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A163269
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T(n,k) = largest coefficient in the expansion of (1 + ... + x^(n-1))^(2*k).
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2
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1, 1, 2, 1, 6, 3, 1, 20, 19, 4, 1, 70, 141, 44, 5, 1, 252, 1107, 580, 85, 6, 1, 924, 8953, 8092, 1751, 146, 7, 1, 3432, 73789, 116304, 38165, 4332, 231, 8, 1, 12870, 616227, 1703636, 856945, 135954, 9331, 344, 9, 1, 48620, 5196627, 25288120, 19611175, 4395456
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OFFSET
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1,3
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COMMENTS
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T(n,k) = number of ways the sums of all components of two 1..n k-vectors can be equal.
T(n,k) is an odd polynomial in n of order 2*k-1.
Examples:
T(n,1) = n.
T(n,2) = (2/3)*n^3 + (1/3)*n.
T(n,3) = (11/20)*n^5 + (1/4)*n^3 + (1/5)*n.
T(n,4) = (151/315)*n^7 + (2/9)*n^5 + (7/45)*n^3 + (1/7)*n.
Table starts:
1 1 1 1 1 ...
2 6 20 70 252 ...
3 19 141 1107 8953 ...
4 44 580 8092 116304 ...
5 85 1751 38165 856945 ...
...
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LINKS
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FORMULA
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EXAMPLE
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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.
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PROG
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(PARI) T(n, k) = polcoef(sum(i=0, n-1, x^i)^(2*k), k*(n-1)); \\ Michel Marcus, Jan 23 2024
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CROSSREFS
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Removing the leftmost column of A349933 generates this sequence.
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KEYWORD
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AUTHOR
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STATUS
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approved
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