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 1-2 of 2 results.

A002875 Sorting numbers (see Motzkin article for details).

Original entry on oeis.org

1, 2, 4, 24, 128, 880, 7440
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

How is the sequence defined (see the links in A000262)? Also more terms would be welcome.
Based on the Motzkin article, where this sequence appears in the last row of the table on p. 173, one would expect that this sequence is the same as A294202. However, they seem to be unrelated. So the true definition of this sequence is a mystery. - Andrew Howroyd and Andrey Zabolotskiy, Oct 25 2017

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000262, A001861, A002872, A002873, A002874, A036073, A294201, A294202.

A294201 Irregular triangle read by rows: T(n,k) is the number of k-partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles (1 <= k <= 3n).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 0, 1, 1, 3, 10, 12, 3, 9, 3, 0, 1, 1, 7, 33, 59, 30, 67, 42, 6, 18, 4, 0, 1, 1, 15, 106, 270, 216, 465, 420, 120, 235, 100, 10, 30, 5, 0, 1, 1, 31, 333, 1187, 1365, 3112, 3675, 1596, 2700, 1655, 330, 605, 195, 15, 45, 6, 0, 1
Offset: 1

Views

Author

Andrew Howroyd, Oct 24 2017

Keywords

Comments

T(n,k) = coefficient of x^k for A(3,n)(x) in Gilbert and Riordan's article. - Robert A. Russell, Jun 13 2018

Examples

			Triangle begins:
  1,  0,   1;
  1,  1,   3,   2,   0,   1;
  1,  3,  10,  12,   3,   9,   3,   0,   1;
  1,  7,  33,  59,  30,  67,  42,   6,  18,   4,  0,  1;
  1, 15, 106, 270, 216, 465, 420, 120, 235, 100, 10, 30, 5, 0, 1;
  ...
Case n=2: Without loss of generality the permutation of two 3-cycles can be taken as (123)(456). The second row is [1, 1, 3, 2, 0, 1] because the set partitions that are invariant under this permutation in increasing order of number of parts are {{1, 2, 3, 4, 5, 6}}; {{1, 2, 3}, {4, 5, 6}}; {{1, 4}, {2, 5}, {3, 6}}, {{1, 5}, {2, 6}, {3, 4}}, {{1, 6}, {2, 4}, {3, 5}}; {{1, 2, 3}, {4}, {5}, {6}}, {{1}, {2}, {3}, {4, 5, 6}}, {{1}, {2}, {3}, {4}, {5}, {6}}.

Links

Crossrefs

Row sums are A002874.
Column k=3 gives A053156.
Maximum row values are A294202.
Unrelated to A002875.
Cf. A002872, A002873, A293181.

Programs

  • Maple
    T:= proc(n, k) option remember; `if`([n, k]=[0, 0], 1, 0)+
     `if`(n>0 and k>0, k*T(n-1, k)+T(n-1, k-1)+T(n-1, k-3), 0)
    end:
seq(seq(T(n, k), k=1..3*n), n=1..8);  # Alois P. Heinz, Sep 20 2019
  • Mathematica
    T[n_, k_] := T[n,k] = If[n>0 && k>0, k T[n-1,k] + T[n-1,k-1] + T[n-1,k-3], Boole[n==0 && k==0]] (* modification of Gilbert & Riordan recursion *)
Table[T[n, k], {n,1,10}, {k,1,3n}] // Flatten (* Robert A. Russell, Jun 13 2018 *)
  • PARI
    \\ see A056391 for Polya enumeration functions
T(n,k)={my(ci=PermCycleIndex(CylinderPerms(3,n)[2])); StructsByCycleIndex(ci,k) - if(k>1,StructsByCycleIndex(ci,k-1))}
for (n=1, 6, for(k=1, 3*n, print1(T(n,k), ", ")); print);
  • PARI
    G(n)={Vec(-1+serlaplace(exp(sumdiv(3, d, y^d*(exp(d*x + O(x*x^n))-1)/d))))}
{ my(A=G(6)); for(n=1, #A, print(Vecrev(A[n]/y))) } \\ Andrew Howroyd, Sep 20 2019

Formula

T(n,k) = [n==0 & k==0] + [n>0 & k>0] * (k*T(n-1,k) + T(n-1,k-1) + T(n-1,k-3)). - Robert A. Russell, Jun 13 2018
T(n,k) = n!*[x^n*y^k] exp(Sum_{d|3} y^d*(exp(d*x) - 1)/d). - Andrew Howroyd, Sep 20 2019
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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