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 51-56 of 56 results.

A177548 Number of permutations of {1,...,n} avoiding adjacent step pattern up, up, down, down, down, down.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5025, 40080, 359640, 3585600, 39322800, 470448000, 6097392939, 85106238492, 1272746220570, 20302567148160, 344103461618400, 6175195781293440, 116975122363277289, 2332451810975205960, 48833870901559002540, 1071107370496773577440
Offset: 0

Views

Author

R. H. Hardin, May 10 2010

Keywords

Links

Crossrefs

Columns k=48,60 of A242784.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(t>6, 0, `if`(u+o+t<7, (u+o)!,
      add(b(u-j, o+j-1, [1, 1, 4, 5, 6, 7][t]), j=1..u)+
      add(b(u+j-1, o-j, [2, 3, 3, 2, 2, 2][t]), j=1..o)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 23 2013
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] = If[t > 6, 0, If[u + o + t < 7, (u + o)!,
     Sum[b[u - j, o + j - 1, {1, 1, 4, 5, 6, 7}[[t]]], {j, 1, u}] +
     Sum[b[u + j - 1, o - j, {2, 3, 3, 2, 2, 2}[[t]]], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 19 2022, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n!, where d = 0.99698626702423025316812958090212389097043667361318991688710688185165..., c = 1.018367520648807878150063272123153629935889863110044893080319840979... . - Vaclav Kotesovec, Jan 17 2015

Extensions

a(17)-a(22) from Alois P. Heinz, Oct 23 2013

A177549 Number of permutations of {1,...,n} avoiding adjacent step pattern up, up, down, down, up, up.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 4969, 39184, 347544, 3424320, 37150741, 439774085, 5639099103, 77873192126, 1152123776419, 18181366630226, 304851804959519, 5412206888619242, 101424438933572112, 2000731009697485843, 41440364401733715980, 899211137893661967405
Offset: 0

Views

Author

R. H. Hardin, May 10 2010

Keywords

Links

Crossrefs

Column k=51 of A242784.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(t>6, 0, `if`(u+o+t<7, (u+o)!,
      add(b(u-j, o+j-1, [1, 1, 4, 5, 1, 1][t]), j=1..u)+
      add(b(u+j-1, o-j, [2, 3, 3, 2, 6, 7][t]), j=1..o)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 23 2013
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] = If[t>6, 0, If[u+o+t<7, (u+o)!, Sum[b[u-j, o+j-1, {1, 1, 4, 5, 1, 1}[[t]]], {j, 1, u}] + Sum[b[u+j-1, o-j, {2, 3, 3, 2, 6, 7}[[t]]], {j, 1, o}]]]; a[n_] := b[n, 0, 1]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Nov 11 2016, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n!, where d = 0.986314277283772995320277545416339641125925..., c = 1.08332315844132327949722334709840176297166... . - Vaclav Kotesovec, Jan 17 2015

Extensions

a(17)-a(22) from Alois P. Heinz, Oct 23 2013

A177550 Number of permutations of {1,...,n} avoiding adjacent step pattern up, up, down, up, down, down.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 4950, 38880, 343440, 3369600, 36352800, 427680000, 5452027218, 74846801304, 1100895311340, 17272089457920, 287920937620800, 5081935953473280, 94681381716805374, 1856848184953043760, 38236452673395920040, 824863858830361247040
Offset: 0

Views

Author

R. H. Hardin, May 10 2010

Keywords

Links

Crossrefs

Column k=52 of A242784.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(t>6, 0, `if`(u+o+t<7, (u+o)!,
      add(b(u-j, o+j-1, [1, 1, 4, 1, 6, 7][t]), j=1..u)+
      add(b(u+j-1, o-j, [2, 3, 3, 5, 3, 2][t]), j=1..o)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 29 2013
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] = If[t > 6, 0, If[u + o + t < 7, (u + o)!,
     Sum[b[u - j, o + j - 1, {1, 1, 4, 1, 6, 7}[[t]]], {j, 1, u}] +
     Sum[b[u + j - 1, o - j, {2, 3, 3, 5, 3, 2}[[t]]], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 19 2022, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n!, where d = 0.98057763883233672986361278986560196505968263650602..., c = 1.129827226571293707156672292645277720979050046894688... . - Vaclav Kotesovec, Jan 17 2015

Extensions

a(17)-a(22) from Alois P. Heinz, Oct 29 2013

A177551 Number of permutations of {1,...,n} avoiding adjacent step pattern up, up, down, up, up, up.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5006, 39776, 355536, 3530880, 38581456, 459905790, 5938902464, 82590382604, 1230586598456, 19557934834104, 330265974019619, 5905113439990368, 111448391640387360, 2214092715361273978, 46185648021443324656, 1009303225985497077402
Offset: 0

Views

Author

R. H. Hardin, May 10 2010

Keywords

Links

Crossrefs

Columns k=55,59 of A242784.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(t>6, 0, `if`(u+o+t<7, (u+o)!,
      add(b(u-j, o+j-1, [1, 1, 4, 1, 1, 4][t]), j=1..u)+
      add(b(u+j-1, o-j, [2, 3, 3, 5, 6, 7][t]), j=1..o)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 23 2013
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] = If[t > 6, 0, If[u + o + t < 7, (u + o)!,
     Sum[b[u - j, o + j - 1, {1, 1, 4, 1, 1, 4}[[t]]], {j, 1, u}] +
     Sum[b[u + j - 1, o - j, {2, 3, 3, 5, 6, 7}[[t]]], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 19 2022, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n!, where d = 0.993326320961568877764783962159580548643374705129260577..., c = 1.0404765828695886736964990440705643676428197038942922... . - Vaclav Kotesovec, Jan 17 2015

A177552 Number of permutations of {1,...,n} avoiding adjacent step pattern up, up, up, down, down, down.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5020, 40000, 358560, 3571200, 39124800, 467596800, 6054250840, 84417778720, 1261161277200, 20097223449600, 340275330912000, 6100262355686400, 115437689217984148, 2299445555596421920, 48093671993708346480, 1053794989665442654080
Offset: 0

Views

Author

R. H. Hardin, May 10 2010

Keywords

Links

Crossrefs

Column k=56 of A242784.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(t>6, 0, `if`(u+o+t<7, (u+o)!,
      add(b(u-j, o+j-1, [1, 1, 1, 5, 6, 7][t]), j=1..u)+
      add(b(u+j-1, o-j, [2, 3, 4, 4, 2, 2][t]), j=1..o)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 30 2013
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] = If[t > 6, 0, If[u + o + t < 7, (u + o)!,
     Sum[b[u - j, o + j - 1, {1, 1, 1, 5, 6, 7}[[t]]], {j, 1, u}] +
     Sum[b[u + j - 1, o - j, {2, 3, 4, 4, 2, 2}[[t]]], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 19 2022, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n!, where d = 0.9959682894155038013878176356538407492626252741099726077392745662726589922..., c = 1.02468007512189851788618819144905616307144561621610927886626291999589... . - Vaclav Kotesovec, Jan 17 2015

Extensions

a(17)-a(22) from Alois P. Heinz, Oct 30 2013

A254523 Number of permutations of [n] avoiding adjacent step pattern {up}^11.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001599, 6227020775, 87178290682, 1307674357710, 20922789683040, 355687423926240, 6402373618334400, 121645098513933120, 2432901965590252800, 51090941178938707200, 1124000703770606323200
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 31 2015

Keywords

Links

Crossrefs

Cf. A049774, A117158, A177523, A177533, A177553, A230051, A230231, A230232, A230233.
Column k=2047 of A242784.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<10, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
      add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..30); # after Alois P. Heinz
  • Mathematica
    CoefficientList[Series[6 / (Exp[-x] + Cos[x] + 2*Cos[x/2] * Cosh[Sqrt[3]*x/2] - Cosh[Sqrt[3]*x/2]*Sin[x/2] - Sin[x] + Cosh[x/2] * (2*Cos[Sqrt[3]*x/2] - Sqrt[3]*Sin[Sqrt[3]*x/2]) - Cos[Sqrt[3]*x/2]*Sinh[x/2] - Sqrt[3]*Cos[x/2]*Sinh[Sqrt[3]*x/2]), {x, 0, 25}], x] * Range[0, 25]!

Formula

E.g.f.: 1 / Sum_{n>=0} (12*n+1-x)*x^(12*n)/(12*n+1)!.
E.g.f.: 6 / (exp(-x) + cos(x) + 2*cos(x/2)*cosh(sqrt(3)*x/2) - cosh(sqrt(3)*x/2)*sin(x/2) - sin(x) + cosh(x/2)*(2*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) - cos(sqrt(3)*x/2)*sinh(x/2) - sqrt(3)*cos(x/2)*sinh(sqrt(3)*x/2)).
a(n)/n! ~ c * (1/r)^n, where r = 1.0000000019270853046730165249753673978954992128247736041276... is the root of the equation Sum_{n>=0} (r^(12*n)/(12*n)! - r^(12*n+1)/(12*n+1)!) = 0, equivalently root of the equation exp(-r) + cos(r) + 2*cos(r/2)*cosh(sqrt(3)*r/2) - cosh(sqrt(3)*r/2)*sin(r/2) - sin(r) + cosh(r/2)*(2*cos(sqrt(3)*r/2) - sqrt(3)*sin(sqrt(3)*r/2)) - cos(sqrt(3)*r/2)*sinh(r/2) - sqrt(3)*cos(r/2)*sinh(sqrt(3)*r/2) = 0, c = 3/(r*sqrt((cosh(sqrt(3)*r/2) * sin(r/2) + sin(r))^2 + 2*sqrt(3)*cosh(r/2) * (cosh(sqrt(3)*r/2) * sin(r/2) + sin(r)) * sin(sqrt(3)*r/2) + 3*cosh(r/2)^2 * sin((sqrt(3)*r)/2)^2)) = 1.0000000210373483515818712802156496756788404534079689145773611990529818919... .
Previous Showing 51-56 of 56 results.

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