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 1-5 of 5 results.

A099141 a(n) = 5^n * T(n,7/5) where T is the Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 7, 73, 847, 10033, 119287, 1419193, 16886527, 200931553, 2390878567, 28449011113, 338514191407, 4027973401873, 47928772841047, 570303484727833, 6786029465163487, 80746825394092993, 960804818888214727
Offset: 0

Views

Author

Paul Barry, Sep 30 2004

Keywords

Comments

In general, r^n * T(n,(r+2)/r) has g.f. (1-(r+2)*x)/(1-2*(r+2)*x + r^2*x^2), e.g.f. exp((r+2)*x)*cosh(2*sqrt(r+1)*x), a(n) = Sum_{k=0..n} (r+1)^k*binomial(2*n,2*k) and a(n) = (1+sqrt(r+1))^(2*n)/2 + (1-sqrt(r+1))^(2*n)/2.

Links

Crossrefs

Column k=6 of A333988.
Cf. A081294, A001541, A090965, A083884, A099140, A099142.

Programs

  • Mathematica
    LinearRecurrence[{14,-25},{1,7},30] (* Harvey P. Dale, Dec 26 2014 *)

Formula

G.f.: (1-7*x)/(1-14*x+25*x^2);
e.g.f.: exp(7*x)*cosh(2*sqrt(6)*x);
a(n) = 5^n * T(n, 7/5) where T is the Chebyshev polynomial of the first kind;
a(n) = Sum_{k=0..n} 6^k * binomial(2n, 2k);
a(n) = (1+sqrt(6))^(2n)/2 + (1-sqrt(6))^(2n)/2.
a(0)=1, a(1)=7, a(n) = 14*a(n-1) - 25*a(n-2) for n > 1. - Philippe Deléham, Sep 08 2009

A099142 a(n) = 6^n * T(n, 4/3) where T is the Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 8, 92, 1184, 15632, 207488, 2757056, 36643328, 487039232, 6473467904, 86042074112, 1143628341248, 15200538791936, 202038000386048, 2685388609667072, 35692849740775424, 474411605904392192
Offset: 0

Views

Author

Paul Barry, Sep 30 2004

Keywords

Comments

In general, r^n * T(n,(r+2)/r) has g.f. (1-(r+2)*x)/(1-2*(r+2)*x + r^2*x^2), e.g.f. exp((r+2)*x)*cosh(2*sqrt(r+1)*x), a(n) = Sum_{k=0..n} (r+1)^k*binomial(2n,2k) and a(n) = (1+sqrt(r+1))^(2*n)/2 + (1-sqrt(r+1))^(2*n)/2.

Links

Crossrefs

Column k=7 of A333988.
Cf. A001541, A081294, A083884, A090965, A099140, A099141.

Programs

  • Mathematica
    LinearRecurrence[{16,-36},{1,8},20] (* Harvey P. Dale, Mar 09 2018 *)
  • PARI
    a(n) = 6^n*polchebyshev(n, 1, 4/3); \\ Michel Marcus, Sep 08 2019

Formula

G.f.: (1-8*x)/(1-16*x+36*x^2);
E.g.f.: exp(8*x)*cosh(2*sqrt(7)*x).
a(n) = 6^n * T(n, 8/6) where T is the Chebyshev polynomial of the first kind.
a(n) = Sum_{k=0..n} 7^k * binomial(2n, 2k).
a(n) = (1+sqrt(7))^(2*n)/2 + (1-sqrt(7))^(2*n)/2.
a(0)=1, a(1)=8, a(n) = 16*a(n-1) - 36*a(n-2) for n > 1. - Philippe Deléham, Sep 08 2009

A165224 a(0)=1, a(1)=9, a(n) = 18*a(n-1) - 49*a(n-2) for n > 1.

Original entry on oeis.org

1, 9, 113, 1593, 23137, 338409, 4957649, 72655641, 1064876737, 15607654857, 228758827313, 3352883803641, 49142725927201, 720277760311209, 10557006115168913, 154732499817791193, 2267891697076964737
Offset: 0

Views

Author

Philippe Deléham, Sep 08 2009

Keywords

Comments

a(n)/a(n-1) tends to 9 + 4*sqrt(2) = 14.65685424... - Klaus Brockhaus, Sep 25 2009

Links

Crossrefs

Column k=8 of A333988.
Cf. A081294, A001541, A090965, A083884, A099140, A099141, A099142, A026244.

Programs

  • Mathematica
    LinearRecurrence[{18,-49},{1,9},20] (* Harvey P. Dale, Sep 30 2016 *)

Formula

G.f.: (1-9x)/(1-18x+49x^2);
e.g.f.: exp(9x)*cosh(4*sqrt(2)x);
a(n) = Sum_{k=0..n} 8^k*binomial(2n,2k) = Sum_{k=0..n} 8^k*A086645(n,k);
a(n) = 7^n*T(n,9/7) where T is the Chebyshev polynomial of the first kind;
a(n) = (1+sqrt(8))^(2n)/2 + (1-sqrt(8))^(2n)/2.
a(n) = ((9-4*sqrt(2))^n + (9+4*sqrt(2))^n)/2. - Klaus Brockhaus, Sep 25 2009

A333990 a(n) = Sum_{k=0..n} n^k * binomial(2*n,2*k).

Original entry on oeis.org

1, 2, 17, 208, 3281, 62976, 1419193, 36643328, 1064876737, 34359869440, 1217844546401, 47005113741312, 1961498610274321, 87961440484327424, 4217109422614386761, 215187913985734475776, 11641533109203575871233, 665430291591787349803008, 40065760383961956327231409
Offset: 0

Views

Author

Seiichi Manyama, Sep 04 2020

Keywords

Links

Crossrefs

Main diagonal of A333988.
Cf. A333991.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[n^k * Binomial[2*n, 2*k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Sep 04 2020 *)
Table[Hypergeometric2F1[1/2 - n, -n, 1/2, n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 05 2020 *)
  • PARI
    {a(n) = sum(k=0, n, n^k*binomial(2*n, 2*k))}

Formula

From Vaclav Kotesovec, Sep 05 2020: (Start)
a(n) = hypergeometric2F1(1/2 - n, -n, 1/2, n).
a(n) = (1 + sqrt(n))^(2*n)/2 + (1 - sqrt(n))^(2*n)/2.
a(n) ~ exp(2*sqrt(n) - 1) * n^n / 2 * (1 + 2/(3*sqrt(n))). (End)

A333989 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of (1+(k-1)*x) / (1+2*(k-1)*x+((k+1)*x)^2).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, -4, 1, 1, -2, -7, 0, 1, 1, -3, -8, 23, 16, 1, 1, -4, -7, 64, 17, 0, 1, 1, -5, -4, 117, -128, -241, -64, 1, 1, -6, 1, 176, -527, -512, 329, 0, 1, 1, -7, 8, 235, -1264, 237, 4096, 1511, 256, 1, 1, -8, 17, 288, -2399, 3776, 11753, -8192, -5983, 0, 1
Offset: 0

Views

Author

Seiichi Manyama, Sep 04 2020

Keywords

Examples

			Square array begins:
  1,  1,    1,    1,    1,     1, ...
  1,  0,   -1,   -2,   -3,    -4, ...
  1, -4,   -7,   -8,   -7,    -4, ...
  1,  0,   23,   64,  117,   176, ...
  1, 16,   17, -128, -527, -1264, ...
  1,  0, -241, -512,  237,  3776, ...

Crossrefs

Main diagonal gives A333991.
Cf. A307884, A333988.

Programs

  • Mathematica
    T[n_, 0] := 1; T[n_, k_] := Sum[(-k)^j * Binomial[2*n, 2*j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Sep 04 2020 *)
  • PARI
    {T(n, k) = sum(j=0, n, (-k)^j*binomial(2*n, 2*j))}

Formula

T(n,k) = Sum_{j=0..n} (-k)^j * binomial(2*n,2*j).
T(0,k) = 1, T(1,k) = 1-k and T(n,k) = -2 * (k-1) * T(n-1,k) - (k+1)^2 * T(n-2,k) for n>1.
Showing 1-5 of 5 results.

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