) +18*(39673125909769*n^4 -598320530478001*n^3 +3228489073613917*n^2 -7321259523567459*n +5788776339353646) *a(n-5) +27*(n-5) *(3102413205331*n^3 -35996479327373*n^2 +114122791959960*n -64735736097804) *a(n-6) -243*(n-6) *(n-7)*(475638134099*n^2 -2399948859181*n +2877042451214) *a(n-7) -243*(45857481910*n -35520400961) *(n-5) *(n-7) *(n-8)*a(n-8)=0. - R. J. Mathar, Aug 29 2023
G.f.: (1/x) * Series_Reversion( x*(1 - x / (1 - x^3)) ). - Seiichi Manyama, Sep 28 2024

A365247 G.f. satisfies A(x) = 1 + x*A(x)^2/(1 - x^3*A(x)^4).

Original entry on oeis.org

1, 1, 2, 5, 15, 50, 177, 650, 2449, 9412, 36761, 145518, 582556, 2354557, 9594898, 39378259, 162619316, 675258452, 2817643240, 11808576745, 49683880754, 209786559004, 888676860191, 3775654643360, 16084818268474, 68694452578325, 294053067958011
Offset: 0

Views

Author

Seiichi Manyama, Aug 28 2023

Keywords

Crossrefs

Cf. A364739, A365246.
Cf. A218251, A364161, A364833.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n-2*k-1, k)*binomial(2*n-2*k+1, n-3*k)/(2*n-2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2*k-1,k) * binomial(2*n-2*k+1,n-3*k)/(2*n-2*k+1).

A218250 G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x^2*A(x))^2.

Original entry on oeis.org

1, 1, 3, 7, 18, 49, 135, 383, 1104, 3228, 9554, 28557, 86095, 261487, 799323, 2457327, 7592620, 23565444, 73437284, 229691620, 720800824, 2268820824, 7161255962, 22661307317, 71878917199, 228487568175, 727779875401, 2322485254421, 7424488376794, 23773398866825
Offset: 0

Views

Author

Paul D. Hanna, Oct 24 2012

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 18*x^4 + 49*x^5 + 135*x^6 + 383*x^7 +...
where
A(x) = 1 + (1+2*x)*x*A(x) + (2+x)*x^3*A(x)^2 + x^5*A(x)^3.

Crossrefs

Cf. A218251, A182053.

Programs

  • Mathematica
    nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1 + x*AGF) * (1 + x^2*AGF)^2 - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 10 2013 *)
  • PARI
    {a(n)=local(A=1); for(i=1, n, A=(1+x*A)*(1+x^2*A)^2+x*O(x^n)); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

Formula

Recurrence: 2*(n+2)*(2*n+5)*(43*n^3 - 48*n^2 - 43*n + 12)*a(n) = 2*(2*n+1)*(2*n+3)*(43*n^3 - 5*n^2 - 94*n + 8)*a(n-1) + 2*(344*n^5 + 132*n^4 - 1303*n^3 - 399*n^2 + 554*n + 168)*a(n-2) + (473*n^5 - 528*n^4 - 1711*n^3 + 1866*n^2 + 1256*n - 960)*a(n-3) - 6*(86*n^5 - 225*n^4 - 321*n^3 + 794*n^2 + 160*n - 416)*a(n-4) + 4*(n-4)*(n-2)*(43*n^3 + 81*n^2 - 10*n - 36)*a(n-5). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = 3.361963061296269297... is the root of the equation -4 + 12*d - 11*d^2 - 16*d^3 - 8*d^4 + 4*d^5 = 0 and c = 2.227460242885392531198808525530878354... - Vaclav Kotesovec, Sep 10 2013

A348666 a(n) is the number of quiddities of 3-periodic dissections of (n + 2)-gons.

Original entry on oeis.org

1, 1, 2, 5, 15, 49, 166, 577, 2050, 7414, 27201, 100984, 378651, 1431901, 5454718, 20912754, 80630085, 312430832, 1216045522, 4752132953, 18638125275, 73340870891, 289463959745, 1145612705905, 4545478673125, 18077348646721, 72048928923617, 287733587217552, 1151233484320195
Offset: 0

Views

Author

Michael De Vlieger, Oct 28 2021

Keywords

Comments

See Conley-Ovsienko paper, p. 6.
a(0) = 1 by convention.

Links

Crossrefs

Cf. A218251.

Programs

  • Mathematica
    {1}~Join~Array[Sum[(3 (k - s) + 2)/(# - s + 1)*Binomial[# - 3 k + s - 2, s]*Binomial[2 # - 3 k - s - 1, # - 3 k - 1], {k, 0, #/3}, {s, 0, k}] &, 29]

Formula

a(n) = Sum_{k=0..n/3} Sum_{s=0..k} ((3*(k-s) + 2)/(n-s+1)) * binomial(n-3*k+s-2,s) * binomial(2*n-3*k-s-1,n-3*k-1).
a(n) ~ c * d^n / n^(3/2), where d = 4.21429839439676340483426656814177802445... is the root of the equation 4 - 12*d^2 - 8*d^3 + 12*d^4 - 20*d^5 + d^7 = 0 and c = 0.590856549086828350357357054105900401452384216047617779361986537... - Vaclav Kotesovec, Nov 04 2021
Showing 1-5 of 5 results.

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