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 1-3 of 3 results.

A188686 Binomial transform of the sequence of binomial(3n,n).

Original entry on oeis.org

1, 4, 22, 139, 934, 6484, 45931, 329893, 2393470, 17499892, 128732992, 951674398, 7064138779, 52616241370, 393052285291, 2943582912904, 22093111508686, 166141033332448, 1251528633163264, 9442096410241438, 71333250226656784
Offset: 0

Views

Author

Emanuele Munarini, Apr 08 2011

Keywords

Comments

Binomial transform of A005809.

Links

Crossrefs

Cf. A005809, A359643.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]Binomial[3k,k],{k,0,n}],{n,0,22}]
  • Maxima
    makelist(sum(binomial(n,k)*binomial(3*k,k),k,0,n),n,0,20);

Formula

G.f.: 2*cos((1/3)*arcsin(3/2*sqrt(3x/(1-x))))/sqrt(4-35x+31x^2).
D-finite recurrence: 2*n*(2*n-1)*a(n) = (39*n^2-43*n+12)*a(n-1) - 2*(n-1)*(33*n-34)*a(n-2) + 31*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 31^(n+1/2)/(6*4^n*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012
a(n) = [x^n] (1 + 4*x + 3*x^2 + x^3)^n. - Ilya Gutkovskiy, Apr 17 2025

A383119 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(4*k,k).

Original entry on oeis.org

1, 3, 21, 147, 1093, 8343, 64869, 510891, 4062277, 32539647, 262181601, 2122581123, 17252278789, 140695104943, 1150670390541, 9433965332127, 77512716483461, 638080242074447, 5261486780929209, 43450477494413751, 359308411992366513, 2974886601163646379, 24657831769475675253
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 17 2025

Keywords

Comments

Inverse binomial transform of A005810.

Crossrefs

Cf. A002426, A005810, A346664, A359643, A383118.
Cf. A317133.

Programs

  • Mathematica
    Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[4 k, k], {k, 0, n}], {n, 0, 22}]
Table[(-1)^n HypergeometricPFQ[{1/4, 1/2, 3/4, -n}, {1/3, 2/3, 1}, 256/27], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(1/x) Sum[Binomial[4 k, k] (x/(1 + x))^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*binomial(4*k, k)); \\ Seiichi Manyama, Apr 17 2025

Formula

G.f.: (1/x) * Sum_{k>=0} binomial(4*k,k) * (x/(1 + x))^(k+1).
a(n) = [x^n] (1 + 3*x + 6*x^2 + 4*x^3 + x^4)^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x/((1+x)^4 - x) ). See A317133. - Seiichi Manyama, Apr 17 2025
a(n) ~ 229^(n + 1/2) / (2^(7/2) * sqrt(Pi*n) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Apr 17 2025

A383120 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k).

Original entry on oeis.org

1, 2, 11, 139, 2885, 82381, 2979565, 130203494, 6664589321, 390857822425, 25832193906761, 1899273577364197, 153741850998047053, 13585520026454056279, 1301210398133681268381, 134270617908678099820891, 14849785991790603714043921, 1752283118795349858851381297
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 17 2025

Keywords

Crossrefs

Cf. A001850, A014062, A026375, A188686, A226391, A359643, A378327, A383121.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k] Binomial[n k, k], {k, 0, n}], {n, 0, 17}]
  • PARI
    a(n) = sum(k=0, n, binomial(n,k) * binomial(n*k,k)); \\ Michel Marcus, Apr 17 2025

Formula

a(n) = [x^n] ((1 + x)^n + x)^n.
a(n) ~ exp(n + exp(-1) - 1/2) * n^n / sqrt(2*Pi*n). - Vaclav Kotesovec, Apr 17 2025
Showing 1-3 of 3 results.

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