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.

A186996 G.f. satisfies: A(x) = 1 + x*A(x) + x^2*A(x)^4.

Original entry on oeis.org

1, 1, 2, 6, 20, 72, 273, 1073, 4333, 17869, 74937, 318601, 1370113, 5949201, 26046727, 114857599, 509669295, 2274146599, 10197234215, 45925646367, 207656685443, 942302814363, 4289903653615, 19588180438263, 89685571667763
Offset: 0

Views

Author

Paul D. Hanna, Mar 01 2011

Keywords

Comments

Compare to a g.f. C(x) of Catalan numbers: C(x) = 1 + x*C(x) + x^2*C(x)^3.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 72*x^5 + 273*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 56*x^4 + 208*x^5 + 806*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 114*x^4 + 438*x^5 + 1739*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 52*x^3 + 201*x^4 + 800*x^5 + 3260*x^6 +...

Links

Crossrefs

Cf. A182454.

Programs

  • Maple
    S:= series(RootOf(y - 1 - x*y - x^2*y^4, y), x, 101):
seq(coeff(S,x,i),i=0..100); # Robert Israel, Apr 23 2015
  • 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 + x^2*AGF^4 - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 10 2013 *)
Table[Sum[Binomial[3n-2k+1,k]*Binomial[k,n-k]/(3n-2k+1),{k,0,n}],{n,0,20}]  (* Vaclav Kotesovec, Apr 27 2015 after Michael D. Weiner *)
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+x*A+x^2*(A+x*O(x^n))^4);polcoeff(A,n)}
  • PARI
    {a(n)=polcoeff((1/x)*serreverse(2*x^3/(1 - sqrt(1-4*x^2-4*x^3 +x^3*O(x^n)))),n)}

Formula

G.f.: A(x) = (1/x)*Series_Reversion( 2*x^3/(1 - sqrt(1-4*x^2-4*x^3)) ).
G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x*A(x)^3) / (1 + x*A(x)^2).
Recurrence: 3*(n-2)*n*(2*n-5)*(3*n-2)*(3*n+2)*a(n) = 3*(2*n-5)*(2*n-1)*(18*n^3 - 54*n^2 + 36*n - 5)*a(n-1) + (2*n-3)*(94*n^4 - 564*n^3 + 1222*n^2 - 1128*n + 331)*a(n-2) + 3*(2*n - 5)*(2*n - 1)*(18*n^3 - 108*n^2 + 198*n - 103)*a(n-3) - 3*(n-3)*(n-1)*(2*n-1)*(3*n-11)*(3*n-7)*a(n-4). - Vaclav Kotesovec, Sep 10 2013
a(n) ~ c*d^n/n^(3/2), where d = (9 + 8*sqrt(3) + 4*sqrt(12 + 9*sqrt(3)))/9 = 4.874032512954972962... is the root of the equation 27 - 108*d - 94*d^2 - 108*d^3 + 27*d^4 = 0 and c = sqrt(11/(8*(-44 + sqrt(748 + 891*sqrt(3)))*Pi)) = 0.336422381089368230542882135982348331566666028... - Vaclav Kotesovec, Sep 10 2013, updated Apr 27 2015
a(n) = sum(k=0..n, binomial(3n-2k+1, k)*binomial(k,n-k)*1/(3n-2k+1)). - Michael D. Weiner, Apr 23 2015
G.f.: A(x) = hypergeom([1/4, 1/2, 3/4], [2/3, 4/3], (256/27)*x^2/(x-1)^4)/(1-x). - Robert Israel, Apr 28 2015

A364472 G.f. satisfies A(x) = 1 + x*A(x) + x^2*A(x)^6.

Original entry on oeis.org

1, 1, 2, 8, 35, 163, 808, 4162, 22041, 119325, 657384, 3673394, 20769983, 118610807, 683131766, 3963486380, 23144000681, 135911263309, 802143851323, 4755506884495, 28306896506651, 169110331570307, 1013643450123455, 6094125091837335, 36739933169338731
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2023

Keywords

Links

Crossrefs

Cf. A000045, A000108, A001006, A182454, A186996, A364476.
Cf. A002295.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(n+4*k, k)*binomial(n+3*k, n-2*k)/(5*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n+4*k,k) * binomial(n+3*k,n-2*k) / (5*k+1) = Sum_{k=0..floor(n/2)} binomial(n+4*k,6*k) * binomial(6*k,k) / (5*k+1).

A364522 G.f. satisfies A(x) = 1 + x*A(x) + x^5*A(x)^5.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 258, 518, 1123, 2718, 7008, 18054, 44969, 108189, 255919, 609179, 1482210, 3689155, 9294440, 23419705, 58639835, 145948111, 362721386, 904673836, 2270287636, 5729191861, 14502873988, 36735974548, 93001413353, 235372519273
Offset: 0

Views

Author

Seiichi Manyama, Jul 27 2023

Keywords

Comments

Number of ordered trees with n edges and having nonleaf nodes of outdegrees 1 or 5. - Emanuele Munarini, Jul 11 2024

Links

Crossrefs

Cf. A001006, A071879, A127902, A364523.
Cf. A182454, A349311.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n, 5*k)*binomial(5*k, k)/(4*k+1));

Formula

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

A364476 G.f. satisfies A(x) = 1 + x*A(x) + x^2*A(x)^7.

Original entry on oeis.org

1, 1, 2, 9, 44, 226, 1241, 7093, 41666, 250260, 1529993, 9488398, 59545909, 377451385, 2413157855, 15542535697, 100753850132, 656856027658, 4303970039402, 28328599504756, 187214549485759, 1241775795647609, 8263989319451514, 55163575187733922
Offset: 0

Views

Author

Seiichi Manyama, Jul 26 2023

Keywords

Links

Crossrefs

Cf. A000045, A000108, A001006, A182454, A186996, A364472.
Cf. A002296, A364477.

Programs

  • PARI
    a(n) = sum(k=0, n\2, binomial(n+5*k, k)*binomial(n+4*k, n-2*k)/(6*k+1));

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n+5*k,k) * binomial(n+4*k,n-2*k) / (6*k+1) = Sum_{k=0..floor(n/2)} binomial(n+5*k,7*k) * binomial(7*k,k) / (6*k+1).

A367030 G.f. satisfies A(x) = 1 + x*A(x) - x^2*A(x)^5.

Original entry on oeis.org

1, 1, 0, -5, -15, 0, 170, 619, 69, -8351, -33601, -8446, 480679, 2078534, 830739, -30221481, -138791261, -76306831, 2009474129, 9742939809, 6811893809, -138835809721, -708398431831, -599833654686, 9861055840019, 52859680413599, 52446605719469
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2023

Keywords

Crossrefs

Cf. A343773, A367028, A367029.
Cf. A182454.

Programs

  • PARI
    a(n) = sum(k=0, n\2, (-1)^k*binomial(n+3*k, k)*binomial(n+2*k, n-2*k)/(4*k+1));

Formula

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

A364539 G.f. satisfies A(x) = 1 + x*A(x) + x^3*A(x)^5.

Original entry on oeis.org

1, 1, 1, 2, 7, 22, 62, 182, 583, 1928, 6358, 21063, 70888, 241889, 831634, 2874584, 9995579, 34966279, 122938956, 434062141, 1538378816, 5471697241, 19525345791, 69880082323, 250767909528, 902123110483, 3252793321513, 11753570922933, 42553831219830
Offset: 0

Views

Author

Seiichi Manyama, Jul 28 2023

Keywords

Crossrefs

Cf. A063019, A182454, A349311, A364522.
Cf. A023431, A071879, A215340.

Programs

  • PARI
    a(n) = sum(k=0, n\3, binomial(n+2*k, 5*k)*binomial(5*k, k)/(4*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(n+2*k,5*k) * binomial(5*k,k) / (4*k+1).

A365798 G.f. satisfies A(x) = 1 + x*A(x)*(1 + x^5*A(x)^4).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 253, 468, 848, 1618, 3433, 8009, 19384, 46264, 106369, 235179, 505955, 1079790, 2332555, 5166405, 11737860, 27086236, 62676956, 144074416, 327837356, 739787486, 1663922487, 3751649542, 8513640107, 19464624667
Offset: 0

Views

Author

Seiichi Manyama, Sep 19 2023

Keywords

Crossrefs

Cf. A063019, A182454, A349311, A364539, A364522.
Cf. A005708, A364523, A365732, A365733.
Cf. A002294, A365736.

Programs

  • PARI
    a(n) = sum(k=0, n\6, binomial(n-5*k, k)*binomial(n-k+1, n-5*k)/(n-k+1));

Formula

a(n) = Sum_{k=0..floor(n/6)} binomial(n-5*k,k) * binomial(n-k+1,n-5*k) / (n-k+1) = Sum_{k=0..floor(n/6)} binomial(n-k,5*k) * binomial(5*k,k) / (4*k+1).
Showing 1-7 of 7 results.

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

Content is available under The OEIS End-User License Agreement.