A005359 -id:A005359 - cp's OEIS frontend

A158757 Expansion of e.g.f. exp(tx)/(1 - x^2/t^2 - t^3 x^3).

1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 6, 0, 0, 0, 7, 24, 0, 0, 0, 12, 0, 0, 0, 25, 0, 0, 120, 0, 0, 0, 260, 0, 0, 0, 61, 720, 0, 0, 0, 360, 0, 0, 0, 1470, 0, 0, 0, 841, 0, 0, 5040, 0, 0, 0, 15960, 0, 0, 0, 5082, 0, 0, 0, 5251, 40320, 0, 0, 0, 20160, 0, 0, 0, 122640, 0, 0, 0, 134456, 0, 0, 0, 20497

Offset: 0

Roger L. Bagula, Mar 25 2009

			Irregular triangle begins as:
      1;
      0, 0,    1;
      2, 0,    0, 0,   1;
      0, 0,    6, 0,   0, 0,     7;
     24, 0,    0, 0,  12, 0,     0, 0,   25;
      0, 0,  120, 0,   0, 0,   260, 0,    0, 0,   61;
    720, 0,    0, 0, 360, 0,     0, 0, 1470, 0,    0, 0, 841;
      0, 0, 5040, 0,   0, 0, 15960, 0,    0, 0, 5082, 0,   0, 0, 5251;

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, page 221.

G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

Cf. A005212, A005359, A158706, A330044.

Table[CoefficientList[Expand[t^n*n!*SeriesCoefficient[Series[Exp[t*x]/(1 - x^2/t^2 - t^3*x^3), {x, 0, 20}], n]], t], {n, 0, 10}]//Flatten

f(x,t) = exp(t*x)/(1 - x^2/t^2 - t^3*x^3)
def A158757(n,k): return ( factorial(n)*t^n*( f(x,t) ).series(x, 20).list()[n] ).series(t,2*n+1).list()[k]
flatten([[A158757(n,k) for k in (0..2*n)] for n in (0..10)]) # G. C. Greubel, Dec 05 2021

T(n, k) = coefficients of e.g.f.: exp(t*x)/(1 - x^2/t^2 - t^3* x^3).
From G. C. Greubel, Dec 05 2021: (Start)
T(n, 2*n) = A330044(n).
T(n, 0) = A005359(n).
T(n, 2) = A005212(n). (End)

Edited by G. C. Greubel, Dec 01 2021

A167568 A triangle related to the GF(z) formulas of the rows of the ED2 array A167560.

Original entry on oeis.org

1, 0, 2, 2, -2, 6, 0, 16, -16, 24, 24, -48, 144, -120, 120, 0, 432, -864, 1392, -960, 720, 720, -2160, 8208, -12816, 14448, -8400, 5040, 0, 23040, -69120, 149760, -184320, 161280, -80640, 40320, 40320, -161280, 760320, -1716480, 2684160, -2695680

Offset: 1

Johannes W. Meijer, Nov 10 2009

The GF(z) formulas given below correspond to the first ten rows of the ED2 array A167560. The polynomials in their numerators lead to the triangle given above.

			Row 1: GF(z) = 1/(1-z).
Row 2: GF(z) = 2/(1-z)^2.
Row 3: GF(z) = (2*z^2 - 2*z + 6)/(1-z)^3.
Row 4: GF(z) = (0*z^3 + 16*z^2 - 16*z + 24)/(1-z)^4.
Row 5: GF(z) = (24*z^4 - 48*z^3 + 144*z^2 - 120*z + 120)/(1-z)^5.
Row 6: GF(z) = (432*z^4 - 864*z^3 + 1392*z^2 - 960*z + 720)/(1-z)^6.
Row 7: GF(z) = (720*z^6 - 2160*z^5 + 8208*z^4 - 12816*z^3 + 14448*z^2 - 8400*z + 5040)/(1-z)^7.
Row 8: GF(z) = (0*z^7 + 23040*z^6 - 69120*z^5 + 149760*z^4 - 184320*z^3 + 161280*z^2 - 80640*z + 40320)/(1-z)^8.
Row 9: GF(z) = (40320*z^8 - 161280*z^7 + 760320*z^6 - 1716480*z^5 + 2684160*z^4 - 2695680*z^3 + 1935360*z^2 - 846720*z + 362880)/(1-z)^9.
Row 10: GF(z) = (0*z^9 + 2016000*z^8 - 8064000*z^7 + 22464000*z^6 - 39168000*z^5 + 48360960*z^4 - 40849920*z^3 + 24917760*z^2 - 9676800*z + 3628800)/(1-z)^10.

A167560 is the ED2 array.
A005359 equals the first left hand column.
A000142(n=>1) and 2*A005990 equal the first two right hand columns.
A000142(n=>1) equals the row sums.

A335873 Total number of points in all permutations of [n] that are fixed or reflected.

Original entry on oeis.org

0, 1, 4, 10, 48, 216, 1440, 9360, 80640, 685440, 7257600, 76204800, 958003200, 11975040000, 174356582400, 2528170444800, 41845579776000, 690452066304000, 12804747411456000, 236887827111936000, 4865804016353280000, 99748982335242240000, 2248001455555215360000

Offset: 0

Alois P. Heinz, Jun 28 2020

A permutation p of [n] has fixed point j if p(j) = j, it has reflected point j if p(n+1-j) = j. A point can be fixed and reflected at the same time.

			a(3) = 10: (1)(2)(3), (1)32, 21(3), 23(1), (3)12, (3)(2)(1).

Alois P. Heinz, Table of n, a(n) for n = 0..450
T. Simpson, Permutations with unique fixed and reflected points, Preprint. (Annotated scanned copy)
Wikipedia, Permutation

Bisection (even part) gives 2 * A010050(n) for n>0.

Cf. A000142, A005359, A306258, A335872.

b:= proc(s, i) option remember; (n-> `if`(n=0, [1, 0],
      add((p-> p+[0, `if`(j in {i, n}, p[1], 0)])(
        b(s minus {j}, i+1)), j=s)))(nops(s))
    end:
a:= n-> b({$1..n}, 1)[2]:
seq(a(n), n=0..14);
# second Maple program:
a:= n-> `if`(n=0, 0, 2*n! -`if`(n::odd, (n-1)!, 0)):
seq(a(n), n=0..22);
# third Maple program:
a:= proc(n) option remember; `if`(n<2, n, (n-1)*
      (4*a(n-1)+(n-2)*(4*n-3)*a(n-2))/(4*n-7))
    end:
seq(a(n), n=0..22);

a[n_] :=  If[n == 0, 0, 2 n! - If[OddQ[n], (n-1)!, 0]];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Aug 24 2021, from 2nd Maple program *)

E.g.f.: 2*x/(1-x) - (log(1+x) - log(1-x))/2.
a(0) = 0, a(n) = 2*n! - (n mod 2)*(n-1)! for n > 0.
a(n) = (n-1)*(4*a(n-1)+(n-2)*(4*n-3)*a(n-2))/(4*n-7) for n >= 2, a(n) = n for n < 2.
a(n) = Sum_{k=1..n} k * A335872(n,k).

A158785 Expansion of e.g.f.: exp(tx)/(1 - x^2/t - t^3x^3).

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 1, 0, 6, 0, 0, 7, 24, 0, 0, 12, 0, 0, 25, 0, 120, 0, 0, 260, 0, 0, 61, 720, 0, 0, 360, 0, 0, 1470, 0, 0, 841, 0, 5040, 0, 0, 15960, 0, 0, 5082, 0, 0, 5251, 40320, 0, 0, 20160, 0, 0, 122640, 0, 0, 134456, 0, 0, 20497, 0, 362880, 0, 0, 1512000

Offset: 0

Roger L. Bagula, Mar 26 2009

			Irregular triangle begins as:
      1;
      0,    1;
      2,    0, 0,     1;
      0,    6, 0,     0,     7;
     24,    0, 0,    12,     0, 0,     25;
      0,  120, 0,     0,   260, 0,      0,   61;
    720,    0, 0,   360,     0, 0,   1470,    0, 0,    841;
      0, 5040, 0,     0, 15960, 0,      0, 5082, 0,      0, 5251;
  40320,    0, 0, 20160,     0, 0, 122640,    0, 0, 134456,    0, 0, 20497;

G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened

Cf. A005212, A005359, A158706, A158757, A330044.

Table[CoefficientList[Expand[t^Floor[n/2]*n!*SeriesCoefficient[Series[Exp[t*x]/(1 - x^2/t - t^3*x^3), {x, 0, 20}], n]], t], {n, 0, 10}]//Flatten

f(x, t) = exp(t*x)/(1 - x^2/t - t^3*x^3)
def A158785(n, k): return ( factorial(n)*t^(n//2)*( f(x, t) ).series(x, 20).list()[n] ).series(t, 2*n+1).list()[k]
flatten([[A158785(n, k) for k in (0..n+(n//2))] for n in (0..10)]) # G. C. Greubel, Dec 05 2021

T(n, k) = coefficients of e.g.f.: t^floor(n/2)*exp(t*x)/(1 - x^2/t - t^3*x^3).
From G. C. Greubel, Dec 05 2021: (Start)
T(n, floor(n/2) + n) = A330044(n).
T(n, 0) = A005359(n).
T(n, 1) = A005212(n). (End)

Edited by G. C. Greubel, Dec 05 2021

A177146 n-th derivative of arctan(x) at x = 1, n >= 4.

Original entry on oeis.org

0, -3, 15, -45, 0, 1260, -11340, 56700, 0, -3742200, 48648600, -340540200, 0, 40864824000, -694702008000, 6252318072000, 0, -1187940433680000, 24946749107280000, -274414240180080000, 0, 75738330289702080000, -1893458257242552000000, 24614957344153176000000, 0

Offset: 4

Michel Lagneau, May 03 2010

sign

d^ny/dx^n = (((-1)^(n-1))*(n-1)!)*sin(n*arctan(1/x)) /(1+x^2)^(n/2) - (proof by recurrence). If n = 1, 2, 3, the values of the derivatives at x=1 are respectively 1/2, -1/2, 1/2.

d^ny/dx^n = n!*sum(k=1..n, (binomial(n-1,k-1)*(-1)^(n-k)*x^(n-k)*(1+x^2)^(-n)*(-1)^((k-1)/2)*(1+(-1)^(k-1)))/(2*k)). - Vladimir Kruchinin, Apr 22 2011

			a(5) = -3 because d^5y/dx^5 = 384*x^4/(1 + x^2)^5 - 288*x^2/(1 + x^2)^4 + 24/(1 + x^2)^3, and for x=1 we obtain 384/32 - 288/16 + 24/8 = -3.

Alois P. Heinz, Table of n, a(n) for n = 4..400

Cf. A005359 (n-th derivatives of arctan(x) at x = 0).

# First program, with the formula:
n0:= 50: T:=array(1..n0+1):for n from 1 to n0 do:T[n]:=(((-1)^(n-1))*(n-1) !)*sin(n*arctan(1)) /(2^(n/2)):od:print(T):
# Second program, with the Maple instruction D(f):
n0:= 50: T:=array(1..n0+1):f:=x->arctan(x):for n from 1 to n0 do:D(f): T[n]:=(D(f)(1)):f:=D(f):od: print(T):
# third Maple program:
a:= n-> n!*coeff(series(arctan(x+1), x, n+1), x, n):
seq(a(n), n=4..40);  # Alois P. Heinz, Feb 14 2015

Table[Abs[Integrate[Sin[x]*E^(-x)*(x^(n - 1)), {x, 0, Infinity}]], {n, 4, 28}] (* John M. Campbell, Jun 21 2011 *)

a(n):=2^(-n-1)*n!*sum((((-1)^(k-1)+1)*(-1)^(n-k+(k-1)/2)*binomial(n-1,k-1))/k,k,1,n); /* Vladimir Kruchinin, Apr 22 2011 */

a(n) = (((-1)^(n-1))*(n-1)!)*sin(n*arctan(1))/2^(n/2).
a(n) = 2^(-n-1)*n!*sum(k=1..n, (((-1)^(k-1)+1)*(-1)^(n-k+(k-1)/2)*binomial(n-1,k-1))/k). - Vladimir Kruchinin, Apr 22 2011
abs(a(n)) = abs(integrate(x=0..infty, sin(x)*exp(-x)*x^(n-1))) (see Mathematica code below). - John M. Campbell, Jun 21 2011
E.g.f.: arctan(x+1). - Alois P. Heinz, Feb 14 2015

A302611 Expansion of e.g.f. -log(1 - x)*arctanh(x).

Original entry on oeis.org

0, 0, 2, 3, 16, 50, 368, 1764, 16896, 109584, 1297152, 10628640, 149944320, 1486442880, 24349317120, 283465647360, 5287713177600, 70734282393600, 1480103564083200, 22376988058521600, 519000166327910400, 8752948036761600000, 222845873874075648000, 4148476779335454720000

Offset: 0

Ilya Gutkovskiy, Apr 10 2018

nonn

			-log(1 - x)*arctanh(x) = 2*x^2/2! + 3*x^3/3! + 16*x^4/4! + 50*x^5/5! + 368*x^6/6! + 1764*x^7/7! + 16896*x^8/8! + ...

Cf. A005359, A009410, A009416, A009429, A009435, A012697, A081358, A104150, A177699, A177700, A202139, A302610.

a:=series(-log(1-x)*arctanh(x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019

nmax = 23; CoefficientList[Series[-Log[1 - x] ArcTanh[x], {x, 0, nmax}], x] Range[0, nmax]!

x='x+O('x^99); concat([0, 0], Vec(serlaplace(log(1-x)*log((1-x)/(1+x))/2))) \\ Altug Alkan, Apr 10 2018

E.g.f.: log(1 - x)*log((1 - x)/(1 + x))/2.

A349645 Triangular array read by rows: T(n,k) is the number of square n-permutations possessing exactly k cycles; n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 11, 0, 1, 0, 24, 0, 35, 0, 1, 0, 0, 184, 0, 85, 0, 1, 0, 720, 0, 994, 0, 175, 0, 1, 0, 0, 9708, 0, 4249, 0, 322, 0, 1, 0, 40320, 0, 72764, 0, 14889, 0, 546, 0, 1, 0, 0, 648576, 0, 402380, 0, 44373, 0, 870, 0, 1

Offset: 0

Steven Finch, Nov 23 2021

A permutation p in S_n is a square if there exists q in S_n with q^2=p.

For such a p, the number of cycles of any even length in its disjoint cycle decomposition must be even.

			The three square 3-permutations are (1, 2, 3) with three cycles (fixed points) and (3, 1, 2) & (2, 3, 1), each with one cycle.
Among the twelve square 4-permutations are {1, 4, 2, 3} & {1, 3, 4, 2} and {3, 4, 1, 2} & {4, 3, 2, 1}, all with two cycles but differing types.
Triangle begins:
[0]   1;
[1]   0,   1;
[2]   0,   0,    1;
[3]   0,   2,    0,   1;
[4]   0,   0,   11,   0,    1;
[5]   0,  24,    0,  35,    0,   1;
[6]   0,   0,  184,   0,   85,   0,   1;
[7]   0, 720,    0, 994,    0, 175,   0,   1;
[8]   0,   0, 9708,   0, 4249,   0, 322,   0,   1;
...

Alois P. Heinz, Rows n = 0..200, flattened
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Columns k=0-1 give: A000007, A005359(n-1).
Row sums give A003483.
T(n+2,n) gives A000914.
Cf. A214851, A246945.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i, 2)=0 and irem(j, 2)=1, 0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1))*x^j, j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Nov 23 2021

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0,
     Sum[If[Mod[i, 2] == 0 && Mod[j, 2] == 1, 0, (i-1)!^j*multinomial[n,
     Join[{n-i*j}, Table[i, {j}]]]/j!*b[n-i*j, i-1]]*x^j, {j, 0, n/i}]]]];
T[n_] := With[{p = b[n, n]}, Table[Coefficient[p, x, i], {i, 0, n}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

A330511 Expansion of e.g.f. Sum_{k>=1} arctan(x^k).

Original entry on oeis.org

1, 2, 4, 24, 144, 480, 4320, 40320, 282240, 4354560, 36288000, 319334400, 6706022400, 74724249600, 1046139494400, 20922789888000, 376610217984000, 4979623993344000, 115242726703104000, 2919482409811968000, 29194824098119680000

Offset: 1

Ilya Gutkovskiy, Dec 16 2019

nonn

Cf. A005359, A050469, A176475, A330504, A330505.

nmax = 21; CoefficientList[Series[Sum[ArcTan[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[(n - 1)! DivisorSum[n, (-1)^((n/# - 1)/2) # &, OddQ[n/#] &], {n, 1, 21}]

a(n) = (n-1)!*sumdiv(n, d, if (n/d % 2, (-1)^((n/d - 1)/2)*d)); \\ Michel Marcus, Dec 17 2019

E.g.f.: Sum_{i>=1} Sum_{j>=1} (-1)^(j + 1) * x^(i*(2*j - 1)) / (2*j - 1).

a(n) = (n - 1)! * Sum_{d|n, n/d odd} (-1)^((n/d - 1)/2) * d.

A331403 E.g.f.: 1 / ((1 + x) * sqrt(1 - 2*x)).

Original entry on oeis.org

1, 0, 3, 6, 81, 540, 7155, 85050, 1346625, 22339800, 431331075, 9004668750, 208178118225, 5199538043700, 140664514065075, 4080315642653250, 126613733680058625, 4180226398201854000, 146399020309066399875, 5419213146765629961750, 211446723837565171580625

Offset: 0

Ilya Gutkovskiy, Jan 16 2020

nonn

Cf. A001147, A005359, A034430, A052585, A317618.

nmax = 20; CoefficientList[Series[1/((1 + x) Sqrt[1 - 2 x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^(n - k) (2 k - 1)!!/k!, {k, 0, n}], {n, 0, 20}]

a(n) = {n! * sum(k=0, n, (-1)^(n - k) * (2*k)! / (2^k*k!^2))} \\ Andrew Howroyd, Jan 16 2020

seq(n) = {Vec(serlaplace(1 / ((1 + x) * sqrt(1 - 2*x + O(x*x^n)))))} \\ Andrew Howroyd, Jan 16 2020

a(n) = n! * Sum_{k=0..n} (-1)^(n - k) * (2*k - 1)!! / k!.
D-finite with recurrence: a(n) +(-n+1)*a(n-1) -(2*n-1)*(n-1)*a(n-2)=0. - R. J. Mathar, Jan 25 2020
a(n) ~ 2^(n + 3/2) * n^n / (3*exp(n)). - Vaclav Kotesovec, Jan 26 2020

cp's OEIS Frontend

A158757 Expansion of e.g.f. exp(t*x)/(1 - x^2/t^2 - t^3* x^3).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A167568 A triangle related to the GF(z) formulas of the rows of the ED2 array A167560.

Views

Author

Keywords

Comments

Examples

Crossrefs

A335873 Total number of points in all permutations of [n] that are fixed or reflected.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A158785 Expansion of e.g.f.: exp(t*x)/(1 - x^2/t - t^3*x^3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A177146 n-th derivative of arctan(x) at x = 1, n >= 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

A302611 Expansion of e.g.f. -log(1 - x)*arctanh(x).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A349645 Triangular array read by rows: T(n,k) is the number of square n-permutations possessing exactly k cycles; n >= 0, 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A158757 Expansion of e.g.f. exp(tx)/(1 - x^2/t^2 - t^3 x^3).

A158785 Expansion of e.g.f.: exp(tx)/(1 - x^2/t - t^3x^3).