OFFSET
0,5
COMMENTS
Conjecture: row sums are the Euler numbers A000364. Second column is A024255. Third column is A117415.
Here, w = w_1,w_2,...,w_(2n) is an alternating permutation if w_1 < w_2 > w_3 < ... > w_(2n-1) < w_2n. An alternating permutation is cyclically alternating if w_1 < w_(2n). Define the cyclically alternating decomposition of w in the following manner: From the set {w_2,w_4,w_6,...,w_(2n)} find the largest i such that w_(2i) > w_1. Then w_1,w_2,...,w_(2i) is the first component in the cyclically alternating decomposition of w. Repeat the process with the set {w_(2i+1),w_(2i+2),...,w_(2n)} to find the successive components. Conjecture: T(n,k) is the number of alternating permutations of [2n] with exactly k cyclically alternating components. - Geoffrey Critzer, Apr 26 2023
LINKS
N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001-2003, page 9.
FORMULA
From Geoffrey Critzer, Apr 26 2023: (Start)
Sum_{n>=0} Sum_{k=0..n} T(n,k)*u^k*z^n/A000680(n) = E((u-1)*z)/E(-z) Where E(z) = Sum_{n>=0} z^n/A000680(n).
Sum_{k=0..n} T(n,k)*k = A086646(n,1). (End)
EXAMPLE
Triangle begins:
1;
0, 1;
0, 4, 1;
0, 48, 12, 1;
0, 1088, 272, 24, 1;
0, 39680, 9920, 880, 40, 1;
0, 2122752, 530688, 47104, 2160, 60, 1;
...
MATHEMATICA
nn = 6; B[n_] := (2 n)!/2^n; e[z_] := Sum[z^n/B[n], {n, 0, nn}];
Map[Select[#, # > 0 &] &, Table[B[n], {n, 0, nn}] CoefficientList[
Series[e[(u - 1) z] 1/e[-z], {z, 0, nn}], {z, u}]] // Grid (* Geoffrey Critzer, Apr 26 2023 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul Barry, Mar 13 2006
STATUS
approved