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.

Previous Showing 11-20 of 26 results. Next

A230341 Number of permutations of [2n] in which the longest increasing run has length n.

Original entry on oeis.org

1, 1, 16, 293, 5811, 133669, 3574957, 109546009, 3788091451, 145957544981, 6201593613645, 288084015576169, 14525808779580645, 790129980885855401, 46120599397152203401, 2875600728737862162481, 190740227037467026439611, 13411608375592255857753781
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2013

Keywords

Links

Crossrefs

A diagonal of A008304.
Cf. A267433.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<5, [1, 1, 16, 293, 5811][n+1],
      (2*(n+1)*(26615475780292426*n^4 +2862121494132556*n^3
        -240402559504315639*n^2 +79488454158677567*n
        +119546195807549142)*a(n-1)
      -n*(406022528821033256*n^4 -1031369150352151615*n^3
        +11028208356875758*n^2 -1654923205028490137*n
        +3900125789057762682)*a(n-2)
      +2*(n-1)*(421508861354067594*n^4 -1543365451253363033*n^3
        -602924004257000736*n^2 +6654606478117189961*n
        -5221800341103267066)*a(n-3)
      -4*(2*n-7)*(n-2)*(26655798868586248*n^3 +401269836638339496*n^2
        -2000296275137853913*n +2124466470996744981)*a(n-4)
      -8*(n-3)*(n-5)*(2*n-7)*(2*n-9)*(8655617328093650*n
        -14323734034655567)*a(n-5)) / (n*(n+2)*(13307737890146213*n^2
        -43906954139467620*n +22672341406878775)))
    end:
seq(a(n), n=0..25);
  • Mathematica
    b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u] + o < k, 0, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}] + Sum[b[u - j, o + j - 1, 1, k], {j, 1, u}]]];
T[n_, k_] := b[0, n, 0, k] - b[0, n, 0, k + 1];
a[n_] := T[2n, n];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jul 19 2018, after Alois P. Heinz *)

Formula

a(n) = A008304(2*n,n).
Recurrence (of order 3): n*(n+2)*(12*n^7 - 101*n^6 + 355*n^5 - 668*n^4 + 631*n^3 - 71*n^2 - 344*n + 174)*a(n) = (n+1)*(48*n^9 - 272*n^8 + 453*n^7 - 10*n^6 - 518*n^5 - 741*n^4 + 4090*n^3 - 5810*n^2 + 3444*n - 720)*a(n-1) - 2*n*(2*n - 3)*(60*n^8 - 277*n^7 + 365*n^6 - 59*n^5 - 549*n^4 + 1619*n^3 - 2101*n^2 + 1228*n - 268)*a(n-2) + 4*(n-1)*(2*n - 5)*(2*n - 3)*(12*n^7 - 17*n^6 + n^5 + 12*n^4 - 91*n^3 + 101*n^2 - 12*n - 12)*a(n-3). - Vaclav Kotesovec, Jul 16 2014
a(n) ~ 2^(2*n+1/2)* n^(n+1) / exp(n). - Vaclav Kotesovec, Jul 16 2014

A230342 Number of permutations of [2n+2] in which the longest increasing run has length n+2.

Original entry on oeis.org

1, 6, 67, 1024, 19710, 456720, 12372360, 383685120, 13406178240, 521194867200, 22318001798400, 1043827513344000, 52949040240096000, 2895555891900672000, 169823181579891840000, 10633812541718446080000, 708077586604965857280000, 49962245750984840232960000
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2013

Keywords

Comments

Also the number of ascending runs of length n+2 in the permutations of [2n+2].

Links

Crossrefs

A diagonal of A008304, A122843.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1+5*n, 2*(n+1)*(2*n+1)*
      (n^3+6*n^2+12*n+11)*a(n-1)/((n+4)*(n^3+3*n^2+3*n+4)))
    end:
seq(a(n), n=0..25);

Formula

a(n) = (n^3+6*n^2+12*n+11)*(2*n+2)!/(n+4)! for n>0, a(0) = 1.
a(n) = A008304(2*n+2,n+2) = A122843(2*n+2,n+2).

A230343 Number of permutations of [2n+3] in which the longest increasing run has length n+3.

Original entry on oeis.org

1, 8, 99, 1602, 32010, 761904, 21064680, 663848640, 23500653120, 923616691200, 39914540709120, 1881558401184000, 96096062174112000, 5286518167746816000, 311689569962010240000, 19608741674518284288000, 1311187373310480906240000, 92868537238628772741120000
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2013

Keywords

Comments

Also the number of ascending runs of length n+3 in the permutations of [2n+3].

Links

Crossrefs

A diagonal of A008304, A122843.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1+7*n, 2*(2*n+3)*(n+1)*
      (n^3+8*n^2+20*n+19)*a(n-1)/((n+5)*(n^3+5*n^2+7*n+6)))
    end:
seq(a(n), n=0..25);

Formula

a(n) = (n^3+8*n^2+20*n+19)*(2*n+3)!/(n+5)! for n>0, a(0) = 1.
a(n) = A008304(2*n+3,n+3) = A122843(2*n+3,n+3).

A230344 Number of permutations of [2n+4] in which the longest increasing run has length n+4.

Original entry on oeis.org

1, 10, 137, 2360, 49236, 1209936, 34288800, 1102187520, 39656131200, 1579837754880, 69064610186880, 3288126441600000, 169388400557376000, 9389435419203840000, 557323393281887232000, 35272416767753797632000, 2371290445442664345600000
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2013

Keywords

Comments

Also the number of ascending runs of length n+4 in the permutations of [2n+4].

Links

Crossrefs

A diagonal of A008304, A122843.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1+9*n, 2*(2*n+3)*(n+2)*
      (n^3+10*n^2+30*n+29)*a(n-1)/((n+6)*(n^3+7*n^2+13*n+8)))
    end:
seq(a(n), n=0..25);

Formula

a(n) = (n^3+10*n^2+30*n+29)*(2*n+4)!/(n+6)! for n>0, a(0) = 1.
a(n) = A008304(2*n+4,n+4) = A122843(2*n+4,n+4).

A230345 Number of permutations of [2n+5] in which the longest increasing run has length n+5.

Original entry on oeis.org

1, 12, 181, 3322, 72540, 1845480, 53749920, 1766525760, 64739122560, 2619453513600, 116043825744000, 5588681114016000, 290812286052288000, 16263827918642304000, 973009916329651200000, 62017234027123415040000, 4195886889891954216960000
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2013

Keywords

Comments

Also the number of ascending runs of length n+5 in the permutations of [2n+5].

Links

Crossrefs

A diagonal of A008304, A122843.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1+11*n, 2*(2*n+5)*(n+2)*
      (n^3+12*n^2+42*n+41)*a(n-1)/((n+7)*(n^3+9*n^2+21*n+10)))
    end:
seq(a(n), n=0..25);

Formula

a(n) = (n^3+12*n^2+42*n+41)*(2*n+5)!/(n+7)! for n>0, a(0) = 1.
a(n) = A008304(2*n+5,n+5) = A122843(2*n+5,n+5).

A230346 Number of permutations of [2n+6] in which the longest increasing run has length n+6.

Original entry on oeis.org

1, 14, 231, 4512, 103194, 2721600, 81591840, 2746068480, 102661518960, 4224849995520, 189917647920000, 9263565222912000, 487461283781472000, 27533206366009344000, 1661865400404937728000, 106768864984887705600000, 7275718977990226283520000
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2013

Keywords

Comments

Also the number of ascending runs of length n+6 in the permutations of [2n+6].

Links

Crossrefs

A diagonal of A008304, A122843.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1+13*n, 2*(2*n+5)*(n+5)*
      (n+3)*(n^2+9*n+11)*a(n-1)/((n+4)*(n+8)*(n^2+7*n+3)))
    end:
seq(a(n), n=0..25);

Formula

a(n) = (n+5)*(n^2+9*n+11)*(2*n+6)!/(n+8)! for n>0, a(0) = 1.
a(n) = A008304(2*n+6,n+6) = A122843(2*n+6,n+6).

A230347 Number of permutations of [2n+7] in which the longest increasing run has length n+7.

Original entry on oeis.org

1, 16, 287, 5954, 142590, 3900480, 120466080, 4156079760, 158664456720, 6647965632000, 303540020784000, 15009431909472000, 799414492260384000, 45641465547245568000, 2781538377619921920000, 180263592116387619840000, 12381113998069012804608000
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2013

Keywords

Comments

Also the number of ascending runs of length n+7 in the permutations of [2n+7].

Links

Crossrefs

A diagonal of A008304, A122843.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1+15*n, 2*(n+3)*(2*n+7)*
      (n^3+16*n^2+72*n+71)*a(n-1)/((n+9)*(n^3+13*n^2+43*n+14)))
    end:
seq(a(n), n=0..25);

Formula

a(n) = (n^3+16*n^2+72*n+71)*(2*n+7)!/(n+9)! for n>0, a(0) = 1.
a(n) = A008304(2*n+7,n+7) = A122843(2*n+7,n+7).

A230348 Number of permutations of [2n+8] in which the longest increasing run has length n+8.

Original entry on oeis.org

1, 18, 349, 7672, 192240, 5454144, 173606040, 6143195520, 239656253760, 10231052832000, 474832908950400, 23819880180096000, 1284985968634368000, 74207855717259264000, 4569213387521502720000, 298885288012537901875200, 20702796608070625112064000
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2013

Keywords

Comments

Also the number of ascending runs of length n+8 in the permutations of [2n+8].

Links

Crossrefs

A diagonal of A008304, A122843.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1+17*n, 2*(n+4)*(2*n+7)*
      (n^3+18*n^2+90*n+89)*a(n-1)/((n+10)*(n^3+15*n^2+57*n+16)))
    end:
seq(a(n), n=0..25);

Formula

a(n) = (n^3+18*n^2+90*n+89)*(2*n+8)!/(n+10)! for n>0, a(0) = 1.
a(n) = A008304(2*n+8,n+8) = A122843(2*n+8,n+8).

A230349 Number of permutations of [2n+9] in which the longest increasing run has length n+9.

Original entry on oeis.org

1, 20, 417, 9690, 253776, 7465176, 244906200, 8891411760, 354610872000, 15432114297600, 728406536457600, 37090538241120000, 2027740775284224000, 118512161081233920000, 7376476698319125196800, 487273386402209523916800, 34055074238462266429440000
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2013

Keywords

Comments

Also the number of ascending runs of length n+9 in the permutations of [2n+9].

Links

Crossrefs

A diagonal of A008304, A122843.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1+19*n, 2*(2*n+9)*(n+4)*
      (n^3+20*n^2+110*n+109)*a(n-1)/((n+11)*(n^3+17*n^2+73*n+18)))
    end:
seq(a(n), n=0..25);

Formula

a(n) = (n^3+20*n^2+110*n+109)*(2*n+9)!/(n+11)! for n>0, a(0) = 1.
a(n) = A008304(2*n+9,n+9) = A122843(2*n+9,n+9).

A230350 Number of permutations of [2n+10] in which the longest increasing run has length n+10.

Original entry on oeis.org

1, 22, 491, 12032, 328950, 10027440, 339006360, 12628788480, 515033719200, 22855760928000, 1097589192336000, 56754471481344000, 3145763658989952000, 186150029203211673600, 11717355323144959488000, 781981263963810054144000, 55165533654753963657216000
Offset: 0

Views

Author

Alois P. Heinz, Oct 16 2013

Keywords

Comments

Also the number of ascending runs of length n+10 in the permutations of [2n+10].

Links

Crossrefs

A diagonal of A008304, A122843.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1+21*n, 2*(n+5)*(2*n+9)*
      (n^3+22*n^2+132*n+131)*a(n-1)/((n+12)*(n^3+19*n^2+91*n+20)))
    end:
seq(a(n), n=0..25);

Formula

a(n) = (n^3+22*n^2+132*n+131)*(2*n+10)!/(n+12)! for n>0, a(0) = 1.
a(n) = A008304(2*n+10,n+10) = A122843(2*n+10,n+10).
Previous Showing 11-20 of 26 results. Next

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

Content is available under The OEIS End-User License Agreement.