OFFSET
0,3
COMMENTS
a(n) = A181985(4,n).
LINKS
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-François 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
KEYWORD
nonn
AUTHOR
Peter Luschny, Apr 04 2012
STATUS
approved