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.

Previous Showing 21-27 of 27 results.

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).

A379546 Expansion of (1/x) * Series_Reversion( x / ( (1+x)^2 * (1+2*x)^3 ) ).

Original entry on oeis.org

1, 8, 89, 1150, 16190, 240966, 3729185, 59404934, 967608590, 16041857672, 269807678442, 4592326407908, 78954271935856, 1369136489157250, 23918810207745777, 420575805001923782, 7437459126200243030, 132190772588551036800, 2360148586461490077870
Offset: 0

Views

Author

Seiichi Manyama, Dec 25 2024

Keywords

Links

Crossrefs

Cf. A003168, A371406, A379547.
Cf. A371398, A379522.
Cf. A371669.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(1+2*x)^3))/x)
  • PARI
    a(n) = sum(k=0, n\2, 2^(n-2*k)*binomial(n+1, k)*binomial(4*(n+1)-k, n-2*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^k * binomial(3*(n+1),k) * binomial(2*(n+1),n-k).
a(n) = A371669(n+1)/2 = (1/(n+1)) * Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n+1,k) * binomial(4*(n+1)-k,n-2*k).
a(n) = (1/(n+1)) * [x^n] ( (1+x)^2 * (1+2*x)^3 )^(n+1).

A379547 Expansion of (1/x) * Series_Reversion( x / ( (1+x)^2 * (1+2*x)^4 ) ).

Original entry on oeis.org

1, 10, 141, 2318, 41586, 789404, 15588677, 316957910, 6591000606, 139521610540, 2996554128002, 65135251885164, 1430214488595340, 31676376789702720, 706819317765805461, 15874751837921964646, 358585244386746378166, 8141109472248910295708
Offset: 0

Views

Author

Seiichi Manyama, Dec 25 2024

Keywords

Links

Crossrefs

Cf. A003168, A371406, A379546.

Programs

  • Mathematica
    CoefficientList[InverseSeries[Series[x / ( (1+x)^2 * (1+2*x)^4 ),{x,0,18}],x]/x,x] (* Stefano Spezia, Aug 25 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(1+2*x)^4))/x)
  • PARI
    a(n) = sum(k=0, n\2, 2^(n-2*k)*binomial(n+1, k)*binomial(5*(n+1)-k, n-2*k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^k * binomial(4*(n+1),k) * binomial(2*(n+1),n-k).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(n+1,k) * binomial(5*(n+1)-k,n-2*k).
a(n) = (1/(n+1)) * [x^n] ( (1+x)^2 * (1+2*x)^4 )^(n+1).

A007169 Number of Q-graphs rooted at a polygon.

Original entry on oeis.org

0, 1, 2, 8, 43, 283, 1946, 14010, 103292, 776784, 5931402, 45870248, 358501321, 2827227425, 22469524972, 179785568562, 1447059691048, 11708296894260, 95176461591230, 776935668014674, 6366225809581766, 52344041948585714
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003168.

Formula

G.f. Sum_{r >= 2} Z(C_{2*r}; B(x)) where Z(C_{2*r}) is the cycle index of the cyclic group C_{2*r} and B(x) is the g.f. for A003168. - Sean A. Irvine, Nov 07 2017

Extensions

a(9) corrected and more terms from Sean A. Irvine, Nov 07 2017

A243693 Number of Hyposylvester classes of 3-multiparking functions of length n.

Original entry on oeis.org

1, 1, 5, 32, 233, 1833, 15180, 130392, 1151057, 10378883, 95182445, 885053524, 8324942620, 79071217228, 757310811912, 7305728683824, 70923966744609, 692370887676567, 6792525607165935, 66933512163735000, 662190712902022017, 6574831459429388169, 65494637699437417584
Offset: 0

Views

Author

N. J. A. Sloane, Jun 14 2014

Keywords

Links

Crossrefs

Cf. A243694, A243695.
Cf. A007564, A003168, A364923, A364924.

Programs

  • Maple
    a := proc(n) option remember; if n <= 1 then return 1 fi;
(a(n - 2)*(-800*n^3 + 3024*n^2 - 3184*n + 672) + a(n - 1)*(3275*n^3 - 7467*n^2 +
5038*n - 1008))/(300*n^3 - 234*n^2 - 192*n) end:
seq(a(n), n = 0..22);  # Peter Luschny, Apr 13 2024
  • Mathematica
    a[n_] := 3^(n - Boole[n>0]) Hypergeometric2F1[1 - n, -2 n, 2, 1/3];
Table[a[n], {n, 0, 22}]  (* Peter Luschny, Apr 12 2024 *)
  • PARI
    a(n) = sum(k=0, n, 3^k*(-2)^(n-k)*binomial(n, k)*binomial(2*n+k+1, n)/(2*n+k+1)); \\ Seiichi Manyama, Aug 12 2023

Formula

From Seiichi Manyama, Aug 12 2023: (Start)
The following statements are equivalent:
The g.f. satisfies A(x) = 1 + x*A(x)^3 / (1 - 2*x*A(x)^2).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(n, k) * binomial(2*n+k+1, n) / (2*n + k + 1).
a(n) = (1/n) * Sum_{k=1..n} 3^(n-k) * binomial(n, k) * binomial(2*n, k-1) for n > 0.
a(n) = (1/n) * Sum_{k=0..n-1} 2^k * binomial(n, k)*binomial(3*n-k, n-1-k) for n > 0.
(End)
The above formula is proved in Theorem 4.1 of the Jun Yan link to be the number of Hyposylvester classes of 3-multiparking functions of length n. - Jun Yan, Apr 12 2024
a(n) ~ 2^(5*n+1) / (sqrt(5*Pi) * n^(3/2) * 3^(n+1)). - Vaclav Kotesovec, Apr 12 2024
a(n) = 3^(n - 1) * hypergeom([1 - n, -2*n], [2], 1/3) for n > 0. - Peter Luschny, Apr 12 2024
G.f. A(x) = 1 + series_reversion( x/((1 + 3*x)*(1 + x)^2) ). - Peter Bala, Sep 10 2024

Extensions

Name clarified by Jun Yan, Apr 12 2024

A294724 Number of Q graphs with 2*n vertices rooted at an internal edge.

Original entry on oeis.org

0, 0, 0, 1, 11, 96, 785, 6283, 49987, 397768, 3174084, 25426290, 204538114, 1652327820, 13402442651, 109129475455, 891793354235, 7312118010384, 60141701374424, 496095022537998, 4103183675407098, 34022136992384720
Offset: 1

Views

Author

Sean A. Irvine, Nov 07 2017

Keywords

Links

Crossrefs

Cf. A003168, A003169.

Formula

G.f.: P(x) * (B(x) - x)^2 where P(x) is the g.f. for A003169 and B(x) is the g.f. for A003168.

A294728 Number of Q graphs with 2*n vertices symmetrical about a distinguished edge.

Original entry on oeis.org

0, 0, 0, 1, 1, 6, 7, 37, 47, 242, 324, 1658, 2300, 11764, 16741, 85741, 124383, 638250, 939880, 4832446, 7202214, 37101556, 55842670, 288180066, 437325262, 2260516372, 3454370352, 17881862308, 27489383818, 142491830992, 220185726817
Offset: 1

Views

Author

Sean A. Irvine, Nov 07 2017

Keywords

Links

Crossrefs

Cf. A003168, A007165.

Formula

G.f.: H(x) * (B(x^2) - x^2) where H(x) is the g.f. for A007165 and B(x) is the g.f. for A003168.
Previous Showing 21-27 of 27 results.

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

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