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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A268840 Number of sequences with n copies each of 1,2,3,4 and longest increasing subsequence of length 4.

Original entry on oeis.org

1, 641, 195709, 46922017, 10258694241, 2176464012941, 460827731023773, 98540942707986273, 21364658238692907265, 4697818999010952011441, 1046430770756355786405517, 235755137688345453796236397, 53640184515807269993604743389, 12308974812428409561104536925709
Offset: 1

Views

Author

Alois P. Heinz, Feb 14 2016

Keywords

Links

Crossrefs

Column k=4 of A047909.

Formula

a(n) ~ 2^(8*n-1/2) / (Pi*n)^(3/2). - Vaclav Kotesovec, Feb 21 2016

A268841 Number of sequences with n copies each of 1,2,...,5 and longest increasing subsequence of length 5.

Original entry on oeis.org

1, 11389, 50775091, 162588279629, 449363984934526, 1162145520205261219, 2931247600219365331976, 7370846583668954571029069, 18683332440278067962764855531, 47964531978782851644184417448714, 124871404619023570844557764310152386
Offset: 1

Views

Author

Alois P. Heinz, Feb 14 2016

Keywords

Links

  • Vaclav Kotesovec and Alois P. Heinz, Table of n, a(n) for n = 1..150 (terms n=1..80 from Vaclav Kotesovec)
  • J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Crossrefs

Column k=5 of A047909.

Formula

a(n) ~ 5^(5*n+1/2) / (2*Pi*n)^2. - Vaclav Kotesovec, Feb 21 2016

A268842 Number of sequences with n copies each of 1,2,...,6 and longest increasing subsequence of length 6.

Original entry on oeis.org

1, 248749, 20117051281, 1077273394836829, 47342758641593552281, 1878320344216429026862153, 70803267480031877368227941803, 2612508237897293571677286548812861, 96042041352156959435669839199503441435, 3553102771891168237056005934820411063204249
Offset: 1

Views

Author

Alois P. Heinz, Feb 14 2016

Keywords

Links

  • Alois P. Heinz, Table of n, a(n) for n = 1..100 (terms n=1..50 from Vaclav Kotesovec)
  • J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Crossrefs

Column k=6 of A047909.

Formula

a(n) ~ 6^(6*n + 1/2) / (2*Pi*n)^(5/2). - Vaclav Kotesovec, Feb 21 2016

A268843 Number of sequences with n copies each of 1,2,...,7 and longest increasing subsequence of length 7.

Original entry on oeis.org

1, 6439075, 11260558754404, 12084070123028603391, 10162884447920460534301136, 7465237877942551321425443305798, 5078529731893937404909347067888886466, 3315159778348807570604149155371730111763599, 2124172213523649116114190361767338538457819064671
Offset: 1

Views

Author

Alois P. Heinz, Feb 14 2016

Keywords

Links

  • Vaclav Kotesovec and Alois P. Heinz, Table of n, a(n) for n = 1..100 (terms n=1..36 from Vaclav Kotesovec)
  • J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Crossrefs

Column k=7 of A047909.

Formula

a(n) ~ 7^(7*n + 1/2) / (2*Pi*n)^3. - Vaclav Kotesovec, Feb 21 2016

A268844 Number of sequences with n copies each of 1,2,...,8 and longest increasing subsequence of length 8.

Original entry on oeis.org

1, 192621953, 8445885515991841, 211301962987912098409729, 3969183064899133655031651559801, 63178476289432401423971737795658030945, 909546798992441266072332791609067485208949369, 12324197596430667064913735085330208112438377122058241
Offset: 1

Views

Author

Alois P. Heinz, Feb 14 2016

Keywords

Links

Crossrefs

Column k=8 of A047909.

Formula

a(n) ~ 8^(8*n + 1/2) / (2*Pi*n)^(7/2). - Vaclav Kotesovec, Mar 03 2016

A268845 Number of sequences with n copies each of 1,2,...,9 and longest increasing subsequence of length 9.

Original entry on oeis.org

1, 6536413529, 8167981106765263789, 5426679072605204732028894233, 2599293828638212400913690945686101111, 1025794060996626005769021866749636185341527229, 358281333933096129012031117609647623312585201668494007
Offset: 1

Views

Author

Alois P. Heinz, Feb 14 2016

Keywords

Links

Crossrefs

Column k=9 of A047909.

Formula

a(n) ~ 9^(9*n + 1/2) / (2*Pi*n)^4. - Vaclav Kotesovec, Mar 03 2016

A268846 Number of sequences with n copies each of 1,2,...,10 and longest increasing subsequence of length 10.

Original entry on oeis.org

1, 248040482741, 9891092676022013399311, 195676681342450229063393365876181, 2683885055441747960475755652405552969614101, 29539005031390270063835072245497576346701114916209911, 282011782951614089942684801199121868144180995938610087493133121
Offset: 1

Views

Author

Alois P. Heinz, Feb 14 2016

Keywords

Links

Crossrefs

Column k=10 of A047909.

Formula

a(n) ~ 10^(10*n + 1/2) / (2*Pi*n)^(9/2). - Vaclav Kotesovec, Mar 03 2016

A268847 Number of sequences with 4 copies each of 1,2,...,n and longest increasing subsequence of length n.

Original entry on oeis.org

1, 1, 69, 31451, 46922017, 162588279629, 1077273394836829, 12084070123028603391, 211301962987912098409729, 5426679072605204732028894233, 195676681342450229063393365876181, 9562449832974304724626743446267704131, 615516610914323638585463757154352054695009
Offset: 0

Views

Author

Alois P. Heinz, Feb 14 2016

Keywords

Examples

			a(2) = binomial(8,4) - 1 = 69 because there are binomial(8,4) = 70 sequences with 4 copies of 1 and 4 copies of 2 and only 22221111 does not have an increasing subsequence of length 2.

Links

  • Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..145 (terms 0..70 from Alois P. Heinz)
  • J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Crossrefs

Row n=4 of A047909.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[k!/(i1!*i2!*i3!*(k - i1 - i2 - i3)!)*(4*k)!/(i1 + 2*i2 + 3*i3 + 4*(k - i1 - i2 - i3))!*(-1)^(i1 + 2*i2 + 3*i3 + 4*(k - i1 - i2 - i3) - k)/(6^i1*2^i2), {i3, 0, k - i1 - i2}], {i2, 0, k - i1}], {i1, 0, k}], {k, 0, 20}] (* Vaclav Kotesovec, Mar 02 2016, after Horton and Kurn *)

Formula

a(n) ~ 2^(7*n+1) * n^(3*n) / (3^n * exp(3*n+3)). - Vaclav Kotesovec, Feb 21 2016
Recurrence: 81*(n-4)*(n-3)^2*(n-2)^3*a(n) = 27*(n-4)*(n-3)^2*(57*n^7 - 328*n^6 + 560*n^5 + 159*n^4 - 1591*n^3 + 1942*n^2 - 994*n + 192)*a(n-1) - 18*(n-4)*(n-1)^3*(2*n - 3)*(4*n - 7)*(4*n - 5)*(18*n^7 - 111*n^6 - 76*n^5 + 2183*n^4 - 6887*n^3 + 9632*n^2 - 6371*n + 1620)*a(n-2) + 24*(n-2)^3*(n-1)^4*(2*n - 5)*(2*n - 3)*(4*n - 11)*(4*n - 9)*(4*n - 7)*(4*n - 5)*(n^5 + 6*n^4 - 115*n^3 + 440*n^2 - 626*n + 288)*a(n-3) - 32*(n-3)^3*(n-2)^4*(n-1)^5*(2*n - 7)*(2*n - 5)*(2*n - 3)*(4*n - 15)*(4*n - 13)*(4*n - 11)*(4*n - 9)*(4*n - 7)*(4*n - 5)*a(n-4). - Vaclav Kotesovec, Mar 03 2016

A268848 Number of sequences with 5 copies each of 1,2,...,n and longest increasing subsequence of length n.

Original entry on oeis.org

1, 1, 251, 729811, 10258694241, 449363984934526, 47342758641593552281, 10162884447920460534301136, 3969183064899133655031651559801, 2599293828638212400913690945686101111, 2683885055441747960475755652405552969614101
Offset: 0

Views

Author

Alois P. Heinz, Feb 14 2016

Keywords

Links

  • Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..100 (terms 0..50 from Alois P. Heinz)
  • J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Crossrefs

Row n=5 of A047909.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[k!/(i1!*i2!*i3!*i4!*(k - i1 - i2 - i3 - i4)!)*(5*k)!/(i1 + 2*i2 + 3*i3 + 4*i4 + 5*(k - i1 - i2 - i3 - i4))!*(-1)^(i1 + 2*i2 + 3*i3 + 4*i4 + 5*(k - i1 - i2 - i3 - i4) - k)/(24^i1*6^i2*2^ i3), {i4, 0, k - i1 - i2 - i3}], {i3, 0, k - i1 - i2}], {i2, 0, k - i1}], {i1, 0, k}], {k, 0, 15}] (* Vaclav Kotesovec, Mar 02 2016, after Horton and Kurn *)

Formula

a(n) ~ sqrt(5) * (3125/24)^n * n^(4*n) / exp(4*n+4). - Vaclav Kotesovec, Feb 21 2016

A268849 Number of sequences with 6 copies each of 1,2,...,n and longest increasing subsequence of length n.

Original entry on oeis.org

1, 1, 923, 16928840, 2176464012941, 1162145520205261219, 1878320344216429026862153, 7465237877942551321425443305798, 63178476289432401423971737795658030945, 1025794060996626005769021866749636185341527229, 29539005031390270063835072245497576346701114916209911
Offset: 0

Views

Author

Alois P. Heinz, Feb 14 2016

Keywords

Links

  • Vaclav Kotesovec and Alois P. Heinz, Table of n, a(n) for n = 0..97 (terms n=0..34 from Vaclav Kotesovec)
  • J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Crossrefs

Row n=6 of A047909.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[k!/(i1!*i2!*i3!*i4!*i5!*(k - i1 - i2 - i3 - i4 - i5)!)*(6*k)!/(i1 + 2*i2 + 3*i3 + 4*i4 + 5*i5 + 6*(k - i1 - i2 - i3 - i4 - i5))!*(-1)^(i1 + 2*i2 + 3*i3 + 4*i4 + 5*i5 + 6*(k - i1 - i2 - i3 - i4 - i5) - k)/(120^ i1*24^i2*6^i3*2^i4), {i5, 0, k - i1 - i2 - i3 - i4}], {i4, 0, k - i1 - i2 - i3}], {i3, 0, k - i1 - i2}], {i2, 0, k - i1}], {i1, 0, k}], {k, 0, 10}] (* Vaclav Kotesovec, Mar 02 2016, after Horton and Kurn *)

Formula

a(n) ~ 2^(3*n + 1/2) * 3^(5*n + 1/2) * n^(5*n) / (5^n * exp(5*(n+1))). - Vaclav Kotesovec, Feb 21 2016
Previous Showing 11-20 of 25 results. Next

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