A242784 -id:A242784 - cp's OEIS frontend

A230051 Number of permutations of [n] avoiding adjacent step pattern {up}^7.

1, 1, 2, 6, 24, 120, 720, 5040, 40319, 362863, 3628550, 39913170, 478947480, 6226179960, 87164597520, 1307440134000, 20918580896069, 355608034188517, 6400803479701178, 121612584595293870, 2432198062707745560, 51075033128533094520, 1123625953230764250960

Offset: 0

Alois P. Heinz, Oct 07 2013

nonn

			a(8) = 40319 = 8!-1: only permutation 12345678 does not avoid {up}^7.

R. E. L. Aldred, M. D. Atkinson, D. J. McCaughan, Avoiding consecutive patterns in permutations. Adv. in Appl. Math., 45(3), 449-461, 2010.

Alois P. Heinz, Table of n, a(n) for n = 0..450
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.

Cf. A049774, A117158, A177523, A177533, A177553.

Column k=127 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<6, 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=7;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[4/(E^(-x) + Cos[x] - Sin[x] + 2*Cos[x/Sqrt[2]] * Cosh[x/Sqrt[2]] - Sqrt[2] * Cos[x/Sqrt[2]] * Sinh[x/Sqrt[2]] - Sqrt[2] * Cosh[x/Sqrt[2]] * Sin[x/Sqrt[2]]),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Aug 23 2014 *)

E.g.f.: 1 / Sum_{n>=0} (8*n+1-x)*x^(8*n)/(8*n+1)!.
E.g.f. (Aldred, Atkinson, McCaughan, 2010): 4/(exp(-x) + cos(x) - sin(x) + 2*cos(x/sqrt(2))*cosh(x/sqrt(2)) - sqrt(2)*cos(x/sqrt(2))*sinh(x/sqrt(2)) - sqrt(2)*cosh(x/sqrt(2))*sin(x/sqrt(2))). - Vaclav Kotesovec, Aug 23 2014
a(n)/n! ~ c / r^n, where r = 1.0000220496837836995332841475679738951237308817759821845322... is the root of the equation exp(-r) + cos(r) - sin(r) + 2*cos(r/sqrt(2)) * cosh(r/sqrt(2)) - sqrt(2)*cos(r/sqrt(2)) * sinh(r/sqrt(2)) - sqrt(2) * cosh(r/sqrt(2)) * sin(r/sqrt(2)) = 0, c = 2*sqrt(2) / (r*sqrt(2 + cosh(sqrt(2)*r) - cos(2*r) + 2*cosh(r/sqrt(2)) * (2*sqrt(2)*sin(r) * sin(r/sqrt(2)) - cos(sqrt(2)*r) * cosh(r/sqrt(2))))) = 1.0001516144914746839400607922657094772985420791612537... . - Vaclav Kotesovec, Aug 23 2014, updated Feb 01 2015

A177553 Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up, up, up.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5039, 40305, 362682, 3626190, 39881160, 478490760, 6219298800, 87055051511, 1305598835941, 20885951018102, 354999461960226, 6388879812001704, 121367620532150280, 2426930566055020080, 50956684690331669759, 1120852238721212726609

Offset: 0

R. H. Hardin, May 10 2010

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.
Mingjia Yang, An experimental walk in patterns, partitions, and words, Ph. D. Dissertation, Rutgers University (2020).

Column k=63 of A242784.

Cf. A080635, A049774, A117158, A177533, A177523.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<5, 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

nn=20;r=6;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 *)
Table[n!*SeriesCoefficient[1/(Sum[x^(7*k)/(7*k)!-x^(7*k+1)/(7*k+1)!,{k,0,n}]),{x,0,n}],{n,1,20}] (* Vaclav Kotesovec, Aug 29 2014 *)

a(n)/n! ~ c * (1/r)^n, where r = 1.0001738181531504504518260962714687775785823593018886... is the root of the equation Sum_{n>=0} (r^(7*n)/(7*n)! - r^(7*n+1)/(7*n+1)!) = 0, c = 1.0010191104259450282450770594076722424772755532278.... - Vaclav Kotesovec, Aug 29 2014
E.g.f.: -(7/(2*((-cos(x*cos(3*Pi/14)))*cosh(x*sin(3*Pi/14)) + cos(x*cos(3*Pi/14))*cosh(x*sin(3*Pi/14))* sin(3*Pi/14) - cosh(x*sin(Pi/14))* (cos(x*cos(Pi/14))*(1 + sin(Pi/14)) - cos(Pi/14)*sin(x*cos(Pi/14))) + cos(3*Pi/14)*cosh(x*sin(3*Pi/14))* sin(x*cos(3*Pi/14)) - cosh(x*cos(Pi/7))* ((1 + cos(Pi/7))*cos(x*sin(Pi/7)) - sin(Pi/7)*sin(x*sin(Pi/7))) + cos(x*sin(Pi/7))* sinh(x*cos(Pi/7)) + cos(Pi/7)*cos(x*sin(Pi/7))* sinh(x*cos(Pi/7)) - sin(Pi/7)*sin(x*sin(Pi/7))* sinh(x*cos(Pi/7)) + cos(x*cos(Pi/14))* sinh(x*sin(Pi/14)) + cos(x*cos(Pi/14))*sin(Pi/14)* sinh(x*sin(Pi/14)) - cos(Pi/14)*sin(x*cos(Pi/14))* sinh(x*sin(Pi/14)) - cos(x*cos(3*Pi/14))* sinh(x*sin(3*Pi/14)) + cos(x*cos(3*Pi/14))* sin(3*Pi/14)*sinh(x*sin(3*Pi/14)) + cos(3*Pi/14)*sin(x*cos(3*Pi/14))* sinh(x*sin(3*Pi/14))))). - Vaclav Kotesovec, Jan 31 2015
In closed form, c = 7 / (r * (2*cos(r*sin(Pi/7))*cosh(r*cos(Pi/7)) + cos(Pi/7 - r*sin(Pi/7)) * cosh(r*cos(Pi/7)) + cos(Pi/7 - r*sin(Pi/7)) * cosh(r*cos(Pi/7)) + 2*cos(r*cos(Pi/14)) * cosh(r*sin(Pi/14)) + 2*cos(r*cos(3*Pi/14)) * cosh(r*sin(3*Pi/14)) + 2*cosh(r*sin(Pi/14)) * sin(Pi/14 + r*cos(Pi/14)) - 2*cosh(r*sin(3*Pi/14)) * sin(3*Pi/14 - r*cos(3*Pi/14)) - 2*cos(r*sin(Pi/7)) * sinh(r*cos(Pi/7)) - cos(Pi/7 - r*sin(Pi/7)) * sinh(r*cos(Pi/7)) - cos(Pi/7 - r*sin(Pi/7)) * sinh(r*cos(Pi/7)) - 2*cos(r*cos(Pi/14)) * sinh(r*sin(Pi/14)) - 2*sin(Pi/14 + r*cos(Pi/14))*sinh(r*sin(Pi/14)) + 2*cos(r*cos(3*Pi/14)) * sinh(r*sin(3*Pi/14)) - 2*sin((3*Pi)/14 - r*cos(3*Pi/14)) * sinh(r*sin(3*Pi/14)))). - Vaclav Kotesovec, Feb 01 2015

a(18)-a(22) from Alois P. Heinz, Oct 07 2013

a(0)=1 prepended by Alois P. Heinz, Aug 08 2018

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

A230797 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern up, down, up, down; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-3)/2)), read by rows.

Original entry on oeis.org

1, 1, 2, 6, 24, 104, 16, 528, 192, 3296, 1472, 272, 23168, 12800, 4352, 179712, 132352, 42880, 7936, 1573632, 1366016, 530432, 158720, 15207424, 14781952, 7662336, 1911296, 353792, 158880768, 178102272, 101713920, 31813632, 8491008, 1801996288, 2282645504

Offset: 0

Alois P. Heinz, Oct 30 2013

			T(5,1) = 16: 13254, 14253, 14352, 15243, 15342, 23154, 24153, 24351, 25143, 25341, 34152, 34251, 35142, 35241, 45132, 45231.
T(7,2) = 272: 1325476, 1326475, 1326574, ..., 6735241, 6745132, 6745231.
Triangle T(n,k) begins:
:  0 :       1;
:  1 :       1;
:  2 :       2;
:  3 :       6;
:  4 :      24;
:  5 :     104,      16;
:  6 :     528,     192;
:  7 :    3296,    1472,    272;
:  8 :   23168,   12800,   4352;
:  9 :  179712,  132352,  42880,   7936;
: 10 : 1573632, 1366016, 530432, 158720;

Alois P. Heinz, Rows n = 0..170, flattened

Columns k=0-2 give: A177520, A230832, A264077.
T(2n-1,n-2) gives A000182(n) for n>=3.
Row sums give: A000142.
Cf. A242783, A242784, A295987.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
      add(b(u-j, o+j-1, [1, 3, 1, 3][t])*`if`(t=4, x, 1), j=1..u)+
      add(b(u+j-1, o-j, [2, 2, 4, 2][t]), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..15);  # Alois P. Heinz, Oct 30 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[Sum[b[u-j, o+j-1, {1, 3, 1, 3}[[t]]]*If[t == 4, x, 1], {j, 1, u}] + Sum[b[u+j-1, o-j, {2, 2, 4, 2}[[t]]], {j, 1, o}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Oct 24 2016, after Alois P. Heinz *)

A227884 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern up, down, up; triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2)-1), read by rows.

Original entry on oeis.org

1, 1, 2, 6, 19, 5, 70, 50, 331, 328, 61, 1863, 2154, 1023, 11637, 16751, 10547, 1385, 81110, 144840, 102030, 34900, 635550, 1314149, 1109973, 518607, 50521, 5495339, 12735722, 13046040, 6858598, 1781101, 51590494, 134159743, 157195762, 97348436, 36004400

Offset: 0

Alois P. Heinz, Oct 25 2013

			T(4,1) = 5: 1324, 1423, 2314, 2413, 3412.
Triangle T(n,k) begins:
:  0 :      1;
:  1 :      1;
:  2 :      2;
:  3 :      6;
:  4 :     19,       5;
:  5 :     70,      50;
:  6 :    331,     328,      61;
:  7 :   1863,    2154,    1023;
:  8 :  11637,   16751,   10547,   1385;
:  9 :  81110,  144840,  102030,  34900;
: 10 : 635550, 1314149, 1109973, 518607, 50521;

Alois P. Heinz, Rows n = 0..170, flattened

Columns k=0-1 give: A177477, A227883.
T(2n,n-1) gives A000364(n) for n>=2.
Row sums give: A000142.
Cf. A000111, A242783, A242784, A295987.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
      add(b(u-j, o+j-1, [1, 3, 1][t]), j=1..u)+
      add(b(u+j-1, o-j, 2)*`if`(t=3, x, 1), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..15);

b[u_, o_, t_] := b[u, o, t] = If[u+o==0, 1, Expand[Sum[b[u-j, o+j-1, {1, 3, 1}[[t]]], {j, 1, u}]+Sum[b[u+j-1, o-j, 2]*If[t==3, x, 1], {j, 1, o}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, 0, 1]];
Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Mar 29 2017, translated from Maple *)

A335308 Number of permutations p of [n] such that the sequence of ascents and descents of p is encoded by the 0's and 1's, respectively, in the binary expansion of n (read from right to left and using leading 0's if necessary).

Original entry on oeis.org

1, 0, 0, 1, 3, 16, 26, 20, 69, 370, 1006, 945, 1266, 3015, 2365, 1001, 4367, 24736, 76960, 69615, 138397, 322944, 286824, 133056, 159391, 546504, 978054, 674245, 531530, 957320, 495495, 142506, 906191, 5537808, 18828096, 16231039, 37000909, 81351936, 71761536

Offset: 0

Alois P. Heinz, Sep 12 2020

			a(0) = 1: (), the empty permutation.
a(3) = 1: 321 (down, down).
a(4) = 3: 1243, 1342, 2341 (up, up, down).
a(5) = 16: 21435, 21534, 31425, 31524, 32415, 32514, 41325, 41523, 42315, 42513, 43512, 51324, 51423, 52314, 52413, 53412 (down, up, down, up).
a(6) = 26: 143256, 153246, 154236, 163245, 164235, 165234, 243156, 253146, 254136, 263145, 264135, 265134, 342156, 352146, 354126, 362145, 364125, 365124, 452136, 453126, 462135, 463125, 465123, 562134, 563124, 564123 (up, down, down, up, up).
a(7) = 20: 4321567, 5321467, 5421367, 5431267, 6321457, 6421357, 6431257, 6521347, 6531247, 6541237, 7321456, 7421356, 7431256, 7521346, 7531246, 7541236, 7621345, 7631245, 7641235, 7651234 (down^3, up^3).

Alois P. Heinz, Table of n, a(n) for n = 0..4095

Cf. A060351, A242783, A242784, A242785.

b:= proc(u, o, t) option remember; `if`(u+o=0, `if`(t=0, 1, 0),
     `if`(irem(t, 2)=0, add(b(u-j, o+j-1, iquo(t, 2)), j=1..u),
      add(b(u+j-1, o-j, iquo(t, 2)), j=1..o)))
    end:
a:= n-> b(n, 0, 2*n):
seq(a(n), n=0..42);

a(n) = A060351(n,n).
a(2^n-1) = binomial(2^n-2,n).
a(2^n) = binomial(2^n,n+1)-1.

A230695 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern up, down, down, up; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-2)/3)), read by rows.

Original entry on oeis.org

1, 1, 2, 6, 24, 109, 11, 588, 132, 3654, 1386, 26125, 13606, 589, 209863, 139714, 13303, 1876502, 1508756, 243542, 18441367, 17429745, 3953529, 92159, 197776850, 214536114, 63334182, 3354454, 2297242583, 2815529811, 1020982869, 93265537, 28739304385

Offset: 0

Alois P. Heinz, Oct 27 2013

			T(5,1) = 11: 14325, 15324, 15423, 24315, 25314, 25413, 34215, 35214, 35412, 45213, 45312.
T(8,2) = 589: 14327658, 14328657, 14328756, ..., 78635412, 78645213, 78645312.
Triangle T(n,k) begins:
:  0 :        1;
:  1 :        1;
:  2 :        2;
:  3 :        6;
:  4 :       24;
:  5 :      109,       11;
:  6 :      588,      132;
:  7 :     3654,     1386;
:  8 :    26125,    13606,     589;
:  9 :   209863,   139714,   13303;
: 10 :  1876502,  1508756,  243542;
: 11 : 18441367, 17429745, 3953529, 92159;

Alois P. Heinz, Rows n = 0..100, flattened

Column k=0 gives: A177519.
Row sums give: A000142.
Cf. A242783, A242784, A295987.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
      add(b(u-j, o+j-1, [1, 3, 4, 1][t]), j=1..u)+
      add(b(u+j-1, o-j, 2)*`if`(t=4, x, 1), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..15);

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Expand[
     Sum[b[u - j, o + j - 1, {1, 3, 4, 1}[[t]]], {j, 1, u}] +
     Sum[b[u + j - 1, o - j, 2]*If[t == 4, x, 1], {j, 1, o}]]];
T[n_] := CoefficientList[b[n, 0, 1], x];
T /@ Range[0, 15] // Flatten (* Jean-François Alcover, Mar 22 2021, after Alois P. Heinz *)

A242785 Number of permutations of [n] avoiding the consecutive step pattern given by the binary expansion of n, where 1=up and 0=down.

Original entry on oeis.org

1, 1, 2, 5, 21, 70, 450, 4326, 34944, 209863, 1573632, 21824925, 302273664, 2854894485, 60269056512, 1207441809209, 19346879737625, 252773481889854, 2918333808555034, 69792946997645295, 982945842995115000, 16085109561896603059, 402131210857811703926

Offset: 0

Alois P. Heinz, May 22 2014

nonn

			a(4) = 21 because there are 4! = 24 permutations of {1,2,3,4} and only 3 of them do not avoid the consecutive step pattern up, down, down given by the binary expansion of 4 = 100_2: (1,4,3,2), (2,4,3,1), (3,4,2,1).

Alois P. Heinz, Table of n, a(n) for n = 0..150

Column k=0 of A242783.
Main diagonal of A242784.
Cf. A335308.

a:= proc(n) option remember; local b, m, r, h;
      if n<2 then return 1 fi;
      m:= iquo(n, 2, 'r'); h:= 2^ilog2(n);
      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(a(n), n=0..30);

a[n_] := a[n] = Module[{b, m, r, h},
     If[n < 2, Return[1]]; {m, r} = QuotientRemainder[n, 2];
     h = 2^(Length@IntegerDigits[n, 2] - 1);
     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[2t, h]], {j, 1, u}]] +
     If[t == m && r == 1, 0,
          Sum[b[u + j - 1, o - j, Mod[2t+1, h]], {j, 1, o}]]];
     b[n, 0, 0]];
a /@ Range[0, 30] (* Jean-François Alcover, Mar 23 2021, after Alois P. Heinz *)

cp's OEIS Frontend

A230051 Number of permutations of [n] avoiding adjacent step pattern {up}^7.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A177553 Number of permutations of 1..n avoiding adjacent step pattern up, up, up, up, up, up.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A230797 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern up, down, up, down; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-3)/2)), read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A227884 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern up, down, up; triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2)-1), read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A335308 Number of permutations p of [n] such that the sequence of ascents and descents of p is encoded by the 0's and 1's, respectively, in the binary expansion of n (read from right to left and using leading 0's if necessary).

Views

Author

Keywords

Examples

Links