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.

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

A365246 G.f. satisfies A(x) = 1 + x*A(x)^2/(1 - x^2*A(x)^4).

Original entry on oeis.org

1, 1, 2, 6, 22, 88, 370, 1613, 7230, 33117, 154330, 729369, 3487470, 16840346, 82007012, 402269702, 1985867630, 9858739759, 49187798158, 246506563980, 1240337033398, 6263601365616, 31734939452116, 161270637750264, 821802841072422, 4198348868249768
Offset: 0

Views

Author

Seiichi Manyama, Aug 28 2023

Keywords

Crossrefs

Cf. A364739, A365247.

Programs

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

Formula

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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