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A334773 Array read by antidiagonals: T(n,k) is the number of permutations of k indistinguishable copies of 1..n with exactly 2 local maxima.

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