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.

A337167 a(n) = 1 + 3 * Sum_{k=0..n-1} a(k) * a(n-k-1).

Original entry on oeis.org

1, 4, 25, 199, 1795, 17422, 177463, 1870960, 20241403, 223438852, 2506596547, 28494103183, 327507800725, 3799735202218, 44440058006593, 523388751658831, 6201937444137619, 73888034816382820, 884517283667145259, 10634234680321209373, 128347834921058404249
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 28 2021

Keywords

Comments

Binomial transform of A005159.

Links

Crossrefs

Column k=3 of A340968.
Cf. A000108, A005159, A007317, A162326, A337169, A338979.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + 3 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n, k] 3^k CatalanNumber[k], {k, 0, n}], {n, 0, 20}]
Table[Hypergeometric2F1[1/2, -n, 2, -12], {n, 0, 20}]
  • PARI
    {a(n) = sum(k=0, n, 3^k*binomial(n, k)*(2*k)!/(k!*(k+1)!))} \\ Seiichi Manyama, Jan 31 2021
  • PARI
    my(N=20, x='x+O('x^N)); Vec(2/(1-x+sqrt((1-x)*(1-13*x)))) \\ Seiichi Manyama, Feb 01 2021

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x) + 3*x*A(x)^2.
G.f.: (1 - sqrt(1 - 12*x / (1 - x))) / (6*x).
E.g.f.: exp(7*x) * (BesselI(0,6*x) - BesselI(1,6*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * Catalan(k).
a(n) = 2F1([1/2, -n], [2], -12), where 2F1 is the hypergeometric function.
D-finite with recurrence (n+1) * a(n) = 2 * (7*n-3) * a(n-1) - 13 * (n-1) * a(n-2) for n > 1. - Seiichi Manyama, Jan 31 2021
a(n) ~ 13^(n + 3/2) / (8 * 3^(3/2) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 14 2021

A349256 G.f. A(x) satisfies A(x) = 1 / ((1 + x) * (1 - 3 * x * A(x)^2)).

Original entry on oeis.org

1, 2, 19, 206, 2563, 34415, 486370, 7128488, 107364421, 1651615568, 25840137724, 409898503763, 6577319627506, 106571487893024, 1741193467526782, 28653852176675324, 474521786894159593, 7902112425718228064, 132243695376774536755, 2222925664652778182060
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 12 2021

Keywords

Links

Crossrefs

Cf. A001764, A107841, A217363, A337169, A346627, A349253, A349254, A349255.

Programs

  • Mathematica
    nmax = 19; A[] = 0; Do[A[x] = 1/((1 + x) (1 - 3 x A[x]^2)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = (-1)^n + 3 Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 19}]
Table[Sum[(-1)^(n - k) Binomial[n + k, n - k] 3^k Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 19}]
a[n_] := (-1)^n*HypergeometricPFQ[{1/3, 2/3, -n, n + 1}, {1/2, 1, 3/2}, (3/2)^4]; Table[a[n], {n, 0, 19}] (* Peter Luschny, Nov 12 2021 *)

Formula

a(n) = (-1)^n + 3 * Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+k,n-k) * 3^k * binomial(3*k,k) / (2*k+1).
a(n) = (-1)^n*hypergeom([1/3, 2/3, -n, n + 1], [1/2, 1, 3/2], (3/2)^4). - Peter Luschny, Nov 12 2021
a(n) ~ sqrt(585 + 73*sqrt(65)) * (73 + 9*sqrt(65))^n / (3^(5/2) * sqrt(Pi) * n^(3/2) * 2^(3*n + 5/2)). - Vaclav Kotesovec, Nov 13 2021

A337168 a(n) = (-1)^n + 2 * Sum_{k=0..n-1} a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 5, 21, 105, 553, 3053, 17405, 101713, 606033, 3667797, 22485477, 139340985, 871429497, 5492959293, 34862161869, 222592918689, 1428814897825, 9215016141989, 59684122637237, 388045493943049, 2531696701375689, 16569559364596365, 108758426952823709
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 28 2021

Keywords

Comments

Inverse binomial transform of A151374.

Links

Crossrefs

Cf. A000108, A005043, A052701, A151374, A162326, A337169.

Programs

  • Mathematica
    a[n_] := a[n] = (-1)^n + 2 Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k CatalanNumber[k], {k, 0, n}], {n, 0, 23}]
Table[(-1)^n Hypergeometric2F1[1/2, -n, 2, 8], {n, 0, 23}]

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 + x) + 2*x*A(x)^2.
G.f.: (1 - sqrt(1 - 8*x / (1 + x))) / (4*x).
E.g.f.: exp(3*x) * (BesselI(0,4*x) - BesselI(1,4*x)).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 2^k * Catalan(k).
a(n) ~ 7^(n + 3/2) / (2^(9/2) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Nov 13 2021

A339001 a(n) = (-1)^n * Sum_{k=0..n} (-n)^k * binomial(n,k) * Catalan(k).

Original entry on oeis.org

1, 0, 5, 89, 2481, 93274, 4450645, 258297570, 17689681345, 1397903887808, 125286890408901, 12562851683433765, 1393925069404093105, 169595051215441936902, 22454465186157134883285
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2021

Keywords

Links

Crossrefs

Cf. A000108, A005043, A337168, A337169, A338979.

Programs

  • Mathematica
    a[0] = 1; a[n_] := (-1)^n * Sum[(-n)^k * Binomial[n, k] * CatalanNumber[k], {k, 0, n}]; Array[a, 15, 0] (* Amiram Eldar, Feb 01 2021 *)
  • PARI
    {a(n) = (-1)^n*sum(k=0, n, (-n)^k*binomial(n, k)*(2*k)!/(k!*(k+1)!))}

Formula

a(n) = n! * [x^n] exp((2*n-1)*x) * (BesselI(0,2*n*x) - BesselI(1,2*n*x)). - Ilya Gutkovskiy, Feb 01 2021
a(n) ~ exp(-1/4) * 4^n * n^(n - 3/2) / sqrt(Pi). - Vaclav Kotesovec, Feb 14 2021
Showing 1-4 of 4 results.

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