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.

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A302285 Expansion of 1/(1 - x - x/(1 - 2*x - x/(1 - 3*x - x/(1 - 4*x - x/(1 - 5*x - x/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 2, 7, 33, 185, 1170, 8121, 60846, 486753, 4125852, 36846557, 345205559, 3381126995, 34524194712, 366635359887, 4041180951473, 46149726728969, 545161967955668, 6652026230285141, 83730953689450825, 1085924693069106823, 14494802798426546660, 198918641942013097723
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 04 2018

Keywords

Comments

a(n) is the number of paths from (0,0) to (2n,0) on or above the x-axis with steps U=(1,1), D=(1,-1), and L=(2,0), where the level steps L at height k have k+1 colors for all k>=0. - Alexander Burstein, Apr 10 2025

Examples

			G.f. A(x) = 1 + 2*x + 7*x^2 + 33*x^3 + 185*x^4 + 1170*x^5 + 8121*x^6 + 60846*x^7 + 486753*x^8 + ...

Links

Crossrefs

Cf. A001515, A006318, A006351, A006789, A258173, A295289.

Programs

  • Maple
    b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, b(x-1, y-1)+b(x-1, y+1)+b(x-2, y)*(y+1)))
    end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 12 2025
  • Mathematica
    nmax = 22; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-x, 1 - (k + 1) x, {k, 1, nmax}]), {x, 0, nmax}], x]

A305536 Expansion of 1/(1 - x/(1 - x - 1*x/(1 - x - 2*x/(1 - x - 3*x/(1 - x - 4*x/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 1, 3, 12, 62, 410, 3426, 35360, 438390, 6358306, 105544388, 1970997142, 40860191470, 930482058472, 23079257369054, 619157277351618, 17860295754328884, 551188620179519302, 18119420989759583998, 632069815329176122584, 23318435171385786420958, 907077442499274638005314
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 04 2018

Keywords

Comments

Invert transform of A001515, shifted right one place.

Links

Crossrefs

Cf. A001003, A001515, A144301, A258173.

Programs

  • Maple
    b:= proc(n) option remember;
     `if`(n<2, n+1, (2*n-1)*b(n-1)+b(n-2))
    end:
a:= proc(n) option remember;
     `if`(n=0, 1, add(b(j-1)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 11 2023
  • Mathematica
    nmax = 21; CoefficientList[Series[1/(1 - x/(1 - x + ContinuedFractionK[-k x, 1 - x, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 21; CoefficientList[Series[1/(1 - Sum[HypergeometricPFQ[{k, 1 - k}, {}, -1/2] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[HypergeometricPFQ[{k, 1 - k}, {}, -1/2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

Formula

a(n) ~ 2^(n - 1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Sep 18 2021

A371567 Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + (k+1)*Sum_{j=0..k} A(n-1,j) with A(0,k) = k+1, n >= 0, k >= 0.

Original entry on oeis.org

1, 2, 3, 3, 9, 12, 4, 22, 46, 58, 5, 45, 147, 263, 321, 6, 81, 397, 1012, 1654, 1975, 7, 133, 933, 3341, 7340, 11290, 13265, 8, 204, 1962, 9637, 28333, 56278, 82808, 96073, 9, 297, 3776, 24758, 96313, 246905, 455534, 647680, 743753, 10, 415, 6767, 57678, 292092, 961897, 2227689, 3882510, 5370016, 6113769
Offset: 0

Views

Author

Mikhail Kurkov, Mar 28 2024

Keywords

Examples

			Array begins:
==============================================================
n\k|     0     1      2       3       4        5         6 ...
---+----------------------------------------------------------
0  |     1     2      3       4       5        6         7 ...
1  |     3     9     22      45      81      133       204 ...
2  |    12    46    147     397     933     1962      3776 ...
3  |    58   263   1012    3341    9637    24758     57678 ...
4  |   321  1654   7340   28333   96313   292092    800991 ...
5  |  1975 11290  56278  246905  961897  3357309  10601156 ...
6  | 13265 82808 455534 2227689 9749034 38415080 137251108 ...
  ...

Crossrefs

Cf. A258173.

Programs

  • PARI
    A(m, n=m)={my(r=vectorv(m+1), v=vector(n+m+1, k, k)); r[1] = v[1..n+1];
for(i=1, m, v=vector(#v-1, k, v[k+1] + k*sum(j=1, k, v[j])); r[1+i] = v[1..n+1]); Mat(r)}
{ A(6) }

Formula

Conjecture: A(n,0) = A258173(n+1). - Mikhail Kurkov, Oct 27 2024
A(n,k) = A(n,k-1) + (A(n,k-1) - A(n-1,k))/k + k*A(n-1,k) + A(n-1,k+1) with A(n,0) = A(n-1,0) + A(n-1,1), A(0,k) = k+1. - Mikhail Kurkov, Nov 24 2024
Previous Showing 11-13 of 13 results.

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