A181936 Number of 5-alternating permutations.

1, 1, 1, 1, 1, 1, 5, 20, 55, 125, 251, 2300, 15775, 70500, 249250, 750751, 10006375, 97226875, 601638125, 2886735625, 11593285251, 202808749375, 2550175096250, 20163891580625, 122209131374375, 613498040952501, 13287626090593750, 205055676105734375

Offset: 0

Author

Peter Luschny, Apr 03 2012

Keywords

• nonn

Comments

For an integer n>0, a permutation s = s_1...s_k is a n-alternating permutation if it has the property that s_i < s_{i+1} if and only if n divides i.

References

• Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• R. J. Cano, PARI Sequencer program.
• Peter Luschny, An old operation on sequences: the Seidel transform.
• Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
• Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]

Crossrefs

Number of m-alternating permutations: A000012 (m=1), A000111 (m=2), A178963 (m=3), A178964 (m=4), this sequence (m=5), A250283 (m=6), A250284 (m=7), A250285 (m=8), A250286 (m=9), A250287 (m=10).

Row n=5 of A181937.

Programs

Maple

A181936_list := proc(dim) local E,DIM,n,k;
DIM := dim-1; E := array(0..DIM, 0..DIM); E[0,0] := 1;
for n from 1 to DIM do
if n mod 5 = 0 then E[n,0] := 0 ;
   for k from n-1 by -1 to 0 do E[k,n-k] := E[k+1,n-k-1] + E[k,n-k-1] od;
else E[0,n] := 0;
   for k from 1 by 1 to n do E[k,n-k] := E[k-1,n-k+1] + E[k-1,n-k] od;
fi od; [E[0,0],seq(E[k,0]+E[0,k],k=1..DIM)] end:
A181936_list(28);
# Alternatively, using an exponential generating function:
A181936_list := proc(n) local H,F,i; H := (r,s) -> hypergeom(r,s/5,-(t/5)^5);
F := t -> 1+(t^5*H([1],[6,7,8,9,10])+5*t^4*H([],[6,7,8,9])+20*t^3*H([],[4,6,7,8])+60*t^2*H([],[3,4,6,7])+120*t^1*H([],[2,3,4,6]))/(120*H([],[2,3,4,1])); seq(i!*coeff(series(F(t),t,n+1),t,i),i=0..n-1) end:

Mathematica

dim = 27; e[0, 0] = 1; e[n_ /; Mod[n, 5] == 0 && 0 <= n <= dim, 0] = 0; e[k_ /; 0 <= k <= dim, n_ /; 0 <= n <= dim] /; Mod[n+k, 5] == 0 := e[k, n] = e[k, n-1] + e[k+1, n-1]; e[0, n_ /; Mod[n, 5] == 0 && 0 <= n <= dim] = 0; e[k_ /; 0 <= k <= dim, n_ /; 0 <= n <= dim] /; Mod[n+k, 5] != 0 := e[k, n] = e[k-1, n] + e[k-1, n+1]; e[, ] = 0; a[0] = 1; a[n_] := e[n, 0] + e[0, n]; Table[a[n], {n, 0, dim}] (* Jean-François Alcover, Jun 27 2013, translated and adapted from Maple *)
b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, If[t == 0,
     Sum[b[u - j, o + j - 1, Mod[t + 1, 5]], {j, 1, u}],
     Sum[b[u + j - 1, o - j, Mod[t + 1, 5]], {j, 1, o}]]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 35] (* Jean-François Alcover, Apr 21 2021, after Alois P. Heinz in A250283 *)
nmax = 30; CoefficientList[Series[1 + Sum[(x^(5 - k) * HypergeometricPFQ[{1}, {6/5 - k/5, 7/5 - k/5, 8/5 - k/5, 9/5 - k/5, 2 - k/5}, -x^5/3125])/(5 - k)!, {k, 0, 4}] / HypergeometricPFQ[{}, {1/5, 2/5, 3/5, 4/5}, -x^5/3125], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 21 2021 *)

Sage

@cached_function
def A(m,n):
    if n == 0: return 1
    s = -1 if m.divides(n) else 1
    t = [m*k for k in (0..(n-1)//m)]
    return s*add(binomial(n,k)*A(m,k) for k in t)
A181936 = lambda n: (-1)^int(is_odd(n//5))*A(5,n)
print([A181936(n) for n in (0..30)]) # Peter Luschny, Jan 24 2017