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A183270 T(n,k) is the number of singly defective permutations of 1..n+2*k-2 with exactly k maxima.

%I A183270 #13 May 12 2020 19:57:22
%S A183270 0,3,2,120,80,15,4760,3552,860,64,249984,199168,57064,6576,220,
%T A183270 17512704,14548480,4643712,681984,42112,672,1599330304,1367568384,
%U A183270 469942528,80506880,6849792,242688,1904,185616337920,162107703296,58754129408
%N A183270 T(n,k) is the number of singly defective permutations of 1..n+2*k-2 with exactly k maxima.
%C A183270 A singly defective permutation omits one value and repeats another value.
%C A183270 T(1,1) is zero because there are no defective permutations of a single element.
%C A183270 T(n,k) is divisible by n + 2*k - 2. - _Andrew Howroyd_, May 12 2020
%H A183270 Andrew Howroyd, <a href="/A183270/b183270.txt">Table of n, a(n) for n = 1..1275</a> (terms 1..36 from R. H. Hardin)
%F A183270 A001804(n) = Sum_{k=1..2*n+1} T(n+2-2*k, k). - _Andrew Howroyd_, May 12 2020
%e A183270 Table starts:
%e A183270      0      3     120     4760    249984   17512704 1599330304 ...
%e A183270      2     80    3552   199168  14548480 1367568384 ...
%e A183270     15    860   57064  4643712 469942528 ...
%e A183270     64   6576  681984 80506880 ...
%e A183270    220  42112 6849792 ...
%e A183270    672 242688 ...
%e A183270   1904 ...
%e A183270   ...
%e A183270 Some solutions for n=4 with 2 maxima:
%e A183270 (6,1,4,4,3,2) (4,3,1,5,6,6) (4,2,1,2,3,5) (3,2,1,6,4,3) (5,5,6,1,2,3).
%o A183270 (PARI) \\ PeaksBySig defined in A334774.
%o A183270 T(n,k) = {my(m=n+2*k-3); (m+1)*sum(i=1, m, PeaksBySig(vector(m,j,if(i==j,2,1)), [k-1])[1])} \\ _Andrew Howroyd_, May 12 2020
%Y A183270 Columns 1..6 are A183264, A183265, A183266, A183267, A183268, A183269.
%Y A183270 Cf. A001804, A008303, A334774.
%K A183270 nonn,tabl
%O A183270 1,2
%A A183270 _R. H. Hardin_, Jan 03 2011