A001470 -id:A001470 - cp's OEIS frontend

A053500 Number of degree-n permutations of order dividing 10.

1, 1, 2, 4, 10, 50, 220, 1240, 6140, 32860, 602200, 5668400, 62030200, 522328600, 4487190800, 62591332000, 715163146000, 9573774122000, 105731659828000, 1187355279592000, 29205778751300000, 481597207656340000, 9086318388933400000, 132525988426667120000

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A001472, A053495-A053505, A005388.

Column k=10 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 + x^5/5 + x^10/10) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 2, 5, 10])))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 14 2013

a[n_]:= a[n] = If[n<0, 0, If[n==0, 1, Sum[Product[n-i, {i, 1, j-1}] *a[n-j], {j, {1, 2, 5, 10}}]]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 24 2014, after Alois P. Heinz *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^5/5 +x^10/10], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 15 2019 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x + x^2/2 + x^5/5 + x^10/10) )) \\ G. C. Greubel, May 15 2019

m = 30; T = taylor(exp(x + x^2/2 + x^5/5 + x^10/10), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

E.g.f.: exp(x + x^2/2 + x^5/5 + x^10/10).

A053501 Number of degree-n permutations of order dividing 11.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3628801, 43545601, 283046401, 1320883201, 4953312001, 15850598401, 44910028801, 115482931201, 274271961601, 609493248001, 1279935820801, 4644633666390681601, 106826520356358566401, 1281918194457262387201

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A001472, A053495-A053505, A005388.

Column k=11 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^11/11) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 11])))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 14 2013

a[n_]:= n!*Sum[If[Mod[11*k-n, 10] == 0, Binomial[k, (11*k-n)/10]*11^((k-n)/10)/k!, 0], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 20 2014, after Vladimir Kruchinin *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^11/11], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 15 2019 *)

a(n):=n!*sum(if mod(11*k-n,10)=0 then binomial(k,(11*k-n)/10)*(11)^((k-n)/10)/k! else 0,k,1,n); /* Vladimir Kruchinin, Sep 10 2010 */

my(x='x+O('x^30)); Vec(serlaplace( exp(x +x^11/11) )) \\ G. C. Greubel, May 15 2019

m = 30; T = taylor(exp(x +x^11/11), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

E.g.f.: exp(x + x^11/11).

a(n) = n!*Sum_{k=1..n} (if mod(11*k-n,10)=0 then C(k,(11*k-n)/10)*(11)^((k-n)/10)/k!, else 0), n>0. - Vladimir Kruchinin, Sep 10 2010

A053503 Number of degree-n permutations of order dividing 16.

Original entry on oeis.org

1, 1, 2, 4, 16, 56, 256, 1072, 11264, 78976, 672256, 4653056, 49810432, 433429504, 4448608256, 39221579776, 1914926104576, 29475151020032, 501759779405824, 6238907914387456, 120652091860975616, 1751735807564578816, 29062253310781161472, 398033706586943258624

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

Differs from A005388 first at n=32. - Alois P. Heinz, Feb 14 2013

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A001472, A053495-A053505, A005388.

Column k=16 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 + x^4/4 + x^8/8 + x^16/16) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..2^j-1)*a(n-2^j), j=0..4)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 14 2013

a[n_]:= a[n] =If[n<0, 0, If[n==0, 1, Sum[Product[n-i, {i, 1, 2^j-1}]* a[n-2^j], {j, 0, 4}]]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 19 2014, after Alois P. Heinz *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^4/4 +x^8/8 + x^16/16], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 15 2019 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x + x^2/2 + x^4/4 + x^8/8 + x^16/16) )) \\ G. C. Greubel, May 15 2019

m = 30; T = taylor(exp(x + x^2/2 + x^4/4 + x^8/8 + x^16/16), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019

E.g.f.: exp(x + x^2/2 + x^4/4 + x^8/8 + x^16/16).

A055814 Expansion of e.g.f.: exp(x^3/3 + x^2/2).

Original entry on oeis.org

1, 0, 1, 2, 3, 20, 55, 210, 1225, 4760, 26145, 157850, 811195, 5345340, 35170135, 222472250, 1650073425, 12000388400, 88563700225, 720929459250, 5786843137075, 48072795270500, 424314078763575, 3731123025279650, 34084058218435225, 323768324084205000

Offset: 0

Karol A. Penson, Mar 05 2003

nonn

a(n) is the number of n-permutations in which all cycles have length two or three. - Geoffrey Critzer, Feb 21 2010

			a(4) = 3 because there are 3 permutations of {1,2,3,4} that have cycle length two or three: (1,2)(3,4);(1,3)(2,4);(1,4)(2,3). - _Geoffrey Critzer_, Feb 21 2010

Miklos Bona, A Walk Through Combinatorics, World Scientific Publishing Co., 2002, page 169. - Geoffrey Critzer, Feb 21 2010

Alois P. Heinz, Table of n, a(n) for n = 0..625

Cf. A081096.

Cf. A000085, A001470. - Joerg Arndt, Oct 02 2009

a:=[1,0,1];; for n in [4..30] do a[n]:=(n-2)*(a[n-2]+(n-3)*a[n-3]); od; a; # G. C. Greubel, Jan 23 2020

I:=[1,0,1]; [n le 3 select I[n] else (n-2)*(Self(n-2) +(n-3)*Self(n-3)): n in [1..30]]; // G. C. Greubel, Jan 23 2020

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *binomial(n-1, j-1)*(j-1)!, j=2..min(3, n)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 25 2018

With[{m=30}, CoefficientList[Series[Exp[x^2/2 + x^3/3], {x,0,m}], x]*Range[0, m]!] (* Geoffrey Critzer, Feb 21 2010 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x^3/3 + x^2/2) )) \\ G. C. Greubel, Jan 23 2020

[factorial(n)*( exp(x^3/3 + x^2/2) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jan 23 2020

a(n) = subs(x=0, (d^n/dx^n)exp(x^3/3 + x^2/2)), n=0, 1, 2, ...
a(n) = (n-1)*a(n-2) + (n-1)*(n-2)*a(n-3). - Joerg Arndt, Oct 02 2009
a(n) ~ n^(2*n/3)*exp(1/18 - 2*n/3 - n^(1/3)/6 + n^(2/3)/2)/sqrt(3) * (1 + 49/(324*n^(1/3)) - 72451/(1049760*n^(2/3))). - Vaclav Kotesovec, Jun 26 2013

Improved definition, as proposed by Joerg Arndt, from R. J. Mathar, Oct 23 2009

a(0)=1 prepended by Alois P. Heinz, Jan 25 2018

A061134 Number of degree-n even permutations of order exactly 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 226800, 2494800, 29937600, 259459200, 1816214400, 10897286400, 301491590400, 4419628012800, 51209462304000, 482551041772800, 6979977625420800, 92611036249804800, 2078225819199129600

Offset: 1

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135.

E.g.f.: - 1/2*exp(x + 1/2*x^2 + 1/4*x^4) - 1/2*exp(x - 1/2*x^2 - 1/4*x^4) + 1/2*exp(x + 1/2*x^2 + 1/4*x^4 + 1/8*x^8) + 1/2*exp(x - 1/2*x^2 - 1/4*x^4 - 1/8*x^8).

A061137 Number of degree-n odd permutations of order dividing 6.

Original entry on oeis.org

0, 0, 1, 3, 6, 30, 270, 1386, 6048, 46656, 387180, 2469060, 17204616, 158065128, 1903506696, 18887563800, 163657221120, 2095170230016, 30792968596368, 346564643468976, 3905503235814240, 58609511127871200, 866032039742528736

Offset: 0

Vladeta Jovovic, Apr 14 2001

Robert Israel, Table of n, a(n) for n = 0..522
Lev Glebsky, Melany Licón, Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.
Robert Israel, Recurrence for this sequence

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128, A000704, A061129-A061132, A048099, A051695, A061133-A061135, A001465, A061136-A061140.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^3/3)*Sinh(x^2/2 + x^6/6) )); [0,0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Jul 02 2019

Egf:= exp(x + x^3/3)*sinh(x^2/2 + x^6/6):
S:= series(Egf,x,31):
seq(coeff(S,x,j)*j!,j=0..30); # Robert Israel, Jul 13 2018

With[{m=30}, CoefficientList[Series[Exp[x + x^3/3]*Sinh[x^2/2 + x^6/6], {x, 0, m}], x]*Range[0,m]!] (* Vincenzo Librandi, Jul 02 2019 *)

my(x='x+O('x^30)); concat([0,0], Vec(serlaplace( exp(x + x^3/3)*sinh(x^2/2 + x^6/6) ))) \\ G. C. Greubel, Jul 02 2019

m = 30; T = taylor(exp(x + x^3/3)*sinh(x^2/2 + x^6/6), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jul 02 2019

E.g.f.: exp(x + x^3/3)*sinh(x^2/2 + x^6/6).

Linear recurrence of order 12 whose coefficients are polynomials in n of degree up to 15: see link. - Robert Israel, Jul 13 2018

A245958 Number T(n,k) of endofunctions f on [n] satisfying f^3(i) = i for all i in [k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 27, 11, 5, 3, 256, 88, 36, 18, 9, 3125, 925, 335, 141, 57, 21, 46656, 12096, 3912, 1440, 516, 186, 81, 823543, 189679, 55377, 18279, 6003, 2079, 837, 351, 16777216, 3473408, 924160, 277824, 84624, 27672, 10116, 3690, 1233

Offset: 0

Alois P. Heinz, Aug 08 2014

			Triangle T(n,k) begins:
0 :       1;
1 :       1,      1;
2 :       4,      2,     1;
3 :      27,     11,     5,     3;
4 :     256,     88,    36,    18,    9;
5 :    3125,    925,   335,   141,   57,   21;
6 :   46656,  12096,  3912,  1440,  516,  186,  81;
7 :  823543, 189679, 55377, 18279, 6003, 2079, 837, 351;
     ...

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

Column k=0 gives A000312.
T(2n,n) gives A245959.
Main diagonal gives A001470.
Cf. A241015.

with(combinat): M:=multinomial:
T:= proc(n, k) local l, g; l, g:= [1, 3],
      proc(k, m, i, t) option remember; local d, j; d:= l[i];
        `if`(i=1, n^m, add(M(k, k-(d-t)*j, (d-t)$j)/j!*
         (d-1)!^j *M(m, m-t*j, t$j) *g(k-(d-t)*j, m-t*j,
        `if`(d-t=1, [i-1, 0], [i, t+1])[]), j=0..min(k/(d-t),
        `if`(t=0, [][], m/t))))
      end; g(k, n-k, nops(l), 0)
    end:
seq(seq(T(n, k), k=0..n), n=0..12);

M[n_, m_, k_List] := n!/Times @@ (Join[{m}, k]!);
T[0, 0] = 1; T[n_, k_] := T[n, k] = Module[{l = {1, 3}, g}, g[k0_, m_, {i_, t_}] := g[k0, m, i, t]; g[k0_, m_, i_, t_] := g[k0, m, i, t] = Module[ {d}, d = l[[i]]; If[i == 1, n^m, Sum[M[k0, k0 - (d-t)*j, Table[(d-t), {j}]]/j!*(d-1)!^j*M[m, m - t*j, Table[t, {j}]]*g[k0 - (d-t)*j, m - t*j, If[d-t == 1, {i-1, 0}, {i, t+1}]], {j, 0, Min[k0/(d-t), If[t == 0, Infinity, m/t]]}]]]; g[k, n-k, Length[l], 0]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 03 2019, after Alois P. Heinz *)

A293588 E.g.f.: exp(x + x^6/6).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 121, 841, 3361, 10081, 25201, 55441, 6763681, 86692321, 605765161, 3027624601, 12109056961, 41169011521, 5063607974881, 94197184734241, 939457659787201, 6572292677455681, 36141156689382361, 166238526616664041, 20612479896229156321

Offset: 0

Eric M. Schmidt, Oct 12 2017

nonn

These are the telephone numbers T^(6)_n of [Artioli et al., p. 7].

Seiichi Manyama, Table of n, a(n) for n = 0..524
Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017.

Sequences with e.g.f. exp(x + x^m/m): A000079 (m=1), A000085 (m=2), A001470 (m=3), A118934 (m=4), A052501 (m=5), this sequence (m=6), A053497 (m=7).

Cf. A000085, A001470, A052501, A118934.

F:= Factorial;
[(&+[ F(n)/(6^k*F(k)*F(n-6*k)): k in [0..Floor(n/3)]]): n in [0..30]]; // G. C. Greubel, Mar 07 2021

With[{nn=30},CoefficientList[Series[Exp[x+x^6/6],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Dec 11 2017 *)
Table[Sum[n!/(6^k*k!*(n-6*k)!), {k, 0, n/6}], {n, 0, 30}] (* G. C. Greubel, Mar 07 2021 *)

my(x = 'x + O('x^30)); Vec(serlaplace(exp(x + x^6/6))) \\ Michel Marcus, Oct 13 2017

f=factorial;
[sum( f(n)/(6^k*f(k)*f(n-6*k)) for k in [0..n/3]) for n in [0..30]] # G. C. Greubel, Mar 07 2021

a(n) = a(n-1) + (n-1)!/(n-6)! * a(n-6).

a(n) = Sum_{j=0..floor(n/6)} n!/(6^j*j!*(n-6*j)!). - G. C. Greubel, Mar 07 2021

A362390 E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)).

Original entry on oeis.org

1, 1, 1, 3, 17, 81, 441, 3641, 33825, 318753, 3505521, 45095601, 616484001, 9013086369, 145909533225, 2556431401161, 47388760825281, 937507626246081, 19840711661183457, 443937299529447009, 10456231167451597761, 259738234024404363201

Offset: 0

Seiichi Manyama, Apr 20 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..434
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=2 of A362378.

Cf. A001470, A362478.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3/3*exp(x)))))

E.g.f.: exp(x - LambertW(-x^3/3 * exp(x))) = -3 * LambertW(-x^3/3 * exp(x))/x^3.

a(n) = n! * Sum_{k=0..floor(n/3)} (1/3)^k * (k+1)^(n-2*k-1) / (k! * (n-3*k)!).

A344912 Irregular triangle read by rows, Trow(n) = Seq_{k=0..n/3} Seq_{j=0..n-3k} (n! binomial(n - 3k, j)) / (k!(n - 3k)!3^k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 1, 4, 6, 4, 1, 8, 8, 1, 5, 10, 10, 5, 1, 20, 40, 20, 1, 6, 15, 20, 15, 6, 1, 40, 120, 120, 40, 40, 1, 7, 21, 35, 35, 21, 7, 1, 70, 280, 420, 280, 70, 280, 280, 1, 8, 28, 56, 70, 56, 28, 8, 1, 112, 560, 1120, 1120, 560, 112, 1120, 2240, 1120

Offset: 0

Peter Luschny, Jun 04 2021

Consider a sequence of Pascal tetrahedrons (depending on a parameter m >= 1), where the slices of the pyramid are scaled. They are given by the e.g.f.s exp(t^m / m) * exp(t*(x + y)), which provide a sequence of bivariate polynomials in x and y, whose monomials are to be ordered in degree-lexicographic order. For m = 1 one gets A109649 (resp. A046816), for m = 2 one gets A344911 (resp. A344678), and for m = 3 the current triangle. The row sums have an unexpected interpretation in A336614 (see the link).

			Triangle begins:
[0] 1;
[1] 1, 1;
[2] 1, 2,  1;
[3] 1, 3,  3,  1,  2;
[4] 1, 4,  6,  4,  1,  8,  8;
[5] 1, 5, 10, 10,  5,  1, 20, 40,  20;
[6] 1, 6, 15, 20, 15,  6,  1, 40, 120, 120,  40,  40;
[7] 1, 7, 21, 35, 35, 21,  7,  1,  70, 280, 420, 280, 70, 280, 280.
.
p_{6}(x, y) = x^6 + 6*x^5*y + 15*x^4*y^2 + 20*x^3*y^3 + 15*x^2*y^4 + 6*x*y^5 + y^6 + 40*x^3 + 120*x^2*y + 120*x*y^2 + 40*y^3 + 40.

mjqxxxx, Proof of conjectured formulas for A336614, Mathematics Stack Exchange.

m=1: A109649, (A046816) [row sums A000244], scaling A007318 [row sums A000079].
m=2: A344911, (A344678) [row sums A005425], scaling A100861 [row sums A000085].
m=3: this triangle      [row sums A336614], scaling A118931 [row sums A001470].

B := (n, k) -> n!/(k!*(n - 3*k)!*(3^k)): C := n -> seq(binomial(n, j), j=0..n):
T := (n, k) -> B(n, k)*C(n - 3*k): seq(seq(T(n, k), k = 0..n/3), n = 0..8);

gf := Exp[t^3 / 3] Exp[t (x + y)]; ser := Series[gf, {t, 0, 9}];
P[n_] := Expand[n! Coefficient[ser, t, n]];
DegLexList[p_] := MonomialList[p, {x, y}, "DegreeLexicographic"] /. x->1 /. y->1;
Table[DegLexList[P[n]], {n, 0, 7}] // Flatten

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A053500 Number of degree-n permutations of order dividing 10.

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A053501 Number of degree-n permutations of order dividing 11.

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A053503 Number of degree-n permutations of order dividing 16.

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A055814 Expansion of e.g.f.: exp(x^3/3 + x^2/2).

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A061134 Number of degree-n even permutations of order exactly 8.

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A061137 Number of degree-n odd permutations of order dividing 6.

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A344912 Irregular triangle read by rows, Trow(n) = Seq_{k=0..n/3} Seq_{j=0..n-3k} (n! binomial(n - 3k, j)) / (k!(n - 3k)!3^k).