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.

A171180 a(n) = (4*n + 1)^(1/2)/(4*n + 1)*((1 - p)*q^n - (1 - q)*p^n), where p = (1 - (4*n + 1)^(1/2))/2 and q = (1 + (4*n + 1)^(1/2))/2.

Original entry on oeis.org

1, 3, 7, 29, 96, 463, 1905, 10233, 49159, 287891, 1557744, 9814741, 58451849, 392539575, 2532516511, 17999936497, 124360077816, 930257069563, 6822980957481, 53470578301581, 413527226164711, 3382254701784223, 27432377661111360, 233410016529114601
Offset: 1

Views

Author

Gary Detlefs, Dec 04 2009

Keywords

Comments

If a sequence (s(n): n >= 0) is of the form s(0) = x, s(1) = x, and s(n) = s(n-1) + k*s(n-2) for n >= 2 (for some integer k >= 1 and some number x), then s(k) = a(k)*x. For example, if k = 6 and x = 3, then (s(n): n = 0..6) = (3, 3, 21, 39, 165, 399, 1389) and s(6) = 1389 = 463*3 = a(6)*x. [Edited by Petros Hadjicostas, Dec 26 2019]

Links

Crossrefs

Cf. A350467.

Programs

  • Mathematica
    Table[Sum[Binomial[n - k, k]*n^k, {k, 0, n}], {n, 1, 25}] (* Vaclav Kotesovec, Jan 08 2024 *)
Table[Hypergeometric2F1[(1 - n)/2, -n/2, -n, -4*n], {n, 1, 25}] (* Vaclav Kotesovec, Jan 08 2024 *)
  • PARI
    {a(n)=polcoeff(1/(1-x-n*x^2+x*O(x^n)), n)} \\ Paul D. Hanna, Dec 27 2012

Formula

a(n) = A193376(n,n). - Olivier Gérard, Jul 25 2011
a(n) = [x^n] 1/(1 - x - n*x^2). - Paul D. Hanna, Dec 27 2012
From Vaclav Kotesovec, Jan 08 2024: (Start)
a(n) = Sum_{k=0..n} binomial(n-k,k) * n^k.
a(n) ~ exp(sqrt(n)/2) * n^(n/2) / 2 * (1 + 23/(48*sqrt(n))). (End)

A350470 Array read by ascending antidiagonals. T(n, k) = J(k, n) where J are the Jacobsthal polynomials.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 5, 1, 1, 1, 7, 9, 11, 1, 1, 1, 9, 13, 29, 21, 1, 1, 1, 11, 17, 55, 65, 43, 1, 1, 1, 13, 21, 89, 133, 181, 85, 1, 1, 1, 15, 25, 131, 225, 463, 441, 171, 1, 1, 1, 17, 29, 181, 341, 937, 1261, 1165, 341, 1
Offset: 0

Views

Author

Peter Luschny, Mar 19 2022

Keywords

Examples

			Array starts:
n\k 0, 1,  2,  3,   4,    5,    6,     7,      8,      9, ...
---------------------------------------------------------------------
[0] 1, 1,  1,  1,   1,    1,    1,     1,      1,      1, ... A000012
[1] 1, 1,  3,  5,  11,   21,   43,    85,    171,    341, ... A001045
[2] 1, 1,  5,  9,  29,   65,  181,   441,   1165,   2929, ... A006131
[3] 1, 1,  7, 13,  55,  133,  463,  1261,   4039,  11605, ... A015441
[4] 1, 1,  9, 17,  89,  225,  937,  2737,  10233,  32129, ... A015443
[5] 1, 1, 11, 21, 131,  341, 1651,  5061,  21571,  72181, ... A015446
[6] 1, 1, 13, 25, 181,  481, 2653,  8425,  40261, 141361, ... A053404
[7] 1, 1, 15, 29, 239,  645, 3991, 13021,  68895, 251189, ... A350468
[8] 1, 1, 17, 33, 305,  833, 5713, 19041, 110449, 415105, ... A168579
[9] 1, 1, 19, 37, 379, 1045, 7867, 26677, 168283, 648469, ... A350469
      A005408 | A082108 |
           A016813   A014641

Crossrefs

Rows: A000012, A001045, A006131, A015441, A015443, A015446, A053404, A350468, A168579, A350469.
Columns: A000012, A005408, A016813, A082108, A014641.
Cf. A350467 (main diagonal), A352361 (Fibonacci polynomials), A352362 (Lucas polynomials).

Programs

  • Maple
    J := (n, x) -> add(2^k*binomial(n - k, k)*x^k, k = 0..n):
seq(seq(J(k, n-k), k = 0..n), n = 0..10);
  • Mathematica
    T[n_, k_] := Hypergeometric2F1[(1 - k)/2, -k/2, -k, -8 n];
Table[T[n, k], {n, 0, 9}, {k, 0, 9}] // TableForm
(* or *)
T[n_, k_] := With[{s = Sqrt[8*n+1]}, ((1+s)^(k+1) - (1-s)^(k+1)) / (2^(k+1)*s)];
Table[Simplify[T[n, k]], {n, 0, 9}, {k, 0, 9}] // TableForm
  • PARI
    T(n, k) = ([1, 2; k, 0]^n)[1, 1] ;
export(T)
for(k = 0, 9, print(parvector(10, n, T(n - 1, k))))

Formula

T(n, k) = Sum_{j=0..k} binomial(k - j, j)*(2*n)^j.
T(n, k) = ((1+s)^(k+1) - (1-s)^(k+1)) / (2^(k+1)*s) where s = sqrt(8*n + 1).
T(n, k) = [x^k] (1 / (1 - x - 2*n*x^2)).
T(n, k) = hypergeom([1/2 - k/2, -k/2], [-k], -8*n).

A368888 a(n) = Sum_{k=0..floor(n/2)} n^(2*k) * binomial(n-k,k).

Original entry on oeis.org

1, 1, 5, 19, 305, 1976, 54613, 494901, 19460545, 226000855, 11535280901, 163226844144, 10246715573041, 170910034261721, 12736193619206485, 244588264748170651, 21100437309369290497, 458426839205360652760, 44935948904379592796101
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2024

Keywords

Crossrefs

Cf. A171180, A368889.
Cf. A176233, A350467.

Programs

  • Mathematica
    Table[Hypergeometric2F1[1/2 - n/2, -n/2, -n, -4*n^2], {n, 0, 20}] (* Vaclav Kotesovec, Jan 09 2024 *)
  • PARI
    a(n) = sum(k=0, n\2, n^(2*k)*binomial(n-k, k));

Formula

a(n) = [x^n] 1/(1 - x - (n*x)^2).
a(n) ~ (exp(1/2) + (-1)^n*exp(-1/2)) * n^n / 2. - Vaclav Kotesovec, Jan 09 2024
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.