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 1-7 of 7 results.

A215341 Expansion of series_reversion( x/(1+x^4*sum(k>=0, x^k)) ) / x.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 5, 10, 16, 23, 53, 118, 232, 411, 813, 1718, 3568, 7012, 13925, 28603, 59533, 121878, 247915, 509136, 1057278, 2194138, 4536943, 9394145, 19552639, 40803472, 85131237, 177640486, 371426592, 778275264, 1632420197, 3425607187, 7195476245, 15134138683, 31866093569
Offset: 0

Views

Author

Joerg Arndt, Aug 19 2012

Keywords

Comments

Number of Dyck words of semilength n with substrings UUU...UU (ascents) only of lengths >= 4. See A215340 for an explanation. [Joerg Arndt, Apr 16 2013]

Links

Crossrefs

Cf. A000108 (rev. of x/(1+1*sum(k>=1,x^k)) ), A005043 (rev. of x/(1+x*sum(k>=1,x^k)) ), A114997 (rev. of x/(1+x^2*sum(k>=1,x^k)) ).
Cf. A001003 (rev. of x*(1-1*sum(k=1,N,x^k)) ), A046736 (rev. of x*(1-x*sum(k=1,N,x^k)) ), A054514 (rev. of x*(1-x^2*sum(k=1,N,x^k)) ), A215342 (rev. of x*(1-x^3*sum(k=1,N,x^k)) ).

Programs

  • Maple
    b:= proc(x, y, t) option remember; `if`(y0 and t in [0, 4],
       b(x-1, y, 0), 0) +b(x, y-1, min(t+1, 4))))
    end:
a:= n-> b(n, n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 16 2013
  • Mathematica
    InverseSeries[x/(1+x^4/(1-x)) + O[x]^50] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Mar 29 2017 *)
  • Maxima
    a(n):=sum(binomial(n+1,i)*binomial(n-3*i-1,n-4*i),i,0,floor(n/4))/(n+1); /* Vladimir Kruchinin, Apr 01 2019 */
  • PARI
    N=66; Vec( serreverse(x/(1+x^4*sum(k=0,N,x^k))+O(x^N)) / x )

Formula

G.f. A(x) satisfies 0 = -x^4*A(x)^4 - x*A(x)^2 + (x + 1)*A(x) - 1. [Joerg Arndt, Mar 01 2014]
Recurrence: 2*n*(n+1)*(2*n+3)*(16204*n^4 - 82948*n^3 + 139973*n^2 - 85643*n + 10674)*a(n) = - (n-1)*n*(307876*n^5 - 960260*n^4 + 288863*n^3 + 582749*n^2 + 5406*n + 12696)*a(n-1) + 4*(n-2)*(129632*n^6 - 469136*n^5 + 354226*n^4 + 317255*n^3 - 469674*n^2 + 176517*n - 21420)*a(n-2) - 2*(n-3)*(n-2)*(16204*n^5 - 34336*n^4 + 82943*n^3 - 208775*n^2 + 192120*n - 40656)*a(n-3) + 6*(n-3)*(n-2)*(97224*n^5 - 351852*n^4 + 179198*n^3 + 540009*n^2 - 571727*n + 92968)*a(n-4) + 229*(n-4)*(n-3)*(n-2)*(16204*n^4 - 18132*n^3 - 11647*n^2 + 10275*n - 1740)*a(n-5). - Vaclav Kotesovec, Mar 22 2014
a(n) ~ sqrt((s-1)*s^3/(6-8*s+3*s^2)) / (2*sqrt(Pi)*n^(3/2)*r^n), where r = 0.4577644245749322..., s = 1.232809919151165... are roots of the system of equations 1 + r*s^2 + r^4*s^4 = (1+r)*s, 1+r = 2*r*s + 4*r^4*s^3. - Vaclav Kotesovec, Mar 22 2014
a(n) = (1/(n+1)) * Sum_{i=0..floor(n/4)} C(n+1,i) * C(n-3*i-1,n-4*i). - Vladimir Kruchinin, Apr 01 2019

Extensions

Modified definition to obtain offset 0 for combinatorial interpretation, Joerg Arndt, Apr 16 2013

A365731 G.f. satisfies A(x) = 1 + x^4*A(x)^5*(1 + x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 5, 11, 6, 0, 35, 120, 136, 51, 285, 1330, 2310, 1771, 3036, 14950, 35100, 40950, 47502, 175392, 503440, 791120, 927520, 2272424, 7037184, 13803405, 18643560, 33997080, 98920536, 226318196, 359255325, 578590155, 1445166360, 3584815443, 6573439928
Offset: 0

Views

Author

Seiichi Manyama, Sep 17 2023

Keywords

Links

Crossrefs

Cf. A365727, A365728, A365729, A365730.
Cf. A001002, A217358, A365725.
Cf. A215342.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(k, n-4*k)*binomial(n+k+1, k)/(n+k+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(k,n-4*k) * binomial(n+k+1,k) / (n+k+1).
G.f.: (1/x) * Series_Reversion( x*(1 - x^4*(1 + x)) ). - Seiichi Manyama, Sep 24 2024

A368932 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x-x^4) ).

Original entry on oeis.org

1, 2, 7, 30, 144, 741, 3996, 22287, 127495, 743941, 4410555, 26492349, 160875186, 986007700, 6091548256, 37894543413, 237168491610, 1492323419929, 9434943086870, 59906035386393, 381832957589226, 2442251022673595, 15670578495195870
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2024

Keywords

Crossrefs

Cf. A236339, A368931, A368933.
Cf. A063021, A215342.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(3*n-3*k+1, n-4*k))/(n+1);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x-x^4))/x)

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} binomial(n+k,k) * binomial(3*n-3*k+1,n-4*k).

A365696 G.f. satisfies A(x) = 1 + x^4*A(x)^2 / (1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 3, 6, 10, 15, 26, 49, 92, 165, 294, 535, 994, 1852, 3437, 6379, 11905, 22344, 42058, 79260, 149601, 283038, 536806, 1020066, 1941317, 3699922, 7062308, 13500402, 25842489, 49528164, 95031920, 182545222, 351023451, 675678911, 1301838177
Offset: 0

Views

Author

Seiichi Manyama, Sep 16 2023

Keywords

Crossrefs

Cf. A023427, A215341, A215342, A357308, A365697.
Cf. A112805.

Programs

  • Mathematica
    CoefficientList[Series[(1 + x - Sqrt[1 - 2*x + x^2 - 4*x^4])/(2*x*(1 + x^3)), {x, 0, 40}], x] (* Vaclav Kotesovec, Sep 26 2023 *)
  • PARI
    a(n) = sum(k=0, n\4, binomial(n-3*k-1, n-4*k)*binomial(n-2*k+1, k)/(n-2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k) * binomial(n-2*k+1,k) / (n-2*k+1).
From Vaclav Kotesovec, Sep 26 2023: (Start)
G.f.: (1 + x - sqrt(1 - 2*x + x^2 - 4*x^4)) / (2*x*(1 + x^3)).
a(n) ~ 2^(n + 3/2) / (sqrt(Pi) * 3^(3/2) * n^(3/2)). (End)

A365697 G.f. satisfies A(x) = 1 + x^4*A(x)^3 / (1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 4, 8, 13, 19, 38, 79, 153, 273, 509, 999, 1979, 3818, 7331, 14279, 28189, 55599, 109275, 215165, 426093, 846638, 1683215, 3348212, 6673679, 13333171, 26679522, 53437369, 107151335, 215154204, 432586412, 870678377, 1754094266
Offset: 0

Views

Author

Seiichi Manyama, Sep 16 2023

Keywords

Crossrefs

Cf. A023427, A215341, A215342, A357308, A365696.
Cf. A365245.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(n-3*k-1, n-4*k)*binomial(n-k+1, k)/(n-k+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k) * binomial(n-k+1,k) / (n-k+1).

A366112 Expansion of (1/x) * Series_Reversion( x*(1-x-x^5)/(1-x) ).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 7, 14, 22, 31, 41, 103, 235, 457, 791, 1261, 2399, 5015, 10257, 19676, 35296, 65170, 127520, 256187, 507601, 969495, 1834433, 3534477, 6962249, 13809538, 27061252, 52439361, 101701035, 199152071, 393332277, 776589611, 1525416837
Offset: 0

Views

Author

Seiichi Manyama, Sep 29 2023

Keywords

Links

Crossrefs

Cf. A046736, A054514, A215342.
Cf. A017899.

Programs

  • Mathematica
    CoefficientList[InverseSeries[Series[x*(1-x-x^5)/(1-x),{x,0,41}]]/x,x] (* Stefano Spezia, Aug 21 2025 *)
  • PARI
    a(n) = sum(k=0, n\5, binomial(n+k, k)*binomial(n-4*k-1, n-5*k))/(n+1);

Formula

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

A365758 G.f. satisfies A(x) = 1 + x*A(x) / (1 - x^4*A(x)^5).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 8, 29, 85, 212, 481, 1081, 2627, 7100, 20328, 58023, 160430, 430391, 1140892, 3051678, 8334638, 23199896, 65148939, 182781853, 510225082, 1419091293, 3948954920, 11034704856, 31001204632, 87466532564, 247303929326, 699572256145
Offset: 0

Views

Author

Seiichi Manyama, Sep 18 2023

Keywords

Crossrefs

Cf. A023427, A112805, A212384, A365245.
Cf. A215342, A365760.

Programs

  • PARI
    a(n) = sum(k=0, n\4, binomial(n-3*k-1, k)*binomial(n+k+1, n-4*k)/(n+k+1));

Formula

a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k-1,k) * binomial(n+k+1,n-4*k) / (n+k+1).
Showing 1-7 of 7 results.

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