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.

A275281 Number T(n,k) of set partitions of [n] with symmetric block size list of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 7, 0, 1, 0, 1, 10, 19, 13, 3, 1, 0, 1, 0, 56, 0, 22, 0, 1, 0, 1, 35, 160, 171, 86, 34, 4, 1, 0, 1, 0, 463, 0, 470, 0, 50, 0, 1, 0, 1, 126, 1337, 2306, 2066, 1035, 250, 70, 5, 1, 0, 1, 0, 3874, 0, 10299, 0, 2160, 0, 95, 0, 1
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2016

Keywords

Examples

			T(4,2) = 3: 12|34, 13|24, 14|23.
T(5,3) = 7: 12|3|45, 13|2|45, 1|234|5, 1|235|4, 14|2|35, 1|245|3, 15|2|34.
T(6,4) = 13: 12|3|4|56, 13|2|4|56, 1|23|45|6, 1|23|46|5, 14|2|3|56, 1|24|35|6, 1|24|36|5, 1|25|34|6, 1|26|34|5, 15|2|3|46, 1|25|36|4, 1|26|35|4, 16|2|3|45.
T(7,5) = 22: 12|3|4|5|67, 13|2|4|5|67, 1|23|4|56|7, 1|23|4|57|6, 14|2|3|5|67, 1|24|3|56|7, 1|24|3|57|6, 1|2|345|6|7, 1|2|346|5|7, 1|2|347|5|6, 15|2|3|4|67, 1|25|3|46|7, 1|25|3|47|6, 1|2|356|4|7, 1|2|357|4|6, 1|26|3|45|7, 1|27|3|45|6, 16|2|3|4|57, 1|26|3|47|5, 1|2|367|4|5, 1|27|3|46|5, 17|2|3|4|56.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 1,   0,    1;
  0, 1,   3,    2,    1;
  0, 1,   0,    7,    0,    1;
  0, 1,  10,   19,   13,    3,    1;
  0, 1,   0,   56,    0,   22,    0,   1;
  0, 1,  35,  160,  171,   86,   34,   4,  1;
  0, 1,   0,  463,    0,  470,    0,  50,  0, 1;
  0, 1, 126, 1337, 2306, 2066, 1035, 250, 70, 5, 1;
  ...

Links

Crossrefs

Columns k=0-1,3,5,7,9 give: A000007, A000012 for n>0, A275289, A275290, A275291, A275292.
Bisections of columns k=2,4,6,8,10 give: A001700(n-1) for n>0, A275293, A275294, A275295, A275296.
Row sums give A275282.
T(n,A004525(n)) gives A305197.
T(2n,n) gives A275283.
T(2n+1,A109613(n)) gives A305198.
T(n,n) gives A000012.
T(n+3,n+1) gives A002623.

Programs

  • Maple
    b:= proc(n, s) option remember; expand(`if`(n>s,
      binomial(n-1, n-s-1)*x, 1)+add(binomial(n-1, j-1)*
      b(n-j, s+j)*binomial(s+j-1, j-1), j=1..(n-s)/2)*x^2)
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..12);
  • Mathematica
    b[n_, s_] := b[n, s] = Expand[If[n>s, Binomial[n-1, n-s-1]*x, 1] + Sum[ Binomial[n-1, j-1]*b[n-j, s+j]*Binomial[s+j-1, j-1], {j, 1, (n-s)/2} ]*x^2]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 03 2017, translated from Maple *)

Formula

T(n,k) = 0 if n is odd and k is even.

A275283 Number of set partitions of [2n] with symmetric block size list of length n.

Original entry on oeis.org

1, 1, 3, 19, 171, 2066, 31346, 559987, 11954993, 282835456, 7785919355, 229359684137, 7731656573016, 272633076900991, 10876116332074739, 446659746000614675, 20580725671071449149, 964732749192326683508, 50418595763262446272127, 2656265906893413392905767
Offset: 0

Views

Author

Alois P. Heinz, Jul 21 2016

Keywords

Examples

			a(0) = 1: {}.
a(1) = 1: 12.
a(2) = 3: 12|34, 13|24, 14|23.
a(3) = 19: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 13|25|46, 13|26|45, 14|23|56, 1|2345|6, 1|2346|5, 15|23|46, 1|2356|4, 16|23|45, 14|25|36, 14|26|35, 15|24|36, 1|2456|3, 16|24|35, 15|26|34, 16|25|34.

Links

Crossrefs

Bisection (even part) of A305197.
Cf. A275281.

Programs

  • Mathematica
    b[n_, s_] := b[n, s] = Expand[If[n>s, Binomial[n-1, n-s-1]*x, 1] + Sum[Binomial[n-1, j-1]*b[n-j, s+j]*Binomial[s+j-1, j-1], {j, 1, (n-s)/2}]*x^2];
T[n_] := T[n] = Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0]];
a[n_] := T[2n][[n+1]];
a /@ Range[0, 20] (* Jean-François Alcover, Aug 21 2021,after Alois P. Heinz in A275281 *)

Formula

a(n) = A275281(2n,n).
a(n) ~ c * n^(n-1/2) * d^n / (exp(n) * 2^(n-3/2)), where d = 5.99720652866734051428..., c = 0.331364442872654716... if n is even and c = 0.32118925729236323... if n is odd. - Vaclav Kotesovec, Aug 08 2016

A305198 Number of set partitions of [2n+1] with symmetric block size list of length A109613(n).

Original entry on oeis.org

1, 1, 7, 56, 470, 10299, 91925, 3939653, 36298007, 2571177913, 24158837489, 2557117944391, 24350208829581, 3601150175699409, 34626777577615921, 6820331445080882282, 66066554102006208712, 16719951521837764142510, 162903256982698962545956
Offset: 0

Views

Author

Alois P. Heinz, May 27 2018

Keywords

Links

Crossrefs

Bisection (odd part) of A305197.
Cf. A109613, A275281, A275283.

Programs

  • Maple
    b:= proc(n, s) option remember; expand(`if`(n>s,
      binomial(n-1, n-s-1)*x, 1)+add(binomial(n-1, j-1)*
      b(n-j, s+j)*binomial(s+j-1, j-1), j=1..(n-s)/2)*x^2)
    end:
a:= n-> coeff(b(2*n+1, 0), x, n+irem(n+1, 2)):
seq(a(n), n=0..20);
  • Mathematica
    b[n_, s_] := b[n, s] = Expand[If[n > s, Binomial[n - 1, n - s - 1] x, 1] + Sum[Binomial[n - 1, j - 1] b[n - j, s + j] Binomial[s + j - 1, j - 1], {j, 1, (n - s)/2}] x^2];
a[n_] := Coefficient[b[2n + 1, 0], x, n + Mod[n + 1, 2]];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

Formula

a(n) = A275281(2n+1,A109613(n)).
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.