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 1-6 of 6 results.

A364475 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x)^3.

Original entry on oeis.org

1, 1, 4, 18, 94, 529, 3135, 19270, 121732, 785496, 5155167, 34304706, 230923653, 1569684910, 10759159000, 74281473504, 516089542684, 3605685460750, 25316226436086, 178538289189108, 1264131169628799, 8982889404251721, 64041351551534215
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2023

Keywords

Crossrefs

Column k=1 of A378323.
Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364478.

Programs

  • Maple
    A364475 := proc(n)
    add( binomial(3*n-3*k,k) * binomial(3*n-4*k,n-2*k)/(2*n-2*k+1),k=0..n/2) ;
end proc:
seq(A364475(n),n=0..80); # R. J. Mathar, Jul 27 2023
  • PARI
    a(n) = sum(k=0, n\2, binomial(3*n-3*k, k)*binomial(3*n-4*k, n-2*k)/(2*n-2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-3*k,k) * binomial(3*n-4*k,n-2*k) / (2*n-2*k+1).
D-finite with recurrence 2*n*(2*n+1)*a(n) -(5*n+1)*(3*n-2)*a(n-1) +4*(-25*n^2+75*n-59) *a(n-2) +9*(-15*n^2+69*n-80)*a(n-3) -6*(3*n-8)*(3*n-10) *a(n-4)=0. - R. J. Mathar, Jul 27 2023

A364474 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x).

Original entry on oeis.org

1, 1, 4, 16, 77, 403, 2228, 12800, 75653, 457022, 2809266, 17514200, 110480475, 703850686, 4522217364, 29268545416, 190645760149, 1248817411471, 8221323983431, 54365667330636, 360954069730636, 2405225494066647, 16080210766344354, 107828663888705292
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2023

Keywords

Links

Crossrefs

Cf. A002293, A104979, A186997, A255673, A361245, A364475, A364478.

Programs

  • Maple
    A364474 := proc(n)
    add( binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k)/(2*n-4*k+1),k=0..n/2) ;
end proc:
seq(A364474(n),n=0..80); # R. J. Mathar, Jul 27 2023
  • Mathematica
    Table[Sum[Binomial[3*n - 5*k, k]*Binomial[3*n - 6*k, n - 2*k]/(2*n - 4*k + 1), {k, 0, Floor[n/2]}], {n, 0, 25}] (* Wesley Ivan Hurt, May 25 2024 *)
  • PARI
    a(n) = sum(k=0, n\2, binomial(3*n-5*k, k)*binomial(3*n-6*k, n-2*k)/(2*n-4*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k) / (2*n-4*k+1).
D-finite with recurrence 2*n*(2*n+1)*(3*n-7)*a(n) -3*(3*n-1)*(3*n-7)*(3*n-2) *a(n-1) -2*(n-3)*(18*n^2-33*n+4) *a(n-2) +2*(18*n^3-141*n^2+287*n-64) *a(n-4) -2*(n-4)*(3*n-1)*(2*n-13)*a(n-6)=0. - R. J. Mathar, Jul 27 2023

A364478 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x)^8.

Original entry on oeis.org

1, 1, 4, 23, 154, 1124, 8675, 69626, 575243, 4859778, 41789764, 364565277, 3218581695, 28702642553, 258172627259, 2339496034381, 21337716782873, 195726876816623, 1804472496834650, 16711389876481027, 155395461519245354, 1450298253483719944
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2023

Keywords

Crossrefs

Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364475.

Programs

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

Formula

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

A361244 Number of noncrossing bridgeless cacti with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 1, 6, 13, 57, 169, 673, 2301, 8933, 32747, 127063, 483484, 1889957, 7352241, 29003446, 114481435, 455542880, 1816976042, 7285391071, 29291855748, 118218771203, 478372112363, 1941436590561, 7897802784418, 32205683248225, 131602039333873
Offset: 0

Views

Author

Andrew Howroyd, Mar 08 2023

Keywords

Links

Crossrefs

Cf. A003168, A361242, A361245.

Programs

  • PARI
    seq(n)={my(g=1+O(x)); for(n=1, n\2, g=1/(1 - x^2*g^4/(1 - x*g^2))); Vec(1 + x*g + O(x*x^n))}

Formula

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

A367040 G.f. satisfies A(x) = 1 + x^2 + x*A(x)^3.

Original entry on oeis.org

1, 1, 4, 15, 70, 360, 1953, 11008, 63837, 378390, 2282205, 13960890, 86411232, 540166219, 3405341160, 21625820793, 138216775785, 888371346825, 5738510504979, 37234351046835, 242567430368298, 1585979835198675, 10403866383915844, 68453912880893025
Offset: 0

Views

Author

Seiichi Manyama, Nov 03 2023

Keywords

Crossrefs

Cf. A176697, A367041.
Cf. A366266, A366676.
Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364475, A364478.

Programs

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

Formula

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

A367027 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3 - x^2*A(x)^5.

Original entry on oeis.org

1, 1, 2, 4, 5, -13, -147, -816, -3534, -12650, -35420, -53040, 199056, 2391340, 14555740, 68264112, 261045693, 769660569, 1167906402, -5145668100, -61758940705, -385813067255, -1857144860445, -7266981925560, -21793022441775, -32643056947527, 161919845140752
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2023

Keywords

Crossrefs

Cf. A000108, A137265, A200753, A200755.
Cf. A361245, A368969.

Programs

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

Formula

a(n) = (1/(2*n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-k,k) * binomial(3*n-2*k,n-2*k).
G.f.: ( (1/x) * Series_Reversion( x * (1-x+x^2)^2 ) )^(1/2). - Seiichi Manyama, Mar 08 2025
Showing 1-6 of 6 results.

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