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%I #17 Feb 14 2024 14:48:41
%S 1,1,2,1,6,3,1,20,19,4,1,70,141,44,5,1,252,1107,580,85,6,1,924,8953,
%T 8092,1751,146,7,1,3432,73789,116304,38165,4332,231,8,1,12870,616227,
%U 1703636,856945,135954,9331,344,9,1,48620,5196627,25288120,19611175,4395456
%N T(n,k) = largest coefficient in the expansion of (1 + ... + x^(n-1))^(2*k).
%C T(n,k) = number of ways the sums of all components of two 1..n k-vectors can be equal.
%C T(n,k) is an odd polynomial in n of order 2*k-1.
%C Examples:
%C T(n,1) = n.
%C T(n,2) = (2/3)*n^3 + (1/3)*n.
%C T(n,3) = (11/20)*n^5 + (1/4)*n^3 + (1/5)*n.
%C T(n,4) = (151/315)*n^7 + (2/9)*n^5 + (7/45)*n^3 + (1/7)*n.
%C Table starts:
%C 1 1 1 1 1 ...
%C 2 6 20 70 252 ...
%C 3 19 141 1107 8953 ...
%C 4 44 580 8092 116304 ...
%C 5 85 1751 38165 856945 ...
%C ...
%H R. H. Hardin, <a href="/A163269/b163269.txt">Table of n, a(n) for n = 1..3240</a>
%F T(n,k) = A273975(2*k, n, (n-1)*k). - _Andrey Zabolotskiy_, Jan 23 2024
%e For n = 3 and k = 2, (1 + x + x^2)^(2*2) = x^8 + 4*x^7 + 10*x^6 + 16*x^5 + 19*x^4 + 16*x^3 + 10*x^2 + 4*x + 1, whose largest coefficient is T(3,2) = 19.
%o (PARI) T(n,k) = polcoef(sum(i=0, n-1, x^i)^(2*k), k*(n-1)); \\ _Michel Marcus_, Jan 23 2024
%Y Removing the leftmost column of A349933 generates this sequence.
%Y Cf. A273975.
%K nonn,tabl
%O 1,3
%A _R. H. Hardin_, Jul 24 2009
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