login
This site is supported by donations to The OEIS Foundation.

 

Logo

The October issue of the Notices of the Amer. Math. Soc. has an article about the OEIS.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A187244 Triangle read by rows: T(n,k) is the number of permutations of [n] having k cycles with 2 alternating runs (it is assumed that the smallest element of the cycle is in the first position), 0<=k<=floor(n/3). 5
1, 1, 2, 5, 1, 17, 7, 78, 42, 463, 247, 10, 3315, 1550, 175, 27164, 11049, 2107, 247975, 92596, 22029, 280, 2492539, 906427, 220734, 9100, 27422698, 10044963, 2264724, 184415, 328607417, 122314296, 25036462, 3028025, 15400, 4266367567, 1607778568, 307273681, 44800184, 800800 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of entries in row n is 1+floor(n/3).

Sum of entries in row n is n!.

T(n,0)=A187245(n).

Sum(k*T(n,k), k>=0) = A187246(n).

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

FORMULA

E.g.f.: G(t,z) = exp[(1/4)(t-1)(2z-4exp(z)+exp(2z)+3)]/(1-z).

The 4-variate g.f. H(u,v,w,z) (exponential with respect z), where u marks number of cycles with 1 alternating run, v marks number of cycles with 2 alternating runs, w marks the number of all cycles, and z marks the size of the permutation, is given by

H(u,v,w,z)=exp[(1/4)w((v-1)(exp(2z)+2z)+4(u-v)exp(z)+1-4u+3v)]/(1-z)^w.

We have G(t,z)=H(1,t,1,z).

EXAMPLE

T(4,1)=7 because we have (132)(4), (142)(3), (1)(243), (143)(2), (1432), (1243), and (1342).

Triangle starts:

1;

1;

2;

5,1;

17,7;

78,42;

463, 247, 10;

MAPLE

G := exp((1/4)*(t-1)*(2*z-4*exp(z)+exp(2*z)+3))/(1-z): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 13 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n from 0 to 13 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form

# second Maple program:

b:= proc(n) option remember; expand(

      `if`(n=0, 1, add(b(n-j)*binomial(n-1, j-1)*

      `if`(j=1, 1, (j-1)!+(2^(j-2)-1)*(x-1)), j=1..n)))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):

seq(T(n), n=0..14);  # Alois P. Heinz, Apr 15 2017

MATHEMATICA

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - j]*Binomial[n - 1, j - 1]* If[j == 1, 1, (j - 1)! + (2^(j - 2) - 1)*(x - 1)], {j, 1, n}]]];

T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n]];

Table[T[n], {n, 0, 14}] // Flatten (* Jean-Fran├žois Alcover, May 03 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A187245, A187246.

Sequence in context: A214733 A106852 A162975 * A120294 A186766 A047921

Adjacent sequences:  A187241 A187242 A187243 * A187245 A187246 A187247

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Mar 07 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 22 17:05 EDT 2018. Contains 315270 sequences. (Running on oeis4.)