A230051 -id:A230051 - cp's OEIS frontend

A242784 Number A(n,k) of permutations of [n] avoiding the consecutive step pattern given by the binary expansion of k, where 1=up and 0=down; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 2, 5, 8, 1, 1, 1, 1, 2, 6, 17, 16, 1, 1, 1, 1, 2, 6, 21, 70, 32, 1, 1, 1, 1, 2, 6, 19, 90, 349, 64, 1, 1, 1, 1, 2, 6, 21, 70, 450, 2017, 128, 1, 1, 1, 1, 2, 6, 23, 90, 331, 2619, 13358, 256, 1, 1

Offset: 0

Alois P. Heinz, May 22 2014

			A(4,5) = 19 because there are 4! = 24 permutations of {1,2,3,4} and only 5 of them do not avoid the consecutive step pattern up, down, up given by the binary expansion of 5 = 101_2: (1,3,2,4), (1,4,2,3), (2,3,1,4), (2,4,1,3), (3,4,1,2).
Square array A(n,k) begins:
  1, 1,   1,     1,     1,     1,     1,     1,     1, ...
  1, 1,   1,     1,     1,     1,     1,     1,     1, ...
  1, 1,   2,     2,     2,     2,     2,     2,     2, ...
  1, 1,   4,     5,     6,     6,     6,     6,     6, ...
  1, 1,   8,    17,    21,    19,    21,    23,    24, ...
  1, 1,  16,    70,    90,    70,    90,   111,   116, ...
  1, 1,  32,   349,   450,   331,   450,   642,   672, ...
  1, 1,  64,  2017,  2619,  1863,  2619,  4326,  4536, ...
  1, 1, 128, 13358, 17334, 11637, 17334, 33333, 34944, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns give: 0, 1: A000012, 2: A011782, 3: A049774, 4, 6: A177479, 5: A177477, 7: A117158, 8, 14: A177518, 9: A177519, 10: A177520, 11, 13: A177521, 12: A177522, 15: A177523, 16, 30: A177524, 17: A177525, 18, 22: A177526, 19, 25: A177527, 20, 26: A177528, 21: A177529, 23, 29: A177530, 24, 28: A177531, 27: A177532, 31: A177533, 32, 62: A177534, 33: A177535, 34, 46: A177536, 35, 49: A177537, 36, 54: A177538, 37, 41: A177539, 38: A177540, 39, 57: A177541, 40, 58: A177542, 42: A177543, 43, 53: A177544, 44, 50: A177545, 45: A177546, 47, 61: A177547, 48, 60: A177548, 51: A177549, 52: A177550, 55, 59: A177551, 56: A177552, 63: A177553, 127: A230051, 255: A230231, 511: A230232, 1023: A230233, 2047: A254523.
Main diagonal gives A242785.
Cf. A242783, A335308.

A:= proc(n, k) option remember; local b, m, r, h;
      if k<2 then return 1 fi;
      m:= iquo(k, 2, 'r'); h:= 2^ilog2(k);
      b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t=m and r=0, 0, add(b(u-j, o+j-1, irem(2*t, h)), j=1..u))+
      `if`(t=m and r=1, 0, add(b(u+j-1, o-j, irem(2*t+1, h)), j=1..o)))
      end; forget(b);
      b(n, 0, 0)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..15);

Clear[A]; A[n_, k_] := A[n, k] = Module[{b, m, r, h}, If[k < 2, Return[1]]; {m, r} = QuotientRemainder[k, 2]; h = 2^Floor[Log[2, k]]; b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t == m && r == 0, 0, Sum[b[u - j, o + j - 1, Mod[2*t, h]], {j, 1, u}]] + If[t == m && r == 1, 0, Sum[b[u + j - 1, o - j, Mod[2*t + 1, h]], {j, 1, o}]]]; b[n, 0, 0]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Sep 22 2014, translated from Maple *)

A117158 Number of permutations avoiding the consecutive pattern 1234.

Original entry on oeis.org

1, 1, 2, 6, 23, 111, 642, 4326, 33333, 288901, 2782082, 29471046, 340568843, 4263603891, 57482264322, 830335952166, 12793889924553, 209449977967081, 3630626729775362, 66429958806679686, 1279448352687538463, 25874432578888440471, 548178875969847203202

Offset: 0

Steven Finch, Apr 26 2006

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 1234. It is the same as the number of permutations which avoid 4321.

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, New York, 1962, pages 156-157.

Alois P. Heinz, Table of n, a(n) for n = 0..450 (terms n = 0..60 from Ray Chandler)
A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. Appl. Math. 36 (2006), 138-155.
Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125.
Steven Finch, Pattern-Avoiding Permutations. [Archived version]
Steven Finch, Pattern-Avoiding Permutations. [Cached copy, with permission]
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory A 53 (1990), 257-285.
Ira M. Gessel, Yan Zhuang, Counting permutations by alternating descents, arXiv:1408.1886 [math.CO], 2014. See displayed equation before Eq. (3) and set m=4. - _N. J. A. Sloane_, Aug 11 2014
Kaarel Hänni, Asymptotics of descent functions, arXiv:2011.14360 [math.CO], Nov 29 2020, p. 14.
Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.
Mingjia Yang, An experimental walk in patterns, partitions, and words, Ph. D. Dissertation, Rutgers University (2020).

Cf. A022558, A049774, A111004, A113228, A113229, A117156, A117226, A177523, A177533, A201692, A201693, A230051, A324132.
Column k=0 of A220183.
Column k=7 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<2, 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);  # Alois P. Heinz, Oct 07 2013

a[n_]:=Coefficient[Series[2/(Cos[x]-Sin[x]+Exp[ -x]),{x,0,30}],x^n]*n!
(* second program: *)
b[u_, o_, t_] := b[u, o, t] = If[u+o==0, 1, If[t<2, Sum[b[u+j-1, o-j, t+1], {j, 1, o}], 0] + Sum[b[u-j, o+j-1, 0], {j, 1, u}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *)

From Sergei N. Gladkovskii, Nov 30 2011: (Start)
E.g.f.: 2/(exp(-x) + cos(x) - sin(x)) = 1/W(0) with continued fraction
W(k) = 1 + (x^(2*k))/(f + f*x/(4*k + 1 - x - (4*k + 1)*b/R)), where R := x^(2*k) + b -(x^(4*k+1))/(c + (x^(2*k+1)) + x*c/T); T := 4*k + 3 - x - (4*k + 3)*d/(d +(x^(2*k+1))/W(k+1)), and
f := (4*k)!/(2*k)!; b := (4*k + 1)!/(2*k + 1)!; c := (4*k + 2)!/(2*k + 1)!; and d :=(4*k + 3)!/(2*k + 2)!. (End)
a(n) ~ n! / (sin(r)*r^(n+1)), where r = 1.0384156372665563... is the root of the equation exp(-r) + cos(r) = sin(r). - Vaclav Kotesovec, Dec 11 2013

A230231 Number of permutations of [n] avoiding adjacent step pattern {up}^8.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362879, 3628781, 39916492, 478996716, 6226941864, 87176969880, 1307651304960, 20922368987520, 355679390626560, 6402213152423659, 121641748198554547, 2432828930036156696, 51089280818439941448, 1123961390341566969192

Offset: 0

Alois P. Heinz, Oct 12 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.

Cf. A049774, A117158, A177523, A177533, A177553, A230051, A230232, A230233.

Column k=255 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<7, 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);

nn=20;r=8;a=Apply[Plus,Table[Normal[Series[y x^(r+1)/(1-Sum[y x^i,{i,1,r}]),{x,0,nn}]][[n]]/(n+r)!,{n,1,nn-r}]]/.y->-1;Range[0,nn]! CoefficientList[Series[1/(1-x-a),{x,0,nn}],x] (* Geoffrey Critzer, Feb 25 2014 *)
CoefficientList[Series[1/(HypergeometricPFQ[{}, {1/9, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 8/9}, x^9/387420489] - x*HypergeometricPFQ[{}, {2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 8/9, 10/9}, x^9/387420489]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 01 2015 *)

E.g.f.: 1 / Sum_{n>=0} (9*n+1-x)*x^(9*n)/(9*n+1)!.
a(n)/n! ~ 1.0000195665100891649606434859189953881417919885320660432331680939719... * (1/r)^n, where r = 1.0000024802134092668222044475851121972165291678378389183730077680957571... is the root of the equation Sum_{n>=0} (r^(9*n)/(9*n)! - r^(9*n+1)/(9*n+1)!) = 0. - Vaclav Kotesovec, Jan 17 2015
E.g.f.: 1/(1/3 * cos((sqrt(3)*x)/2) * cosh(x/2) + 2/9 * cos(x * sin(Pi/9)) * cosh(x * cos(Pi/9)) + 2/9 * cos(Pi/9) * cos(x * sin(Pi/9)) * cosh(x * cos(Pi/9)) + 2/9 * cos(x * cos(Pi/18)) * cosh(x * sin(Pi/18)) + 2/9 * cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))) - 1/9 * cos(Pi/9) * cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))) * cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))) - 2/9 * cos(x * cos(Pi/18))* cosh(x * sin(Pi/18)) * sin(Pi/18) - (cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9)))* sin(Pi/9))/(3 * sqrt(3)) - (cosh(x/2) * sin((sqrt(3)*x)/2))/(3 * sqrt(3)) - 2/9 * cos(Pi/18) * cosh(x * sin(Pi/18)) * sin(x * cos(Pi/18)) - (cos(Pi/9) * cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9)))* sin(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))))/ (3 * sqrt(3)) + 1/9 * cosh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9)))* sin(Pi/9) * sin(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))) - 2/9 * cosh(x * cos(Pi/9)) * sin(Pi/9)* sin(x * sin(Pi/9)) - 1/3 * cos((sqrt(3)*x)/2)* sinh(x/2) + (sin((sqrt(3)*x)/2) * sinh(x/2))/ (3 * sqrt(3)) - 2/9 * cos(x * sin(Pi/9))* sinh(x * cos(Pi/9)) - 2/9 * cos(Pi/9) * cos(x * sin(Pi/9))* sinh(x * cos(Pi/9)) + 2/9 * sin(Pi/9) * sin(x * sin(Pi/9))* sinh(x * cos(Pi/9)) + 2/9 * cos(x * cos(Pi/18))* sinh(x * sin(Pi/18)) - 2/9 * cos(x * cos(Pi/18))* sin(Pi/18) * sinh(x * sin(Pi/18)) - 2/9 * cos(Pi/18)* sin(x * cos(Pi/18)) * sinh(x * sin(Pi/18)) + 2/9 * cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))) - 1/9 * cos(Pi/9) * cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))) * sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))) - (cos(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9))) * sin(Pi/9)* sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))))/ (3 * sqrt(3)) - (cos(Pi/9) * sin(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9))))/ (3 * sqrt(3)) + 1/9 * sin(Pi/9)* sin(x/2*(sqrt(3) * cos(Pi/9) - sin(Pi/9)))* sinh(x/2*(cos(Pi/9) + sqrt(3) * sin(Pi/9)))). - Vaclav Kotesovec, Feb 01 2015

A230232 Number of permutations of [n] avoiding adjacent step pattern {up}^9.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628799, 39916779, 479001228, 6227014404, 87178179816, 1307672369640, 20922752672640, 355686706327680, 6402359109968640, 121644792614741760, 2432895242801771955, 51090787299486057355, 1123997039003038423610

Offset: 0

Alois P. Heinz, Oct 12 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A049774, A117158, A177523, A177533, A177553, A230051, A230231, A230233.

Column k=511 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<8, 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);

nn=20;r=9;a=Apply[Plus,Table[Normal[Series[y x^(r+1)/(1-Sum[y x^i,{i,1,r}]),{x,0,nn}]][[n]]/(n+r)!,{n,1,nn-r}]]/.y->-1;Range[0,nn]! CoefficientList[Series[1/(1-x-a),{x,0,nn}],x] (* Geoffrey Critzer, Feb 25 2014 *)
FullSimplify[CoefficientList[Series[10/(2/E^x - Sqrt[2*(5 - Sqrt[5])]* Cosh[(1/4)*(1 + Sqrt[5])*x]* Sin[Sqrt[(1/8)*(5 - Sqrt[5])]*x] - Sqrt[2*(5 + Sqrt[5])]*Cosh[(1/4)*(Sqrt[5] - 1)* x]*Sin[Sqrt[(1/8)*(5 + Sqrt[5])]*x] + Cos[Sqrt[(1/8)*(5 + Sqrt[5])]*x]* (4*Cosh[(1/4)*(Sqrt[5] - 1)*x] - (Sqrt[5] - 1)*Sinh[(1/4)*(Sqrt[5] - 1)*x]) - Cos[Sqrt[(1/8)*(5 - Sqrt[5])]*x]* ((1 + Sqrt[5])*Sinh[(1/4)*(1 + Sqrt[5])*x] - 4*Cosh[(1/4)*(1 + Sqrt[5])*x])), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 31 2015 *)

E.g.f.: 1 / Sum_{n>=0} (10*n+1-x)*x^(10*n)/(10*n+1)!.
a(n)/n! ~ c * (1/r)^n, where r = 1.0000002505217051890946793081039639693008257169189079028339632923816... is the root of the equation Sum_{n>=0} (r^(10*n)/(10*n)! - r^(10*n+1)/(10*n+1)!) = 0, c = 1.000002229648140602899529055054469878816530201510267349345270187155... . - Vaclav Kotesovec, Jan 17 2015
E.g.f.: 10 / (2/exp(x) - sqrt(2*(5 - sqrt(5))) * cosh((1/4)*(1 + sqrt(5))*x) * sin(sqrt((1/8)*(5 - sqrt(5)))*x) - sqrt(2*(5 + sqrt(5))) * cosh((1/4)*(sqrt(5) - 1)*x) * sin(sqrt((1/8)*(5 + sqrt(5)))*x) + cos(sqrt((1/8)*(5 + sqrt(5)))*x) * (4*cosh((1/4)*(sqrt(5) - 1)*x) - (sqrt(5) - 1)*sinh((1/4)*(sqrt(5) - 1)*x)) - cos(sqrt((1/8)*(5 - sqrt(5)))*x) * ((1 + sqrt(5))*sinh((1/4)*(1 + sqrt(5))*x) - 4*cosh((1/4)*(1 + sqrt(5))*x))). - Vaclav Kotesovec, Jan 31 2015
In closed form, c = 5 / (r * (sqrt(10 - 2*sqrt(5)) * cosh((sqrt(5)+1)*r/4) * sin(sqrt((5 - sqrt(5))/2)*r/2) + sqrt(2*(5 + sqrt(5))) * cosh((sqrt(5)-1)*r/4) * sin(sqrt((5 + sqrt(5))/2)*r/2))). - Vaclav Kotesovec, Feb 01 2015

A230233 Number of permutations of [n] avoiding adjacent step pattern {up}^10.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916799, 479001577, 6227020358, 87178283010, 1307674215120, 20922786961440, 355687370176320, 6402372516146880, 121645075013280000, 2432901444395385600, 51090929159028595200, 1124000415686590747031

Offset: 0

Alois P. Heinz, Oct 12 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A049774, A117158, A177523, A177533, A177553, A230051, A230231, A230232.

Column k=1023 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<9, 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);

nn=20;r=10;a=Apply[Plus,Table[Normal[Series[y x^(r+1)/(1-Sum[y x^i,{i,1,r}]),{x,0,nn}]][[n]]/(n+r)!,{n,1,nn-r}]]/.y->-1;Range[0,nn]! CoefficientList[Series[1/(1-x-a),{x,0,nn}],x] (* Geoffrey Critzer, Feb 25 2014 *)
CoefficientList[Series[1/(HypergeometricPFQ[{}, {1/11, 2/11, 3/11, 4/11, 5/11, 6/11, 7/11, 8/11, 9/11, 10/11}, x^11/285311670611] - x*HypergeometricPFQ[{}, {2/11, 3/11, 4/11, 5/11, 6/11, 7/11, 8/11, 9/11, 10/11, 12/11}, x^11/285311670611]), {x, 0, 25}], x] * Range[0, 25]! (* Vaclav Kotesovec, Jan 17 2015 *)

E.g.f.: 1 / Sum_{n>=0} (11*n+1-x)*x^(11*n)/(11*n+1)!.
a(n)/n! ~ 1.000000227556759905306252970186381144189779110025896440589711080508... * (1/r)^n, where r = 1.000000022964438439732421879840792836238519233492197325926442472620564... is the root of the equation Sum_{n>=0} (r^(11*n)/(11*n)! - r^(11*n+1)/(11*n+1)!) = 0. - Vaclav Kotesovec, Jan 17 2015
E.g.f.: -11/(2*((-cos(x*cos(Pi/22)))* cosh(x*sin(Pi/22)) - cos(x*cos(3*Pi/22))* cosh(x*sin(3*Pi/22)) - cos(x*cos(5*Pi/22))* cosh(x*sin(5*Pi/22)) - cos(x*cos(Pi/22))* cosh(x*sin(Pi/22))*sin(Pi/22) + cos(x*cos(3*Pi/22))* cosh(x*sin(3*Pi/22))* sin(3*Pi/22) - cos(x*cos(5*Pi/22))* cosh(x*sin(5*Pi/22))* sin(5*Pi/22) + cos(Pi/22)* cosh(x*sin(Pi/22))* sin(x*cos(Pi/22)) + cos(3*Pi/22)*cosh( x*sin(3*Pi/22))* sin(x*cos(3*Pi/22)) + cos(5*Pi/22)*cosh( x*sin(5*Pi/22))* sin(x*cos(5*Pi/22)) - cosh(x*cos(Pi/11))* ((1 + cos(Pi/11))* cos(x*sin(Pi/11)) - sin(Pi/11)*sin(x*sin(Pi/11))) + cosh(x*cos(2*Pi/11))* ((-1 + cos(2*Pi/11))* cos(x*sin(2*Pi/11)) + sin(2*Pi/11)* sin(x*sin(2*Pi/11))) + cos(x*sin(Pi/11))* sinh(x*cos(Pi/11)) + cos(Pi/11)*cos(x*sin(Pi/11))* sinh(x*cos(Pi/11)) - sin(Pi/11)*sin(x*sin(Pi/11))* sinh(x*cos(Pi/11)) - cos(x*sin(2*Pi/11))* sinh(x*cos(2*Pi/11)) + cos(2*Pi/11)* cos(x*sin(2*Pi/11))* sinh(x*cos(2*Pi/11)) + sin(2*Pi/11)* sin(x*sin(2*Pi/11))* sinh(x*cos(2*Pi/11)) + cos(x*cos(Pi/22))* sinh(x*sin(Pi/22)) + cos(x*cos(Pi/22))*sin(Pi/22)* sinh(x*sin(Pi/22)) - cos(Pi/22)*sin(x*cos(Pi/22))* sinh(x*sin(Pi/22)) - cos(x*cos(3*Pi/22))* sinh(x*sin(3*Pi/22)) + cos(x*cos(3*Pi/22))* sin(3*Pi/22)* sinh(x*sin(3*Pi/22)) + cos(3*Pi/22)* sin(x*cos(3*Pi/22))* sinh(x*sin(3*Pi/22)) + cos(x*cos(5*Pi/22))* sinh(x*sin(5*Pi/22)) + cos(x*cos(5*Pi/22))* sin(5*Pi/22)* sinh(x*sin(5*Pi/22)) - cos(5*Pi/22)* sin(x*cos(5*Pi/22))* sinh(x*sin(5*Pi/22)))). - Vaclav Kotesovec, Jan 31 2015

A230055 Number of permutations of [n] in which the longest increasing run has length 7.

Original entry on oeis.org

1, 14, 181, 2360, 32010, 456720, 6881160, 109546009, 1841298059, 32629877967, 608572228291, 11923667699474, 244964063143590, 5267496652725480, 118348438201424761, 2773714509551524351, 67705791536824698266, 1718769199589362743761, 45314525515737783596251

Offset: 7

Alois P. Heinz, Oct 07 2013

nonn

			a(7) = 1: 1234567.
a(8) = 14: 12345687, 12345786, 12346785, 12356784, 12456783, 13456782, 21345678, 23456781, 31245678, 41235678, 51234678, 61234578, 71234568, 81234567.

Alois P. Heinz, Table of n, a(n) for n = 7..200

Column k=7 of A008304.

b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1,
      `if`(t b(n, 0, 0, 7)-b(n, 0, 0, 6):
seq(a(n), n=7..30);

b[u_, o_, t_, k_] := b[u, o, t, k] = If[u + o == 0, 1, If[t < k - 1, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}], 0] + Sum[b[u - j, o + j - 1, 0, k], {j, 1, u}]];
a[n_] := b[n, 0, 0, 7] - b[n, 0, 0, 6];
Table[a[n], {n, 7, 30}] (* Jean-François Alcover, Jul 19 2018, after Alois P. Heinz *)

a(n) = A230051(n) - A177553(n).

E.g.f.: 1/Sum_{n>=0} (8*n+1-x)*x^(8*n)/(8*n+1)! - 1/Sum_{n>=0} (7*n+1-x)*x^(7*n)/(7*n+1)!.

A230234 Number of permutations of [n] in which the longest increasing run has length 8.

Original entry on oeis.org

1, 16, 231, 3322, 49236, 761904, 12372360, 211170960, 3788091451, 71356438043, 1409672722481, 29163603260677, 630867328411136, 14247689906846928, 335437110802718232, 8220763598490652440, 209435069840238717949, 5539287889970005834349, 151909981369978722092098

Offset: 8

Alois P. Heinz, Oct 12 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..170

Column k=8 of A008304.

b:= proc(u, o, t, k) option remember; `if`(u+o=0, 1,
      `if`(t b(n, 0, 0, 8)-b(n, 0, 0, 7):
seq(a(n), n=8..30);

E.g.f.: 1/Sum_{n>=0} (9*n+1-x)*x^(9*n)/(9*n+1)! - 1/Sum_{n>=0} (8*n+1-x)*x^(8*n)/(8*n+1)!.

a(n) = A230231(n) - A230051(n).

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

Vaclav Kotesovec, Jan 31 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A049774, A117158, A177523, A177533, A177553, A230051, A230231, A230232, A230233.

Column k=2047 of A242784.

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

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]!

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... .

cp's OEIS Frontend

A242784 Number A(n,k) of permutations of [n] avoiding the consecutive step pattern given by the binary expansion of k, where 1=up and 0=down; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A117158 Number of permutations avoiding the consecutive pattern 1234.

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A230231 Number of permutations of [n] avoiding adjacent step pattern {up}^8.

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A230232 Number of permutations of [n] avoiding adjacent step pattern {up}^9.

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A230233 Number of permutations of [n] avoiding adjacent step pattern {up}^10.

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A230055 Number of permutations of [n] in which the longest increasing run has length 7.

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A230234 Number of permutations of [n] in which the longest increasing run has length 8.

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

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