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%I #17 May 09 2018 09:15:59
%S 1,1,2,3,1,4,6,5,18,3,6,44,26,7,91,123,11,8,172,448,136,9,300,1348,
%T 912,51,10,496,3600,4552,838,11,781,8704,18476,7441,283,12,1186,19584,
%U 65324,48116,5930,13,1742,41379,206556,250375,66606,1833,14,2492,83210,600456,1120042,536908,47358
%N Number T(n,k) of involutions of [n] having exactly k peaks; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/2)), read by rows.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Involution_(mathematics)">Involution</a>
%F T(2n+1,n) = A072187(n+1).
%e T(5,0) = 5: 12345, 21345, 32145, 43215, 54321.
%e T(5,1) = 18: 12354, 12435, 12543, 13245, 14325, 14523, 15432, 21354, 21435, 21543, 32154, 34125, 42315, 42513, 45312, 52341, 52431, 53241.
%e T(5,2) = 3: 13254, 15342, 35142.
%e Triangle T(n,k) begins:
%e 1;
%e 1;
%e 2;
%e 3, 1;
%e 4, 6;
%e 5, 18, 3;
%e 6, 44, 26;
%e 7, 91, 123, 11;
%e 8, 172, 448, 136;
%e 9, 300, 1348, 912, 51;
%e 10, 496, 3600, 4552, 838;
%e 11, 781, 8704, 18476, 7441, 283;
%e 12, 1186, 19584, 65324, 48116, 5930;
%e 13, 1742, 41379, 206556, 250375, 66606, 1833;
%Y Row sums give A000085.
%Y Columns k=0-1 give: A028310, A303649.
%Y Cf. A008303 (the same for permutations), A072187, A303564 (the same for derangements).
%K nonn,tabf
%O 0,3
%A _Alois P. Heinz_, Apr 27 2018
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