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.

A089846 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A089867/A089868.

Original entry on oeis.org

1, 1, 2, 5, 13, 35, 100, 296, 903, 2857, 9274, 30852, 104784, 362108, 1269324, 4502624, 16130539, 58271303, 212001890, 776027286, 2855731796
Offset: 0

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Author

Antti Karttunen, Dec 20 2003

Keywords

Comments

The number of orbits to which the corresponding automorphism(s) partitions the set of A000108(n) binary trees with n internal nodes.

Links

A086586 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutations A074681/A074682 & A074683/A074684.

Original entry on oeis.org

1, 1, 2, 5, 9, 28, 57, 253, 842, 3753, 10927, 15014, 130831, 218961, 967104, 3767216, 29715310, 89923607, 314897868, 785059994
Offset: 0

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Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

Shifted once right (beginning as 1,1,1,2,5,9,...) this is maximum cycle size (in the same range) of permutations A085169/A085170, shifted twice right (beginning as 1,1,1,1,2,5,9,...) this is the maximum cycle size in permutations A089867/A089868 and A089869/A089870.

Links

A090826 Convolution of Catalan and Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 5, 12, 31, 85, 248, 762, 2440, 8064, 27300, 94150, 329462, 1166512, 4170414, 15031771, 54559855, 199236416, 731434971, 2697934577, 9993489968, 37157691565, 138633745173, 518851050388, 1947388942885, 7328186394725
Offset: 0

Views

Author

Antti Karttunen, Dec 20 2003

Keywords

Comments

Also (with a(0)=1 instead of 0): Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A089867/A089868, i.e., the number of n-node binary trees fixed by the corresponding automorphism(s).

Links

Crossrefs

Cf. Catalan numbers: A000108, Fibonacci numbers: A000045.

Programs

  • Haskell
    import Data.List (inits)
a090826 n = a090826_list !! n
a090826_list = map (sum . zipWith (*) a000045_list . reverse) $
                   tail $ inits a000108_list
-- Reinhard Zumkeller, Aug 28 2013
  • Mathematica
    CoefficientList[Series[(1-(1-4x)^(1/2))/(2(1-x-x^2)), {x,0,30}], x]  (* Harvey P. Dale, Apr 05 2011 *)

Formula

CONV(A000045, A000108).
G.f.: (1-(1-4x)^(1/2))/(2(1-x-x^2)). The generating function for the convolution of Catalan and Fibonacci numbers is simply the generating functions of the Catalan and Fibonacci numbers multiplied together. - Molly Leonard (maleonard1(AT)stthomas.edu), Aug 04 2006
For n>1, a(n) = a(n-1) + a(n-2) + A000108(n-1). - Gerald McGarvey, Sep 19 2008
Conjecture: n*a(n) + (-5*n+6)*a(n-1) + 3*(n-2)*a(n-2) + 2*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jul 09 2013
a(n) = A139375(n,1) for n > 0. - Reinhard Zumkeller, Aug 28 2013
a(n) ~ 2^(2*n + 2) / (11*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 10 2018

A089867 Permutation of natural numbers induced by the Catalan bijection gma089867 acting on the parenthesizations/binary trees encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 49, 51, 52, 53, 54, 55, 60, 61, 64, 63, 56, 57, 59, 58, 62, 65, 66, 67, 68, 69
Offset: 0

Views

Author

Antti Karttunen, Dec 20 2003

Keywords

Comments

This Catalan bijection arises when we apply the Catalan bijection A085169 to the left subtree and keep the right subtree intact.

Links

Crossrefs

Inverse of A089868.
Number of cycles: A089846. Number of fixed-points: A090826. Max. cycle size: A086586. LCM of cycle sizes: A086587. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A373301 Sum of successive nonnegative integers in a row of length p(n) where p counts integer partitions.

Original entry on oeis.org

0, 3, 12, 40, 98, 253, 540, 1199, 2415, 4893, 9268, 17864, 32421, 59265, 104632, 184338, 315414, 540155, 901845, 1504173, 2461932, 4013511, 6443170, 10314675, 16281749, 25608450, 39838855, 61716941, 94682665, 144726102
Offset: 1

Views

Author

Olivier Gérard, May 31 2024

Keywords

Comments

The length of each row is given by A000041.
As many sequences start like the nonnegative integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A001477 to have the same row sums for at least 8 terms: A089867, A089868, A089869, A089870, A118760, A123719, A130696, A136602, A254109, A258069, A258070, A258071, A266279, A272813, A273885, A273886, A273887, A273888.

Examples

			Illustration of the first few terms
.
0   | 0
3   | 1,  2
12  | 3,  4,  5
40  | 6,  7,  8,  9,  10
98  | 11, 12, 13, 14, 15, 16, 17
253 | 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28
540 | 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43
.

Crossrefs

Cf. A373300, original version, with positive integers A000027.
Cf. A001477, the nonnegative integers.
Cf. A027480, the sequence of row sums for a regular triangle.

Programs

  • Mathematica
    Module[{s = -1},
 Table[s +=
   PartitionsP[
    n - 1]; (s + PartitionsP[n]) (s + PartitionsP[n] - 1)/2 -
   s (s - 1)/2, {n, 1, 30}]]
Showing 1-5 of 5 results.

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