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.

A152297 Alternate binomial partial sums of binomial(2n,n)*binomial(3n,n) (A006480).

Original entry on oeis.org

1, 5, 79, 1427, 28447, 599435, 13100065, 293737085, 6713171455, 155700711995, 3653740285729, 86561367835805, 2067026079739921, 49689509437820933, 1201321507453119103, 29187308928225658787, 712192597620218620735
Offset: 0

Views

Author

Emanuele Munarini, Apr 14 2011

Keywords

Crossrefs

Cf. A006480, A188441, A188918, A188946.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]Binomial[2k,k]Binomial[3k,k](-1)^(n-k),{k,0,n}],{n,0,16}]
  • Maxima
    makelist(sum((-1)^(n-k)*binomial(n,k)*binomial(2*k,k)*binomial(3*k,k),k,0,n),n,0,16);

Formula

a(n) = sum((-1)^(n-k)*binomial(n,k)*binomial(2*k,k)*binomial(3*k,k),k=0..n).
D-finite with recurrence Recurrence: (n+3)^2*a(n+3)-(24*n^2+120*n+149)*a(n+2)-51*(n+2)^2*a(n+1)-26*(n+1)*(n+2)*a(n)=0.
E.g.f.: exp(-x)*F(1/3,2/3;1,1;27*x), where F(a1,a2;b1;z) is a hypergeometric series.
a(n) ~ 13*sqrt(3) * 26^n / (27*Pi*n). - Vaclav Kotesovec, Mar 02 2014

A188918 Alternate partial sums of binomial(2n,n)*binomial(3n,n) (A006480).

Original entry on oeis.org

1, 5, 85, 1595, 33055, 723701, 16429435, 382643525, 9082868245, 218790563255, 5332206228085, 131194789234955, 3253536973286245, 81224561099580155, 2039348104811147845, 51455631680563483835, 1303889832725451598495
Offset: 0

Views

Author

Emanuele Munarini, Apr 14 2011

Keywords

Crossrefs

Cf. A006480, A188441, A188946, A152297.

Programs

  • Mathematica
    Table[Sum[(-1)^(n-k)*Binomial[2k,k]Binomial[3k,k],{k,0,n}],{n,0,16}] (* fixed by Vaclav Kotesovec, Nov 27 2017 *)
  • Maxima
    makelist(sum(binomial(2*k,k)*binomial(3*k,k)*(-1)^(n-k),k,0,n),n,0,16);

Formula

a(n) = sum((-1)^(n-k)*binomial(2*k,k)*binomial(3*k,k),k=0..n).
Recurrence: (n+2)^2*a(n+2)-(26*n^2+77*n+56)*a(n+1)-3*(9*n^2+27*n+20)*a(n)=0.
G.f.: F(1/3,2/3;1;27*x)/(1+x), where F(a1,a2;b1;z) is a hypergeometric series.
a(n) ~ 3^(3*n + 7/2) / (56*Pi*n). - Vaclav Kotesovec, Nov 27 2017

A188946 Binomial partial sums of binomial(2n,n)*binomial(3n,n) (A006480).

Original entry on oeis.org

1, 7, 103, 1969, 41935, 947737, 22248409, 536310271, 13183283743, 328970388985, 8307368234473, 211822788505951, 5444571611722369, 140892128574440887, 3667015053678269095, 95918056089104563489, 2519845343307697266943
Offset: 0

Views

Author

Emanuele Munarini, Apr 14 2011

Keywords

Crossrefs

Cf. A006480, A188441, A188918, A152297.

Programs

  • Magma
    [&+[Binomial(n, k)*Binomial(2*k, k)*Binomial(3*k, k): k in [0..n]]: n in [0.. 18]]; // Vincenzo Librandi, Oct 16 2017
  • Mathematica
    Table[Sum[Binomial[n,k]Binomial[2k,k]Binomial[3k,k],{k,0,n}],{n,0,16}]
Table[HypergeometricPFQ[{1/3, 2/3, -n}, {1, 1}, -27], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2017 *)
  • Maxima
    makelist(sum(binomial(n,k)*binomial(2*k,k)*binomial(3*k,k),k,0,n),n,0,16);
  • PARI
    a(n) = sum(k=0, n, binomial(n,k)*binomial(2*k,k)*binomial(3*k,k)); \\ Michel Marcus, Oct 15 2017

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*binomial(3*k,k).
Recurrence: (n+3)^2*a(n+3)-(30*n^2+150*n+187)*a(n+2)+57*(n+2)^2*a(n+1)-28*(n+1)*(n+2)*a(n)=0.
E.g.f.: exp(x)*F(1/3,2/3;1,1;27*x), where F(a1,a2;b1;z) is a hypergeometric series.
a(n) = hypergeom([1/3, 2/3, -n], [1, 1], -27). - Vladimir Reshetnikov, Oct 15 2017
a(n) ~ 2^(2*n+1) * 7^(n+1) / (3^(5/2)*Pi*n). - Vaclav Kotesovec, Nov 27 2017

A342107 a(n) = Sum_{k=0..n} (4*k)!/k!^4.

Original entry on oeis.org

1, 25, 2545, 372145, 63435145, 11796180169, 2320539673225, 474838887231625, 100035931337622625, 21552788197602942625, 4726913659271173170145, 1051798742538350304851425, 236861100204680963085573025
Offset: 0

Views

Author

Seiichi Manyama, Feb 28 2021

Keywords

Comments

Partial sums of A008977.
In general, for m > 1, Sum_{k=0..n} (m*k)!/k!^m ~ m^(m*n + m + 1/2) / ((m^m - 1) * (2*Pi*n)^((m-1)/2)). - Vaclav Kotesovec, Feb 28 2021

Crossrefs

Cf. A006134, A008977, A188441, A221177.

Programs

  • Maple
    A342107 := proc(n)
    add((4*k)!/k!^4,k=0..n) ;
end proc:
seq(A342107(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • Mathematica
    Table[Sum[(4*k)!/k!^4, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 28 2021 *)
  • PARI
    a(n) = sum(k=0, n, (4*k)!/k!^4);

Formula

a(n) ~ 2^(8*n + 15/2) / (255 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Feb 28 2021
D-finite with recurrence n^3*a(n) +(-257*n^3+384*n^2-176*n+24)*a(n-1) +8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-2)=0. - R. J. Mathar, Dec 04 2023
Showing 1-4 of 4 results.

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