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A362582 Triangular array read by rows. T(n,k) is the number of alternating permutations of [2n+1] having exactly 2k elements to the left of 1, n >= 0, 0 <= k <= n.

%I A362582 #22 Apr 27 2023 11:47:35
%S A362582 1,1,1,5,6,5,61,75,75,61,1385,1708,1750,1708,1385,50521,62325,64050,
%T A362582 64050,62325,50521,2702765,3334386,3427875,3438204,3427875,3334386,
%U A362582 2702765,199360981,245951615,252857605,253708455,253708455,252857605,245951615,199360981
%N A362582 Triangular array read by rows.  T(n,k) is the number of alternating permutations of [2n+1] having exactly 2k elements to the left of 1, n >= 0, 0 <= k <= n.
%C A362582 Here, w = w_1,w_2,...,w_(2n+1) is an alternating permutation if w_1 < w_2 > w_3 < ... < w_(2n) > w_(2n+1).
%H A362582 Alois P. Heinz, <a href="/A362582/b362582.txt">Rows n = 0..140, flattened</a>
%F A362582 Sum_{n>=0} Sum_{k=0..n} T(n,k)*u^k*z^n/A000680(n) = 1/(E(-u*z)*E(-z)) where E(z) = Sum_{n>=0} z^n/A000680(n).
%F A362582 T(n,k) = binomial(2*n,2*k)*A000111(2*k)*A000111(2*(n-k)). - _Alois P. Heinz_, Apr 25 2023
%e A362582 T(2,1) = 6 because we have: {2, 3, 1, 5, 4}, {2, 4, 1, 5, 3}, {2, 5, 1, 4, 3}, {3, 4, 1, 5, 2}, {3, 5, 1, 4, 2}, {4, 5, 1, 3, 2}.
%e A362582 Triangle begins
%e A362582      1;
%e A362582      1,     1;
%e A362582      5,     6,     5;
%e A362582     61,    75,    75,    61;
%e A362582   1385,  1708,  1750,  1708,  1385;
%e A362582  50521, 62325, 64050, 64050, 62325, 50521;
%e A362582  ...
%p A362582 b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p A362582       add(b(o-1+j, u-j), j=1..u))
%p A362582     end:
%p A362582 T:= (n, k)-> binomial(2*n, 2*k)*b(2*k, 0)*b(2*(n-k), 0):
%p A362582 seq(seq(T(n, k), k=0..n), n=0..8);  # _Alois P. Heinz_, Apr 25 2023
%t A362582 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[1/e[-u z]*1/e[-z], {z, 0, nn}], {z, u}]] // Grid
%Y A362582 Cf. A000182 (row sums), A000364 (column k=0), A000680.
%Y A362582 Cf. A000111, A104345.
%K A362582 nonn,tabl
%O A362582 0,4
%A A362582 _Geoffrey Critzer_, Apr 25 2023