A262033 -id:A262033 - cp's OEIS frontend

A262034 Number of permutations of [n] beginning with at least ceiling(n/2) ascents.

1, 0, 1, 1, 4, 5, 30, 42, 336, 504, 5040, 7920, 95040, 154440, 2162160, 3603600, 57657600, 98017920, 1764322560, 3047466240, 60949324800, 106661318400, 2346549004800, 4151586700800, 99638080819200, 177925144320000, 4626053752320000, 8326896754176000

Offset: 0

Alois P. Heinz, Sep 08 2015

nonn

			a(4) = 4: 1234, 1243, 1342, 2341.
a(5) = 5: 12345, 12354, 12453, 13452, 23451.

Alois P. Heinz, Table of n, a(n) for n = 0..733

Cf. A001761, A102693, A262033, A262035.

a:= proc(n) option remember; `if`(n<4, [1, 0, 1$2][n+1],
      2*((n^2-1)*a(n-2)-a(n-1))/(n+3))
    end:
seq(a(n), n=0..30);

np=Rest[With[{nn=30},CoefficientList[Series[(Exp[x^2](x+1)-x^4/2+x^2+x+1)/ x^3,{x,0,nn}],x] Range[0,nn]!]//Quiet];Join[{1},np] (* Harvey P. Dale, May 18 2019 *)

E.g.f.: (exp(x^2)*(x+1)-(x^4/2+x^2+x+1))/x^3.
a(n) = 2*((n^2-1)*a(n-2)-a(n-1))/(n+3) for n>3, a(0)=a(2)=a(3)=1, a(1)=0.
a(n) = n!/(n/2+1)! if n even, a(n) = floor(C(n+1,(n+1)/2)/(n+3)*((n-1)/2)!) if n odd.
a(2n) = A262033(2n) = A001761(n).
a(2n+1) = A102693(n+1).
Sum_{n>=2} 1/a(n) = (39*exp(1/4)*sqrt(Pi)*erf(1/2) - 6)/16, where erf is the error function. - Amiram Eldar, Dec 04 2022

A262035 Number of permutations of [2n+1] beginning with exactly n ascents.

Original entry on oeis.org

1, 2, 15, 168, 2520, 47520, 1081080, 28828800, 882161280, 30474662400, 1173274502400, 49819040409600, 2313026876160000, 116576554558464000, 6338850154116480000, 369890550169620480000, 23056510960573009920000, 1529010726859052236800000

Offset: 0

Alois P. Heinz, Sep 08 2015

nonn

			a(0) = 1: 1.
a(1) = 2: 132, 231.
a(2) = 15: 12435, 12534, 12543, 13425, 13524, 13542, 14523, 14532, 23415, 23514, 23541, 24513, 24531, 34512, 34521.

Alois P. Heinz, Table of n, a(n) for n = 0..365

Cf. A262033, A262034.

a:= proc(n) option remember; `if`(n<2, n+1,
      2*(n+1)*(2*n+1)*a(n-1)/(n+2))
    end:
seq(a(n), n=0..20);

E.g.f.: (1-2*x)/(4*sqrt(1-4*x)*x^2)+(2*x^2-1)/(4*x^2).
a(n) = 2*(n+1)*(2*n+1)*a(n-1)/(n+2) for n>1, a(n) = n+1 for n<=1.
a(n) = (2*n+1)!/(n+1)! - floor((2*n+2)!/((n+1)!*(n+1)*(n+2)*2)).
a(n) = A262033(2n+1) - A262034(2n+1).

A329964 a(n) = (n!/floor(1+n/2)!)^2.

Original entry on oeis.org

1, 1, 1, 9, 16, 400, 900, 44100, 112896, 9144576, 25401600, 3073593600, 9032601600, 1526509670400, 4674935865600, 1051860569760000, 3324398837760000, 960751264112640000, 3112834095724953600, 1123733108556708249600, 3714820193575895040000

Offset: 0

Peter Luschny, Dec 04 2019

nonn

Cf. A262033.

A329964 := n -> (n!/floor(1+n/2)!)^2:
seq(A329964(n), n=0..20);

a[n_] := (n!/Floor[1 + n/2]!)^2; Array[a, 20, 0] (* Amiram Eldar, Dec 04 2022 *)

a(n) = A262033(n)^2.

Sum_{n>=0} 1/a(n) = 29/16 + (3/16)*Pi*StruveL(-1, 1/2) + (57/64)*Pi*StruveL(0, 1/2) + (1/4)*Pi*StruveL(1, 1/2), where StruveL is the modified Struve function. - Amiram Eldar, Dec 04 2022

A329965 a(n) = ((1+n)floor(1+n/2))(n!/floor(1+n/2)!)^2.

Original entry on oeis.org

1, 2, 6, 72, 240, 7200, 25200, 1411200, 5080320, 457228800, 1676505600, 221298739200, 821966745600, 149597947699200, 560992303872000, 134638152929280000, 508633022177280000, 155641704786247680000, 591438478187741184000, 224746621711341649920000

Offset: 0

Peter Luschny, Dec 04 2019

nonn

Cf. A212303, A057977, A052510, A123072, A093005, A262033, A329964.

A329965 := n -> ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2:
seq(A329965(n), n=0..19);

ser := Series[(1 - Sqrt[1 - 4 x^2] - 4 x^2 (1 - x - Sqrt[1 - 4 x^2]))/(2 x^2 (1 - 4 x^2)^(3/2)), {x, 0, 22}]; Table[n! Coefficient[ser, x, n], {n, 0, 20}]
Table[(1+n)Floor[1+n/2](n!/Floor[1+n/2]!)^2,{n,0,30}] (* Harvey P. Dale, Oct 01 2023 *)

from fractions import Fraction
def A329965():
    x, n = 1, Fraction(1)
    while True:
        yield int(x)
        m = n if n % 2 else 4/(n+2)
        n += 1
        x *= m * n
a = A329965(); [next(a) for i in range(36)]

a(n) = n!*A212303(n+1).
a(n) = (n+1)!*A057977(n).
a(n) = A093005(n+1)*A262033(n)^2.
a(n) = A093005(n+1)*A329964(n).
a(2*n) = A052510(n) (n >= 0).
a(2*n+1) = A123072(n+1) (n >= 0).
a(n) = n! [x^n] (1 - sqrt(1 - 4*x^2) - 4*x^2*(1 - x - sqrt(1 - 4*x^2)))/(2*x^2*(1 - 4*x^2)^(3/2)).

cp's OEIS Frontend

A262034 Number of permutations of [n] beginning with at least ceiling(n/2) ascents.

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A262035 Number of permutations of [2n+1] beginning with exactly n ascents.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A329964 a(n) = (n!/floor(1+n/2)!)^2.

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Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A329965 a(n) = ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2.

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Maple

Mathematica

Python

Formula

A329965 a(n) = ((1+n)floor(1+n/2))(n!/floor(1+n/2)!)^2.