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 1-4 of 4 results.

A156886 a(n) = Sum_{k=0..n} C(n,k)*C(3*n+k,k).

Original entry on oeis.org

1, 5, 43, 416, 4239, 44485, 475780, 5156548, 56437231, 622361423, 6904185523, 76964141600, 861408728964, 9673849095708, 108954068684616, 1230185577016156, 13920106205444335, 157814104889538739
Offset: 0

Views

Author

Paul Barry, Feb 17 2009

Keywords

Comments

a(n)=[x^n](1+5x+9x^2+7x^3+2x^4)^n. The coefficients (1,5,9,7,2) are the 5th row of A029635.

Links

Crossrefs

Cf. A001850, A114496, A029635, A156887.

Programs

  • Maple
    A156886 := proc(n)
    add(binomial(n,k)*binomial(3*n+k,k), k = 0..n);
end proc:
seq(A156886(n), n = 0..20); # Peter Bala, Feb 11 2018
  • Mathematica
    a[n_] := Sum[ Binomial[n, k] Binomial[3n + k, k], {k, 0, n}]; Array[a, 21, 0] (* Robert G. Wilson v, Feb 11 2018 *)

Formula

From Peter Bala, Feb 11 2018: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k)*C(n,k)*C(3*n+k,n)*2^k.
a(n) = Sum_{k = 0..n} C(n,k)*C(3*n,k)*2^(n-k),
12*n*(3*n-1)*(3*n-2)*(238*n^2 - 663*n + 457)*a(n) = 2*(150416*n^5 - 644640*n^4 + 1020351*n^3 - 734334*n^2 + 237007*n - 26880)*a(n-1) - (3*n-3)*(3*n-4)*(3*n-5)*(238*n^2 - 187*n + 32)*a(n-2). (End)
a(n) = P_n(0,2*n,3) where P_n(a,b,x) is the n-th Jacobi polynomial with parameters a and b. - Robert Israel, Feb 11 2018
a(n) ~ sqrt(1/3 + 11/(12*sqrt(7))) * ((316 + 119*sqrt(7))/54)^n / sqrt(Pi*n). - Vaclav Kotesovec, Jan 09 2023

A359643 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,k).

Original entry on oeis.org

1, 5, 37, 317, 2885, 27105, 259765, 2523813, 24768069, 244941833, 2437083697, 24367722725, 244639635749, 2464477467769, 24899468129405, 252202062544617, 2560119328830725, 26038134699958233, 265278657849511561, 2706809063101138409, 27657194997231516145, 282941098708193905485
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 09 2023

Keywords

Comments

In general, for m>1, Sum_{k=0..n} binomial(n,k) * binomial(m*k,k) ~ sqrt((m + (1 - 1/m)^(m-1))/(m-1)) * (1 + m^m/(m-1)^(m-1))^n / sqrt(2*Pi*n).

Links

Crossrefs

Cf. A026375, A188686.
Cf. A156887, A346646, A346664.

Programs

  • Maple
    A359643 := proc(n)
    hypergeom([-n,1/4,1/2,3/4],[1/3,2/3,1],-256/27) ;
    simplify(%) ;
end proc:
seq(A359643(n),n=0..40) ; # R. J. Mathar, Jan 10 2023
  • Mathematica
    Table[Sum[Binomial[n, k]*Binomial[4*k, k], {k, 0, n}], {n, 0, 20}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k) * binomial(4*k,k)); \\ Michel Marcus, Jan 09 2023

Formula

a(n) ~ 283^(n + 1/2) / (2^(7/2) * sqrt(Pi*n) * 3^(3*n + 1/2)).
Conjecture D-finite with recurrence +81*n*(3*n-1)*(3*n-2)*a(n) +3*(243*n^3-8433*n^2+14984*n-7064)*a(n-1) +2*(-58607*n^3+297306*n^2-491401*n+269124)*a(n-2) +6*(n-2)*(56663*n^2-237722*n+252221)*a(n-3) -3*(n-2)*(n-3)*(111625*n-286402)*a(n-4) +110653*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 09 2023
a(n) = 4F3( -n,1/4,1/2,3/4 ; 1/3, 2/3,1 ; -256/27). - R. J. Mathar, Jan 10 2023
a(n) = [x^n] (1 + 5*x + 6*x^2 + 4*x^3 + x^4)^n. - Ilya Gutkovskiy, Apr 17 2025

A359646 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(5*n+k,k).

Original entry on oeis.org

1, 7, 89, 1273, 19181, 297662, 4707971, 75459496, 1221388525, 19919031781, 326797222834, 5387618403526, 89178832899887, 1481143718244912, 24671054686539336, 411966653603163008, 6894167059382069485, 115593504497163747167, 1941434442814233362939, 32656575110841643234631
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 09 2023

Keywords

Comments

In general, for m>0, Sum_{k=0..n} binomial(n,k) * binomial(m*n+k,k) ~ (m+c) / sqrt(2*Pi*c*m * (m*(2-c)+c)*n) * d^n, where d = (m+c)^(m+c) / ((1-c)^(1-c) * c^(2*c) * m^m) and c = (sqrt(m^2 + 6*m + 1) + 1 - m)/4.
Equivalently, d = (3 + m + sqrt(1 + m*(6 + m))) * (1 + 3*m + sqrt(1 + m*(6 + m)))^m / (2^(2*m + 1) * m^m).

Links

Crossrefs

Cf. A001850, A114496, A156886, A156887.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]*Binomial[5*n+k, k], {k, 0, n}], {n, 0, 20}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k) * binomial(5*n+k,k)) \\ Andrew Howroyd, Jan 09 2023

Formula

a(n) ~ sqrt(3/10 + 23/(20*sqrt(14))) * ((108007 + 28854*sqrt(14))/12500)^n / sqrt(Pi*n).

A306280 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(n^2+k,k).

Original entry on oeis.org

1, 3, 26, 416, 9708, 297662, 11306572, 512307336, 26968496504, 1617489748394, 108885682104744, 8129721925098468, 666736347200187804, 59582961423951290184, 5762936296492591067968, 599807329803134064385488, 66843498592187788579795440
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2019

Keywords

Links

Crossrefs

Cf. A001850, A114496, A135860, A156886, A156887.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[n,k] * Binomial[n^2+k,k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 03 2019 *)
  • PARI
    {a(n) = sum(k=0, n, binomial(n, k)*binomial(n^2+k, k))}

Formula

From Vaclav Kotesovec, Feb 08 2019: (Start)
a(n) ~ exp(1) * A135860(n).
a(n) ~ exp(n + 3/2) * n^(n - 1/2) / sqrt(2*Pi). (End)
Showing 1-4 of 4 results.

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