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A211212 4-alternating permutations of length 4n. 8
1, 1, 69, 33661, 60376809, 288294050521, 3019098162602349, 60921822444067346581, 2159058013333667522020689, 125339574046311949415000577841, 11289082167259099068433198467575829, 1510335441937894173173702826484473600301 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) = A181985(4,n).

LINKS

Table of n, a(n) for n=0..11.

L. Carlitz, Permutations with prescribed pattern, Math. Nachr. 58 (1973), 31-53.

Peter Luschny, An old operation on sequences: the Seidel transform

FORMULA

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(4*n,4*k) * a(n-k). - Ilya Gutkovskiy, Jan 27 2020

E.g.f.: 1/(cos(x/sqrt(2))*cosh(x/sqrt(2))) = 1 + 1*z^4/4! + 69*z^8/8! + 33661*z^12/12! + ... - Michael Wallner, Nov 17 2020

a(n) ~ 2^(10*n + 9/2) * n^(4*n + 1/2) / (cosh(Pi/2) * Pi^(4*n + 1/2) * exp(4*n)). - Vaclav Kotesovec, Nov 17 2020

MAPLE

A211212 := proc(n) local E, dim, i, k; dim := 4*(n-1);

E := array(0..dim, 0..dim); E[0, 0] := 1;

for i from 1 to dim do

   if i mod 4 = 0 then E[i, 0] := 0 ;

      for k from i-1 by -1 to 0 do E[k, i-k] := E[k+1, i-k-1] + E[k, i-k-1] od;

   else E[0, i] := 0;

      for k from 1 by 1 to i do E[k, i-k] := E[k-1, i-k+1] + E[k-1, i-k] od;

   fi od;

E[0, dim] end:

seq(A211212(i), i = 1..12);

A211212_list := proc(size) local E, S;

E := 2*exp(x*z)/(cosh(z)+cos(z));

S := z -> series(E, z, 4*(size+1));

seq((-1)^n*(4*n)!*subs(x=0, coeff(S(z), z, 4*n)), n=0..size-1) end:

A211212_list(12); # Peter Luschny, Jun 06 2016

MATHEMATICA

A181985[n_, len_] := Module[{e, dim = n (len - 1)}, e[0, 0] = 1; For[i = 1, i <= dim, i++, If[Mod[i, n] == 0, e[i, 0] = 0; For[k = i - 1, k >= 0, k--, e[k, i - k] = e[k + 1, i - k - 1] + e[k, i - k - 1]], e[0, i] = 0; For[k = 1, k <= i, k++, e[k, i - k] = e[k - 1, i - k + 1] + e[k - 1, i - k]]]]; Table[e[0, n k], {k, 0, len - 1}]];

a[n_] := A181985[4, n + 1] // Last;

Table[a[n], {n, 0, 11}] (* Jean-Fran├žois Alcover, Jun 29 2019 *)

PROG

(Sage) # uses[A from A181936]

A211212 = lambda n: A(4, 4*n)*(-1)^n

print([A211212(n) for n in (0..11)]) # Peter Luschny, Jan 24 2017

CROSSREFS

Cf. A000364, A002115, A181985.

Sequence in context: A194612 A297768 A268847 * A175049 A116099 A116238

Adjacent sequences:  A211209 A211210 A211211 * A211213 A211214 A211215

KEYWORD

nonn

AUTHOR

Peter Luschny, Apr 04 2012

STATUS

approved

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Last modified August 11 19:26 EDT 2022. Contains 356066 sequences. (Running on oeis4.)