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 A181936 Number of 5-alternating permutations. 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 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 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 *) PROG (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 CROSSREFS Number of m-alternating permutations: A000012 (m=1), A000111 (m=2), A178963 (m=3), A178964 (m=4), A181936 (m=5). Cf. A181937. Sequence in context: A090133 A289306 A062988 * A226639 A264874 A270092 Adjacent sequences:  A181933 A181934 A181935 * A181937 A181938 A181939 KEYWORD nonn AUTHOR Peter Luschny, Apr 03 2012 STATUS approved

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Last modified April 18 22:08 EDT 2019. Contains 322237 sequences. (Running on oeis4.)