cp's OEIS Frontend

A117414 An Euler triangle.

%I A117414 #17 Apr 27 2023 11:23:11
%S A117414 1,0,1,0,4,1,0,48,12,1,0,1088,272,24,1,0,39680,9920,880,40,1,0,
%T A117414 2122752,530688,47104,2160,60,1,0,156577792,39144448,3474688,159488,
%U A117414 4480,84,1,0,15230058496,3807514624,337979392,15514880,436352,8288,112,1
%N A117414 An Euler triangle.
%C A117414 Conjecture: row sums are the Euler numbers A000364. Second column is A024255. Third column is A117415.
%C A117414 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
%H A117414 N. D. Elkies, <a href="https://arxiv.org/abs/math/0101168">On the sums Sum((4k+1)^(-n),k,-inf,+inf)</a>, arXiv:math/0101168 [math.CA], 2001-2003, page 9.
%F A117414 From _Geoffrey Critzer_, Apr 26 2023: (Start)
%F A117414 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).
%F A117414 Sum_{k=0..n} T(n,k)*k = A086646(n,1). (End)
%e A117414 Triangle begins:
%e A117414   1;
%e A117414   0,       1;
%e A117414   0,       4,      1;
%e A117414   0,      48,     12,     1;
%e A117414   0,    1088,    272,    24,    1;
%e A117414   0,   39680,   9920,   880,   40,  1;
%e A117414   0, 2122752, 530688, 47104, 2160, 60, 1;
%e A117414   ...
%t A117414 nn = 6; B[n_] := (2 n)!/2^n; e[z_] := Sum[z^n/B[n], {n, 0, nn}];
%t A117414 Map[Select[#, # > 0 &] &,Table[B[n], {n, 0, nn}] CoefficientList[
%t A117414 Series[e[(u - 1) z] 1/e[-z], {z, 0, nn}], {z, u}]] // Grid (* _Geoffrey Critzer_, Apr 26 2023 *)
%Y A117414 Cf. A000364, A024255, A117415, A086646, A000680.
%K A117414 nonn,tabl
%O A117414 0,5
%A A117414 _Paul Barry_, Mar 13 2006