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.

A006634 a(n) = 3*binomial(4*n+8, n)/(n+3).

Original entry on oeis.org

1, 9, 72, 570, 4554, 36855, 302064, 2504304, 20974005, 177232627, 1509395976, 12943656180, 111676661460, 968786892675, 8445123522144, 73940567860896, 649942898236596, 5733561315124260, 50744886833898400, 450461491952952690, 4009721145437152530, 35782256673785401065
Offset: 0

Views

Author

Simon Plouffe

Keywords

Comments

Former name: From generalized Catalan numbers.

References

  • H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A006630, A006631, A006632, A006633, A006635, A006636, A006637.

Programs

  • Magma
    A006634:= func< n | 3*Binomial(4*n+8,n)/(n+3) >;
[A006634(n): n in [0..40]]; // G. C. Greubel, Sep 01 2025
  • Maple
    series(RootOf(g = 1+x*g^4,g)^9, x=0, 30); # Mark van Hoeij, Apr 22 2013
  • Mathematica
    f[x_] := HypergeometricPFQ[ {9/4, 5/2, 11/4, 3}, {10/3, 11/3, 4}, 256/27*x]; Series[f[x], {x, 0, 16}] // CoefficientList[#, x]& (* Jean-François Alcover, Apr 23 2013, after Simon Plouffe *)
Table[3*Binomial[4*n+8,n]/(n+3), {n,0,40}] (* G. C. Greubel, Sep 01 2025 *)
  • PARI
    N = 3*66;  x = 'x + O('x^N);
g=serreverse(x-x^4)/x;
gf=g^9;  v=Vec(gf);
vector(#v\3,n,v[3*n-2])
/* Joerg Arndt, Apr 23 2013 */
  • SageMath
    def A006634(n): return 3*binomial(4*(n+2),n)//(n+3)
print([A006634(n) for n in range(41)]) # G. C. Greubel, Sep 01 2025

Formula

G.f.: hypergeom([9/4, 5/2, 11/4, 3], [10/3, 11/3, 4], 256/27*x). - Simon Plouffe, Master's Thesis, UQAM, 1992
G.f.: g^9 where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Apr 22 2013
From G. C. Greubel, Sep 01 2025: (Start)
a(n) = 3*binomial(4*n+8, n)/(n+3).
E.g.f.: hypergeom([9/4, 5/2, 11/4, 3], [1, 10/3, 11/3, 4], 256*x/27). (End)

Extensions

More terms from Joerg Arndt, Apr 23 2013
New name by G. C. Greubel, Sep 01 2025

A006633 Expansion of hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x).

Original entry on oeis.org

1, 6, 39, 272, 1995, 15180, 118755, 949344, 7721604, 63698830, 531697881, 4482448656, 38111876530, 326439471960, 2814095259675, 24397023508416, 212579132600076, 1860620845932216, 16351267454243260, 144222309948974400, 1276307560533365955, 11329053395044653180
Offset: 0

Views

Author

Simon Plouffe

Keywords

Comments

From generalized Catalan numbers.

References

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

Links

Crossrefs

Cf. A006631, A006632, A006634, A006635, A000027, A000260.

Programs

  • Maple
    gf := hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x):
ser := series(gf, x, 22): seq(coeff(ser, x, n), n = 0..21); # Peter Luschny, Feb 22 2024
  • Mathematica
    A006633[n_] := 2*Binomial[4*n+5, n]/(n+2);
Array[A006633, 25, 0] (* Paolo Xausa, Feb 25 2024 *)

Formula

O.g.f.: hypergeom_4F3([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x). - Simon Plouffe, Master's Thesis, UQAM 1992
a(n) = 2*binomial(4*n + 5, n) / (n+2). - Bruno Berselli, Jan 18 2014
a(n) = (n+1) * A000260(n+1). - F. Chapoton, Feb 22 2024

Extensions

New name by using a formula from the author by Peter Luschny, Feb 24 2024

A233667 a(n) = 5*binomial(4*n+10,n)/(2*n+5).

Original entry on oeis.org

1, 10, 85, 700, 5750, 47502, 395560, 3321120, 28102425, 239503550, 2054455634, 17726454200, 153757722300, 1340045361750, 11729338225200, 103068670351552, 908923976461140, 8041606944709800, 71359997110169625, 634978885837495500, 5664526697522326590
Offset: 0

Views

Author

Tim Fulford, Dec 14 2013

Keywords

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=4, r=10.

Links

Crossrefs

Cf. A000108, A002293, A069271, A006632, A196678, A006633, A233658, A233666, A006634, A006635.

Programs

  • Magma
    [5*Binomial(4*n+10,n)/(2*n+5): n in [0..30]];
  • Mathematica
    Table[5 Binomial[4 n + 10, n]/(2 n + 5), {n, 0, 30}]
  • PARI
    a(n) = 5*binomial(4*n+10,n)/(2*n+5);
  • PARI
    {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(2/5))^10+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=4, r=10.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 4F4(5/2,11/4,3,13/4; 1,11/3,4,13/3; 256*x/27).
a(n) ~ 5*2^(8*n+39/2)/(sqrt(Pi)*3^(3*n+21/2)*n^(3/2)). (End)

A233666 a(n) = 2*binomial(4*n + 8, n)/(n + 2).

Original entry on oeis.org

1, 8, 60, 456, 3542, 28080, 226548, 1855040, 15380937, 128896456, 1090119316, 9292881360, 79769043900, 688915123680, 5981962494852, 52193342019456, 457367224685012, 4023551800087200, 35521420783728880, 314608026125871720, 2794654131668318430
Offset: 0

Views

Author

Tim Fulford, Dec 14 2013

Keywords

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=4, r=8.

Links

Crossrefs

Cf. A000108, A002293, A069271, A006632, A196678, A006633, A233658, A006634, A233667, A006635.

Programs

  • Magma
    [2*Binomial(4*n+8,n)/(n+2): n in [0..30]]; // Vincenzo Librandi, Dec 14 2013
  • Mathematica
    Table[2/(n + 2) Binomial[4 n + 8, n], {n, 0, 40}] (* Vincenzo Librandi, Dec 14 2013 *)
  • PARI
    a(n) = 4*binomial(4*n+8,n)/(n+2);
  • PARI
    {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(1/2))^8+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=4, r=8.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 4F4(2,9/4,5/2,11/4; 1,3,10/3,11/3; 256*x/27).
a(n) ~ 2^(8*n+35/2)/(sqrt(Pi)*3^(3*n+17/2)*n^(3/2)). (End)
D-finite with recurrence 3*(3*n+7)*(n+2)*(3*n+8)*a(n) -2*(n+1)*(317*n^2+954*n+709)*a(n-1) +112*(4*n+1)*(2*n+1)*(4*n+3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024

A006637 Expansion of (2 - x)^4/(1 - x)^8.

Original entry on oeis.org

16, 96, 344, 952, 2241, 4712, 9108, 16488, 28314, 46552, 73788, 113360, 169507, 247536, 354008, 496944, 686052, 932976, 1251568, 1658184, 2172005, 2815384, 3614220, 4598360, 5802030, 7264296, 9029556, 11148064, 13676487, 16678496, 20225392, 24396768, 29281208
Offset: 0

Views

Author

Simon Plouffe

Keywords

Comments

Former name: From generalized Catalan numbers. - G. C. Greubel, Sep 03 2025

References

  • H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A006633, A006634, A006635, A006636, A181289.

Programs

  • Magma
    A006637:= func< n | (n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)/5040 >;
[A006637(n): n in [0..40]]; // G. C. Greubel, Sep 03 2025
  • Mathematica
    Table[(n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)/7!, {n,0,40}] (* G. C. Greubel, Sep 03 2025 *)
  • SageMath
    def A006637(n): return (n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2+31*n+192)//5040
print([A006637(n) for n in range(41)]) # G. C. Greubel, Sep 03 2025

Formula

G.f.: (2-x)^4/(1-x)^8. - Sean A. Irvine, May 31 2017
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Wesley Ivan Hurt, Jun 18 2022
From G. C. Greubel, Sep 03 2025: (Start)
a(n) = Sum_{k=0..4} binomial(4, k)*binomial(n+k+3, k+3).
a(n) = (1/7!)*(n+1)*(n+2)*(n+3)*(n+5)*(n+14)*(n^2 + 31*n + 192).
E.g.f.: (1/7!)*(80640 + 403200*x + 423360*x^2 + 161280*x^3 + 27090*x^4 + 2142*x^5 + 77*x^6 + x^7)*exp(x). (End)

Extensions

a(6) and a(8) corrected and more terms from Sean A. Irvine, May 31 2017
New name by G. C. Greubel, Sep 03 2025

A339350 Triangle read by rows. T(n,k) = Sum_{j=0..k} binomial(k-j+2, 2)*T(n-1, j), for n>=0, 0<=k<=n, with T(0,0)=1 and T(n,n)=0 for n>0.

Original entry on oeis.org

1, 1, 0, 1, 3, 0, 1, 6, 15, 0, 1, 9, 39, 91, 0, 1, 12, 72, 272, 612, 0, 1, 15, 114, 570, 1995, 4389, 0, 1, 18, 165, 1012, 4554, 15180, 32890, 0, 1, 21, 225, 1625, 8775, 36855, 118755, 254475, 0, 1, 24, 294, 2436, 15225, 75516, 302064, 949344, 2017356, 0
Offset: 0

Views

Author

Michel Marcus, Dec 01 2020

Keywords

Examples

			Triangle begins:
  1;
  1,  0;
  1,  3,  0;
  1,  6, 15,   0;
  1,  9, 39,  91,   0;
  1, 12, 72, 272, 612, 0;
  ...

Links

Crossrefs

Cf. subdiagonals: A006632, A006633, A006634, A006635.
Cf. A002293 (row sums).

Programs

  • Mathematica
    A339350[n_, k_] := A339350[n, k] = Which[k == 0, 1, n == k, 0, True, Sum[Binomial[k-j+2, 2]*A339350[n-1, j], {j, 0, k}]];
Table[A339350[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Feb 23 2024 *)
  • PARI
    T(n,k) = if ((n==0) && (k==0), 1, if (n<=k, 0, sum(j=0, k, binomial(k-j+2, 2)*T(n-1, j))));
Showing 1-6 of 6 results.

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

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