login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A324429 Number T(n,k) of labeled cyclic chord diagrams having n chords and minimal chord length k (or k=0 if n=0); triangle T(n,k), n>=0, 0<=k<=n, read by rows. 12
1, 0, 1, 0, 2, 1, 0, 11, 3, 1, 0, 74, 24, 6, 1, 0, 652, 225, 57, 10, 1, 0, 7069, 2489, 678, 141, 17, 1, 0, 90946, 32326, 9375, 2107, 352, 28, 1, 0, 1353554, 483968, 146334, 35568, 6722, 832, 46, 1, 0, 22870541, 8211543, 2555228, 661329, 137225, 21510, 1973, 75, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms with 0 <= k <= n. T(n,k) = 0 for k > n.

LINKS

Alois P. Heinz, Rows n = 0..19, flattened

FORMULA

T(n,k) = A324428(n,k) - A324428(n,k+1) for k > 0, T(n,0) = A000007(n).

EXAMPLE

Triangle T(n,k) begins:

  1;

  0,       1;

  0,       2,      1;

  0,      11,      3,      1;

  0,      74,     24,      6,     1;

  0,     652,    225,     57,    10,    1;

  0,    7069,   2489,    678,   141,   17,   1;

  0,   90946,  32326,   9375,  2107,  352,  28,  1;

  0, 1353554, 483968, 146334, 35568, 6722, 832, 46, 1;

  ...

MAPLE

b:= proc(n, f, m, l, j) option remember; (k-> `if`(n<add(i, i=f)+m+

      add(i, i=l), 0, `if`(n=0, 1, add(`if`(f[i]=0, 0, b(n-1,

      subsop(i=0, f), m+l[1], [subsop(1=[][], l)[], 0], max(0, j-1))),

      i=max(1, j+1)..min(k, n-1))+`if`(m=0, 0, m*b(n-1, f, m-1+l[1],

      [subsop(1=[][], l)[], 0], max(0, j-1)))+b(n-1, f, m+l[1],

      [subsop(1=[][], l)[], 1], max(0, j-1)))))(nops(l))

    end:

A:= (n, k)-> `if`(n=0 or k<2, doublefactorial(2*n-1),

              b(2*n-k+1, [1$k-1], 0, [0$k-1], k-1)):

T:= (n, k)-> `if`(n=k, 1, A(n, k)-A(n, k+1)):

seq(seq(T(n, k), k=0..n), n=0..10);

MATHEMATICA

b[n_, f_List, m_, l_List, j_] := b[n, f, m, l, j] = Function[k, If[n < Total[f] + m +  Total[l], 0, If[n == 0, 1, Sum[If[f[[i]] == 0, 0, b[n - 1, ReplacePart[f, i -> 0], m + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 0], Max[0, j - 1]]], {i, Max[1, j + 1], Min[k, n - 1]}] + If[m == 0, 0, m*b[n - 1, f, m - 1 + l[[1]], Append[ReplacePart[l, 1 -> Nothing], 0], Max[0, j-1]]] + b[n-1, f, m + l[[1]], Append[ReplacePart[ l, 1 -> Nothing], 1], Max[0, j - 1]]]]][Length[l]];

A[n_, k_] := If[n == 0 || k < 2, 2^(n-1) Pochhammer[3/2, n-1], b[2n-k+1, Table[1, {k - 1}], 0, Table[0, {k - 1}], k - 1]];

T[n_, k_] := If[n == k, 1, A[n, k] - A[n, k + 1]];

Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Apr 27 2020, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A000007, A324445, A324446, A324447, A324448, A324449, A324450, A324451, A324452, A324453, A324454.

Row sums give A001147.

Main diagonal gives A000012.

T(n+1,n) gives A001610.

Cf. A293157, A293881, A324428.

Sequence in context: A185285 A268434 A010107 * A119830 A268435 A039910

Adjacent sequences:  A324426 A324427 A324428 * A324430 A324431 A324432

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Feb 27 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 25 09:37 EST 2022. Contains 350565 sequences. (Running on oeis4.)