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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A246195 Number of endofunctions on [n] where the smallest cycle length equals 8.

Original entry on oeis.org

5040, 362880, 18144000, 804988800, 34488115200, 1482082842240, 65120246231040, 2955402450000000, 139296260790086400, 6837541748945107200, 349978565353512775680, 18685433917574232157440, 1040347501174674201600000, 60368458528123335777196800
Offset: 8

Views

Author

Alois P. Heinz, Aug 18 2014

Keywords

Crossrefs

Column k=8 of A246049.

Programs

  • Maple
    with(combinat):
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
          add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
          b(n-i*j, i+1), j=0..n/i)))
        end:
    A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, k), j=0..n):
    a:= n-> A(n, 8) -A(n, 9):
    seq(a(n), n=8..25);

Formula

a(n) ~ (exp(-363/140) - exp(-761/280)) * n^n. - Vaclav Kotesovec, Aug 21 2014

A246196 Number of endofunctions on [n] where the smallest cycle length equals 9.

Original entry on oeis.org

40320, 3628800, 219542400, 11496038400, 570031862400, 27908676956160, 1379187810000000, 69648048665395200, 3619848890071814400, 194427690066299289600, 10817193383227574703360, 624124327363867459584000, 37361601302439627939398400, 2320453709453248711940505600
Offset: 9

Views

Author

Alois P. Heinz, Aug 18 2014

Keywords

Crossrefs

Column k=9 of A246049.

Programs

  • Maple
    with(combinat):
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
          add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
          b(n-i*j, i+1), j=0..n/i)))
        end:
    A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, k), j=0..n):
    a:= n-> A(n, 9) -A(n, 10):
    seq(a(n), n=9..25);

Formula

a(n) ~ (exp(-761/280) - exp(-7129/2520)) * n^n. - Vaclav Kotesovec, Aug 21 2014

A246197 Number of endofunctions on [n] where the smallest cycle length equals 10.

Original entry on oeis.org

362880, 39916800, 2874009600, 175394419200, 9967384627200, 551675124000000, 30471021291110400, 1703458301210265600, 97213825272736972800, 5693251850259515942400, 343266609438110040883200, 21349182724237331665228800, 1371132989012608561874534400
Offset: 10

Views

Author

Alois P. Heinz, Aug 18 2014

Keywords

Comments

In general, number of endofunctions on [n] where the smallest cycle length equals k is asymptotic to (exp(-H(k-1)) - exp(-H(k))) * n^n, where H(k) is the harmonic number A001008/A002805, k>=1. - Vaclav Kotesovec, Aug 21 2014

Crossrefs

Column k=10 of A246049.

Programs

  • Maple
    with(combinat):
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
          add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
          b(n-i*j, i+1), j=0..n/i)))
        end:
    A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, k), j=0..n):
    a:= n-> A(n, 10) -A(n, 11):
    seq(a(n), n=10..25);

Formula

a(n) ~ (exp(-7129/2520) - exp(-7381/2520))* n^n. - Vaclav Kotesovec, Aug 21 2014
Previous Showing 11-13 of 13 results.