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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A190927 Number of permutations of n copies of 1..7 introduced in order 1..7 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 47844, 4420321081, 551248360550999, 81644850343968535401, 13519747358522016160671387, 2421032324142610480402567434373, 459408385876250801291447710561829082, 91155245844064069307740171414201519055298
Offset: 1

Views

Author

R. H. Hardin, May 23 2011

Keywords

Examples

			Some solutions for n=2:
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3
  4  4  4  4  4  2  2  4  4  4  2  4  4  4  4  2
  5  5  5  1  5  4  4  2  5  5  4  2  1  3  2  1
  6  6  6  5  6  5  3  5  3  6  5  4  4  5  5  4
  2  7  4  6  7  6  5  6  6  3  6  5  5  6  6  5
  7  3  5  4  2  3  6  5  7  7  7  1  6  4  4  6
  5  4  2  2  1  5  7  6  2  6  3  6  7  5  7  7
  6  5  6  3  6  4  5  1  4  1  7  3  2  7  1  3
  4  6  1  5  7  7  7  3  7  7  5  7  5  2  7  4
  3  2  7  7  5  1  1  7  1  5  1  6  7  1  5  6
  7  7  3  6  4  6  6  4  5  4  6  5  6  7  6  7
  1  1  7  7  3  7  4  7  6  2  4  7  3  6  3  5

Links

Crossrefs

Column k=7 of A322013.
Cf. A000012 (b=2), A190917 (b=3), A190918 (b=4), A190920 (b=5), A190923 (b=6), A190932 (b=8), A321987 (b=9).

Formula

a(n) ~ 343 * sqrt(7) * 2^(7*n-8) * 3^(7*n-3) / (625 * Pi^3 * n^3). - Vaclav Kotesovec, Nov 24 2018

A190932 Number of permutations of n copies of 1..8 introduced in order 1..8 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 721315, 1133879136649, 2536823683737613858, 6945222145021508480249929, 21671513613423101256198918372909, 74115215422015289392187745053216373265, 271259741131895052775392614041761701799270286
Offset: 1

Views

Author

R. H. Hardin, May 23 2011

Keywords

Examples

			Some solutions for n=2
..1....1....1....1....1....1....1....1....1....1....1....1....1....1....1....1
..2....2....2....2....2....2....2....2....2....2....2....2....2....2....2....2
..1....3....3....1....1....1....3....1....3....3....1....1....1....1....3....1
..3....1....1....3....3....3....1....3....1....1....3....3....3....3....1....2
..4....4....4....4....4....4....4....4....2....4....4....4....4....4....4....3
..5....5....5....3....5....5....5....5....3....5....5....5....2....5....5....4
..6....4....3....5....6....6....6....4....4....3....4....6....5....6....2....5
..5....6....6....6....7....7....2....6....5....6....5....7....6....5....6....6
..6....7....7....7....5....2....3....7....6....5....6....8....7....7....7....7
..3....8....8....8....6....8....5....8....7....2....7....7....5....8....3....8
..2....5....2....7....8....6....7....3....4....7....2....5....8....3....8....3
..7....2....6....8....2....3....6....6....6....6....6....2....3....6....4....5
..8....8....7....4....3....4....8....5....8....7....7....3....7....4....6....7
..4....3....4....6....4....5....7....8....5....8....8....4....4....2....7....4
..8....7....5....5....8....8....8....2....7....4....3....8....8....7....8....8
..7....6....8....2....7....7....4....7....8....8....8....6....6....8....5....6

Links

Crossrefs

Column k=8 of A322013.
Cf. A000012, A190917, A190918, A190920, A190923, A190927, A321987.

Formula

a(n) ~ 2 * 7^(8*n-2) / (1215 * sqrt(3) * Pi^(7/2) * n^(7/2)). - Vaclav Kotesovec, Nov 24 2018

A321987 Number of permutations of n copies of 1..9 introduced in order 1..9 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 12310199, 372419001449076, 16904301142107043464659, 967335448974819561548523580438, 64311863997340571475504065539218471107, 4749303210651587675797285013227098386984170468, 379065045836307787068046364731543393514652159389593652
Offset: 1

Views

Author

Seiichi Manyama, Nov 23 2018

Keywords

Links

Crossrefs

Column k=9 of A322013.
Cf. A000012, A190917, A190918, A190920, A190923, A190927, A190932.

Formula

a(n) ~ 2187 * 2^(27*n-14) / (84035 * Pi^4 * n^4). - Vaclav Kotesovec, Nov 24 2018

A369923 Array read by antidiagonals: A(n,k) is the number of permutations of n copies of 1..k with values introduced in order and without cyclically adjacent elements equal.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 31, 22, 1, 0, 1, 293, 1415, 134, 1, 0, 1, 3326, 140343, 75843, 866, 1, 0, 1, 44189, 20167651, 83002866, 4446741, 5812, 1, 0, 1, 673471, 3980871156, 158861646466, 55279816356, 276154969, 40048, 1, 0
Offset: 1

Views

Author

Andrew Howroyd, Feb 05 2024

Keywords

Comments

Also, T(n,k) is the number of generalized chord labeled loopless diagrams with k parts of K_n. See the Krasko reference for a full definition.

Examples

			Array begins:
n\k| 1 2    3         4              5                    6 ...
---+-----------------------------------------------------------
 1 | 0 1    1         1              1                    1 ...
 2 | 0 1    4        31            293                 3326 ...
 3 | 0 1   22      1415         140343             20167651 ...
 4 | 0 1  134     75843       83002866         158861646466 ...
 5 | 0 1  866   4446741    55279816356     1450728060971387 ...
 6 | 0 1 5812 276154969 39738077935264 14571371516350429940 ...
 ...

Links

Crossrefs

Rows 2..6 are A003436, A348813, A348815, A348818, A348821.
Column 3 is A197657, column 4 appears to be A209183(n)/2.
Cf. A322013 (without linearly adjacent elements equal), A322093.

Programs

  • Mathematica
    T[n_, k_] := If[k == 1, 0, Expand[(-1)^(k (n + 1))/(k - 1)! n Hypergeometric1F1[1 - n, 2, x]^k x^(k - 1)] /. x^p_ :> p!] (* Eric W. Weisstein, Feb 20 2025 *)
  • PARI
    \\ compare with A322013.
q(n, x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!)
T(n, k) = if(k > 1, subst(serlaplace(n*q(n, x)^k/x), x, 1)/(k-1)!, 0)

A190836 Number of permutations of 7 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 0, 1, 56620, 22875971289, 36533294879349056, 183095824753841610373405, 2421032324142610480402567434373, 74115215422015289392187745053216373265, 4749303210651587675797285013227098386984170468, 588242979144354234332728292738493758656488275002948671
Offset: 0

Views

Author

R. H. Hardin, May 21 2011

Keywords

Examples

			Some solutions for n=3
..1....1....1....1....1....1....1....1....1....1....1....1....1....1....1....1
..2....2....2....2....2....2....2....2....2....2....2....2....2....2....2....2
..3....1....3....3....3....3....1....3....3....1....1....1....3....1....3....1
..1....3....1....1....2....2....3....2....2....2....3....2....1....3....2....3
..2....1....2....2....3....3....2....3....1....3....2....1....3....1....3....2
..3....2....3....3....2....1....1....1....3....1....3....3....1....2....1....3
..1....3....2....1....1....3....2....3....1....3....1....1....3....1....2....2
..2....2....1....3....3....1....3....2....2....2....3....2....1....3....3....3
..3....3....2....1....1....2....1....1....3....3....2....3....3....1....2....1
..1....2....3....3....2....1....3....2....2....1....3....1....2....2....1....3
..3....1....1....2....3....3....2....3....3....2....1....2....1....3....3....1
..2....3....2....3....1....1....1....2....2....3....3....1....2....1....1....3
..3....2....3....1....3....3....3....1....3....1....2....3....3....3....2....2
..1....3....2....2....2....2....2....3....1....2....3....2....2....1....1....1
..2....1....3....1....1....3....1....2....3....3....2....3....3....2....3....2
..1....2....2....2....3....2....3....1....1....1....1....1....2....3....1....3
..2....1....1....3....1....3....2....3....3....2....2....3....3....2....2....2
..3....3....3....2....3....2....3....1....1....3....1....2....2....3....3....1
..1....1....1....3....2....1....2....3....2....2....2....3....1....2....1....2
..2....2....3....1....1....2....1....1....1....3....3....2....2....3....3....1
..3....3....1....2....2....1....3....2....2....1....1....3....1....2....2....3

Links

Crossrefs

Row n=7 of A322013.

Formula

a(n) ~ sqrt(7) * 117649^n * n^(6*n) / (720^n * exp(6*n + 6)). - Vaclav Kotesovec, Nov 24 2018

Extensions

a(0)=1 prepended by Seiichi Manyama, Nov 16 2018

A190837 Number of permutations of 8 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 0, 1, 400598, 1530622143864, 28920026907938624194, 2070756746775910218326948065, 459408385876250801291447710561829082, 271259741131895052775392614041761701799270286, 379065045836307787068046364731543393514652159389593652
Offset: 0

Views

Author

R. H. Hardin, May 21 2011

Keywords

Examples

			Some solutions for n=3
..1....1....1....1....1....1....1....1....1....1....1....1....1....1....1....1
..2....2....2....2....2....2....2....2....2....2....2....2....2....2....2....2
..1....1....1....1....1....1....1....1....1....1....1....1....1....1....1....1
..2....2....2....3....2....3....2....2....3....2....2....2....3....2....2....2
..3....3....3....1....3....1....1....1....1....1....1....3....1....3....3....3
..2....1....1....3....1....2....2....3....2....3....3....1....3....1....2....1
..3....3....2....1....2....1....3....2....3....1....1....3....1....3....3....2
..1....2....3....2....1....2....1....1....2....3....3....2....2....2....2....3
..3....1....1....1....3....1....2....3....3....2....1....1....3....3....3....2
..1....3....2....2....2....2....1....2....1....1....2....3....1....1....2....3
..2....1....1....1....3....3....3....1....3....3....1....2....3....3....3....1
..1....3....3....3....1....1....1....2....1....2....2....3....2....2....1....3
..2....2....1....2....3....3....2....3....3....3....3....2....3....3....2....2
..3....3....3....3....1....1....3....1....2....2....2....3....2....2....1....3
..1....1....2....2....3....3....2....3....1....3....3....1....1....3....2....1
..3....3....3....3....2....2....3....2....3....2....1....3....3....1....3....3
..2....2....1....1....3....3....1....3....2....1....2....1....2....3....1....2
..1....3....3....3....2....2....3....2....3....3....3....2....1....2....3....1
..3....2....2....2....3....1....1....1....2....2....2....1....2....3....1....2
..2....1....3....3....1....3....3....3....3....3....3....3....3....1....3....3
..3....2....1....2....2....2....2....1....2....1....1....1....2....2....1....1
..1....3....2....3....1....3....3....3....1....3....3....2....1....1....3....3
..2....2....3....1....3....2....2....2....2....2....2....3....3....2....2....2
..3....1....2....2....2....3....3....3....1....1....3....2....2....1....1....1

Links

Crossrefs

Row n=8 of A322013.

Formula

a(n) ~ sqrt(8) * 131072^n * n^(7*n) / (315^n * exp(7*n + 7)). - Vaclav Kotesovec, Nov 24 2018

Extensions

a(0)=1 prepended and a(7)-a(9) added by Seiichi Manyama, Nov 16 2018

A321666 Number of arrangements of n 1's, n 2's, ..., n n's avoiding equal consecutive terms and introduced in ascending order.

Original entry on oeis.org

1, 1, 1, 29, 94376, 66218360625, 16985819072511102549, 2421032324142610480402567434373, 271259741131895052775392614041761701799270286, 32119646666355552112999645991677870426882424139287301894021793
Offset: 0

Views

Author

Seiichi Manyama, Nov 16 2018

Keywords

Links

Crossrefs

Main diagonal of A322013.
Cf. A190826, A190830, A190833, A190835, A190836, A190837, A278990, A321634.

Programs

  • PARI
    {a(n) = sum(i=n, n^2, i!*polcoef(sum(j=1, n, (-1)^(n-j)*binomial(n-1, j-1)*x^j/j!)^n, i))/n!} \\ Seiichi Manyama, May 27 2019

Formula

a(n) = A321634(n)/n!.
a(n) ~ exp(5/12) * n^((n-1)*(2*n-1)/2) / (2*Pi)^(n/2). - Vaclav Kotesovec, Nov 24 2018

A321669 Number of permutations of 9 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 0, 1, 2872754, 104650147201049, 23575497690601916022516, 24302858067615766089801166488125, 91155245844064069307740171414201519055298, 1046031892354833895113128900608177633584652958677057, 32119646666355552112999645991677870426882424139287301894021793
Offset: 0

Views

Author

Seiichi Manyama, Nov 16 2018

Keywords

Links

Crossrefs

Row n=9 of A322013.
Cf. A190826, A190830, A190833, A190835, A190836, A190837, A321670.

Formula

a(n) ~ 9^(7*n + 1/2) * n^(8*n) / (4480^n * exp(8*(n+1))). - Vaclav Kotesovec, Nov 24 2018

A321670 Number of permutations of 10 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 0, 1, 20824778, 7279277647839552, 19672658572012343899666292, 293736218147318801678882792470437721, 18739368045280595665934917472507368174737872589, 4204427313459831775866154680419213479057724331798640498651
Offset: 0

Views

Author

Seiichi Manyama, Nov 16 2018

Keywords

Comments

In general, for r > 1, row r of A322013 is asymptotic to r^(r*n + 1/2) * n^((r-1)*n) / ((r!)^n * exp((r-1)*(n+1))). - Vaclav Kotesovec, Nov 24 2018

Links

Crossrefs

Cf. A190826, A190830, A190833, A190835, A190836, A190837, A321669.
Row r=10 of A322013.

Formula

a(n) ~ 2^(2*n + 1/2) * 5^(8*n + 1/2) * n^(9*n) / (567^n * exp(9*(n+1))). - Vaclav Kotesovec, Nov 24 2018

A322061 Number of permutations of n copies of 1..10 introduced in order 1..10 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 234615096, 152466248712342181, 156690501089429126239232946, 209141786137614009701487336108267723, 330586922756304429697714946501284146322953006, 588242979144354234332728292738493758656488275002948671
Offset: 1

Views

Author

Seiichi Manyama, Nov 25 2018

Keywords

Links

Crossrefs

Column 10 of A322013.

Formula

a(n) ~ 625 * 9^(10*n) / (5350883328 * (Pi*n)^(9/2)). - Vaclav Kotesovec, Nov 25 2018
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