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.

A114496 a(n) = Sum of binomial(n,k)*binomial(2n+k,k) over all k.

Original entry on oeis.org

1, 4, 26, 190, 1462, 11584, 93536, 765314, 6323270, 52638760, 440815036, 3709445084, 31340292076, 265683004240, 2258793820988, 19251776923210, 164440378882630, 1407266585304760, 12063701803046300, 103571977632247076
Offset: 0

Views

Author

Eric Rowland, Dec 01 2005

Keywords

Comments

Modification of A001850 inspired by the Apéry numbers A005259.
From Paul Barry, Feb 17 2009: (Start)
Central coefficient of (1 + 4x + 5x^2 + 2x^3)^n. The coefficients are the 4th row of A029635.
The third row of A029635 corresponds to the central Delannoy numbers A001850. (End)

Links

Crossrefs

Cf. A114497, A114498, A003168, A156894.
Cf. A156886. - Paul Barry, Feb 17 2009

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]*Binomial[2n+k, k], {k, 0, n}], {n,0,25}]
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(2*n,k)*binomial(n,k));
vector(50, n, a(n-1)) \\ Altug Alkan, Oct 05 2015

Formula

a(n) = Sum_{k=0..n} (binomial(n,k)*binomial(2n+k,k)).
Recurrence: 20*n*(2*n - 1)*a(n) = (371*n^2 - 411*n + 120)*a(n-1) -2*(81*n^2 - 299*n + 278)*a(n-2) + 4*(n-2)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ sqrt(1734 + 442*sqrt(17))*((71 + 17*sqrt(17))/16)^n/(68*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 19 2012
From Peter Bala, Oct 05 2015: (Start)
a(n) = Sum_{i = 0..n} 2^(n-i)*binomial(2*n,i)*binomial(n,i).
4*n*(2*n - 1)*(17*n - 23)*a(n) = (1207*n^3 - 2840*n^2 + 1897*n - 360)*a(n-1) - 2*(n - 1)*(17*n - 6)*(2*n - 3)*a(n-2) with a(0) = 1 and a(1) = 4.
1 + x*exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 4*x^2 + 21*x^3 + 126*x^4 + ... is the o.g.f. for A003168. (End)

A156887 a(n) = Sum_{k=0..n} C(n,k)*C(4*n+k,k).

Original entry on oeis.org

1, 6, 64, 768, 9708, 126386, 1676956, 22548168, 306167324, 4188703512, 57649462164, 797294161824, 11071026740964, 154250752864812, 2155368246401224, 30192512693210888, 423859798484668188, 5961793387214958792, 83998039356129372448, 1185277027372535468544
Offset: 0

Views

Author

Paul Barry, Feb 17 2009

Keywords

Comments

a(n)=[x^n] (1+6x+14x^2+16x^3+9x^4+2x^5)^n. The coefficients (1,6,14,16,9,2) are the 6th row of A029635.

Links

Crossrefs

Cf. A001850, A114496, A156886, A359646.

Programs

  • Maple
    A156887 := proc(n)
    add(binomial(n,k)*binomial(4*n+k,k),k=0..n) ;
end proc: # R. J. Mathar, Feb 25 2015
  • Mathematica
    Table[Sum[Binomial[n,k]Binomial[4n+k,k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, Jul 24 2018 *)
  • PARI
    {a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n+k, k))} \\ Seiichi Manyama, Feb 02 2019

Formula

Conjecture: 576*n*(32901928701*n-65877527665)*(4*n-3)*(2*n-1)*(4*n-1)*a(n) +(-8795436181229177*n^5 +35251410418024655*n^4 -47934714902592853*n^3 +29414167990853161*n^2 -9060238526902314*n +1466702211905280)*a(n-1) +8*(10299715469615*n^5 -136961193094719*n^4 +872530072905392*n^3 -2699499511785411*n^2 +3902106377543903*n -2123717948975100)*a(n-2) -64*(2*n-5)*(4*n-9)*(n-2)*(27741827*n-2925269736)*(4*n-11)*a(n-3)=0. - R. J. Mathar, Feb 25 2015
From Peter Bala, Feb 11 2018: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k)*C(n,k)*C(4*n+k,n)*2^k.
a(n) = Sum_{k = 0..n} C(n,k)*C(4*n,k)*2^(n-k).
8*(4*n)*(4*n-1)*(4*n-2)**(4*n-3)*(4633*n^3-19662*n^2+27593*n-12804)*a(n) = (137604733*n^7-859190528*n^6+2179882848*n^5-2890753162*n^4+2144669963*n^3-880916550*n^2+182941416*n-14515200)*a(n-1) - (4*n-4)*(4*n-5)*(4*n-6)*(4*n-7)*(4633*n^3-5763*n^2+2168*n-240)*a(n-2). A proof of Mathar's conjectured third-order recurrence above follows easily using this second-order recurrence. (End)
a(n) ~ sqrt(5 + 33/sqrt(41)) * ((29701 + 4633*sqrt(41)))^n / (sqrt(Pi*n) * 2^(12*n + 2)). - Vaclav Kotesovec, Jan 09 2023

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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