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%I #11 Jan 11 2021 13:28:54
%S 3,12,57,30,360,705,60,1400,7968,7617,105,4170,51750,163584,78357,168,
%T 10437,241080,1830000,3293184,791589,252,23072,894201,13562040,
%U 64168750,65968128,7944321,360,46440,2804480,75278553,759940800,2246625000,1319854080,79541625
%N Array read by antidiagonals: T(n,k) is the number of permutations of k indistinguishable copies of 1..n with exactly 2 local maxima.
%H Andrew Howroyd, <a href="/A334773/b334773.txt">Table of n, a(n) for n = 2..1276</a>
%F T(n,k) = Sum_{j=0..n-2} P(k-1,3) * P(k-2,2) * P(k,2)^(n-2-j) * P(k,4)^j + 2 * (n-j-2) * P(k-1,3)^2 * P(k,2)^(n-3-j) * P(k,4)^j where P(n,k) = binomial(n+k-1,k-1).
%F T(n,k) = 3*((k^2 + 4*k + 1)*binomial(k+3,3)^(n-1) - (2*k^2 + 9*k + 1)*(k+1)^(n-1) - k*(k + 5)*(n-2)*(k+1)^(n-1))/(k + 5)^2.
%e Array begins:
%e ======================================================
%e n\k | 2 3 4 5
%e ----|-------------------------------------------------
%e 2 | 3 12 30 60 ...
%e 3 | 57 360 1400 4170 ...
%e 4 | 705 7968 51750 241080 ...
%e 5 | 7617 163584 1830000 13562040 ...
%e 6 | 78357 3293184 64168750 759940800 ...
%e 7 | 791589 65968128 2246625000 42560067360 ...
%e 8 | 7944321 1319854080 78636093750 2383387566720 ...
%e ...
%e The T(2,2) = 3 permutations of 1122 with 2 local maxima are 1212, 2112, 2121.
%o (PARI) T(n,k) = {3*((k^2 + 4*k + 1)*binomial(k+3,3)^(n-1) - (2*k^2 + 9*k + 1)*(k+1)^(n-1) - k*(k + 5)*(n-2)*(k+1)^(n-1))/(k + 5)^2}
%Y Columns k=2..8 are 3*A152494, 12*A152499, 10*A152504, 30*A152509, 21*A152513, 56*A152517, 36*A152518.
%Y Cf. A334772, A334774, A334778.
%K nonn,tabl
%O 2,1
%A _Andrew Howroyd_, May 10 2020
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