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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A378112 Number A(n,k) of k-tuples (p_1, p_2, ..., p_k) of Dyck paths of semilength n, such that each p_i is never below p_{i-1} and the upper path p_k only touches the x-axis at its endpoints; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 3, 9, 5, 0, 1, 1, 4, 23, 55, 14, 0, 1, 1, 5, 46, 265, 400, 42, 0, 1, 1, 6, 80, 880, 3942, 3266, 132, 0, 1, 1, 7, 127, 2347, 23695, 70395, 28999, 429, 0, 1, 1, 8, 189, 5403, 105554, 824229, 1445700, 274537, 1430, 0
Offset: 0

Views

Author

Alois P. Heinz, Nov 16 2024

Keywords

Examples

			A(3,2) = 9:
                             /\
           /\/\             /  \      /\     /\/\
  (/\/\/\,/    \)  (/\/\/\,/    \)  (/  \/\,/    \)
.
            /\                                /\
    /\     /  \        /\   /\/\        /\   /  \
  (/  \/\,/    \)  (/\/  \,/    \)  (/\/  \,/    \)
.
                             /\        /\     /\
    /\/\   /\/\      /\/\   /  \      /  \   /  \
  (/    \,/    \)  (/    \,/    \)  (/    \,/    \)
.
Square array A(n,k) begins:
  1,  1,    1,     1,      1,       1,        1, ...
  1,  1,    1,     1,      1,       1,        1, ...
  0,  1,    2,     3,      4,       5,        6, ...
  0,  2,    9,    23,     46,      80,      127, ...
  0,  5,   55,   265,    880,    2347,     5403, ...
  0, 14,  400,  3942,  23695,  105554,   382508, ...
  0, 42, 3266, 70395, 824229, 6601728, 40446551, ...

Links

Crossrefs

Columns k=0-3 give: A019590(n+1), A120588, A355281, A378114.
Rows n=0+1,2,3 give: A000012, A001477, A101986.
Main diagonal gives A378113.
Cf. A000108, A078920, A123352, A368025.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1, 2^k*mul(
      (2*(n-i)+2*k-3)/(n+2*k-1-i), i=0..k-1)*b(n-1, k))
    end:
A:= proc(n, k) option remember;
      b(n, k)-add(A(n-i, k)*b(i, k), i=1..n-1)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

Formula

Column k is INVERTi transform of row k of A368025.

A355503 Total number of m-tuples (p_1, p_2, ..., p_m) of Dyck paths of semilength n-m, such that each p_i is never below p_{i-1} for m=0..n.

Original entry on oeis.org

1, 2, 3, 5, 11, 35, 164, 1120, 10969, 152849, 3029650, 85227078, 3400752392, 192644205130, 15470939367651, 1761760468965521, 284641456742538865, 65175288287611738435, 21159611204475209730138, 9743708333490185603430830, 6357930817596444858142966826
Offset: 0

Views

Author

Alois P. Heinz, Jul 04 2022

Keywords

Examples

			a(3) = 5: ( ), (/\/\), (//\\), (/\, /\, /\), (<>, <>, <>, <>).

Links

Crossrefs

Cf. A000108, A074962, A078920, A123352, A355400.
Antidiagonal sums of A368025.

Programs

  • Maple
    a:= n-> add(mul(mul((i+j+2*(n-m))/(i+j), j=i..m-1), i=1..m-1), m=0..n):
seq(a(n), n=0..23);
  • Mathematica
    Table[Sum[Product[Product[(i+j+2*(n-m))/(i+j), {j,i,m-1}], {i,1,m-1}], {m,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 27 2023 *)
Table[Sum[BarnesG[1 + m] * Sqrt[BarnesG[1 + 2*n] * BarnesG[2 - 2*m + 2*n] * Gamma[1 + 2*m] * Gamma[1 + n] / (BarnesG[1 + 2*m] * Gamma[1 + m] * Gamma[1 + 2*n] * Gamma[1 - m + n])] / BarnesG[1 - m + 2*n], {m, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 27 2023 *)

Formula

a(n) = Sum_{m=0..n} Product_{i=1..m-1, j=i..m-1} (i+j+2*(n-m))/(i+j).
a(n) = 1 + Sum_{k=0..n-1} A078920(n-1,k).
a(n) = 1 + Sum_{k=0..n-1} A123352(n-1,k).
a(n) = Sum_{k=0..n} A368025(n-k, k).
From Vaclav Kotesovec, Aug 27 2023: (Start)
a(n) ~ c * exp(1/24) * 3^(n^2 - n/2) / (sqrt(A) * n^(1/24) * 2^((4*n^2-n-1)/3)), where A = A074962 is the Glaisher-Kinkelin constant and
c = Sum_{k,-oo,oo} 2^((k + mod(n,3)/3)/2 - 3*(k + mod(n,3)/3)^2/2).
Numerically, c = 1.78933741155287907159762028... if mod(n,3)=0 or mod(n,3)=1 and c = 1.78893263307672974352375161... if mod(n,3)=2. (End)

A368298 a(n) is the permanent of the n-th order Hankel matrix of Catalan numbers M(n) whose generic element is given by M(i,j) = A000108(i+j+n) with i,j = 0, ..., n-1.

Original entry on oeis.org

1, 1, 53, 490614, 930744290905, 386735380538157813864, 36494318768452684668237864399892, 800075179375382235705309991148469060609055210, 4138855242465150993428071754285859188133806122546895149328625, 5109461743591866972924602083859433690113667142460933537037028649653229023827000
Offset: 0

Views

Author

Stefano Spezia, Dec 20 2023

Keywords

Examples

			a(3) = 490614:
   5,  14,  42;
  14,  42, 132;
  42, 132, 429.

Links

Crossrefs

Cf. A000108, A355400.
Diagonal of A368025.

Programs

  • Maple
    with(LinearAlgebra):
C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> C(i+j+n-2)))):
seq(a(n), n=0..10);  # Alois P. Heinz, Dec 20 2023
  • Mathematica
    a[n_]:=If[n==0, 1, Permanent[Table[CatalanNumber[i+j+n], {i, 0, n-1}, {j, 0, n-1}]]]; Array[a,10,0]
Previous Showing 11-13 of 13 results.

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