|
%I #37 Nov 17 2020 15:35:54
%S 1,1,69,33661,60376809,288294050521,3019098162602349,
%T 60921822444067346581,2159058013333667522020689,
%U 125339574046311949415000577841,11289082167259099068433198467575829,1510335441937894173173702826484473600301
%N 4-alternating permutations of length 4n.
%C a(n) = A181985(4,n).
%H L. Carlitz, <a href="https://doi.org/10.1002/mana.19730580104">Permutations with prescribed pattern</a>, Math. Nachr. 58 (1973), 31-53.
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SeidelTransform">An old operation on sequences: the Seidel transform</a>
%F 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
%F 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
%F 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
%p A211212 := proc(n) local E, dim, i, k; dim := 4*(n-1);
%p E := array(0..dim, 0..dim); E[0, 0] := 1;
%p for i from 1 to dim do
%p if i mod 4 = 0 then E[i, 0] := 0 ;
%p 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;
%p else E[0, i] := 0;
%p 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;
%p fi od;
%p E[0, dim] end:
%p seq(A211212(i), i = 1..12);
%p A211212_list := proc(size) local E, S;
%p E := 2*exp(x*z)/(cosh(z)+cos(z));
%p S := z -> series(E, z, 4*(size+1));
%p seq((-1)^n*(4*n)!*subs(x=0, coeff(S(z), z, 4*n)), n=0..size-1) end:
%p A211212_list(12); # _Peter Luschny_, Jun 06 2016
%t 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}]];
%t a[n_] := A181985[4, n + 1] // Last;
%t Table[a[n], {n, 0, 11}] (* _Jean-François Alcover_, Jun 29 2019 *)
%o (Sage) # uses[A from A181936]
%o A211212 = lambda n: A(4,4*n)*(-1)^n
%o print([A211212(n) for n in (0..11)]) # _Peter Luschny_, Jan 24 2017
%Y Cf. A000364, A002115, A181985.
%K nonn
%O 0,3
%A _Peter Luschny_, Apr 04 2012
|
