A163269 T(n,k) = largest coefficient in the expansion of (1 + ... + x^(n-1))^(2*k).

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

Offset: 1

Author

R. H. Hardin, Jul 24 2009

Keywords

• nonn
• tabl

Comments

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 ...

...

Examples

			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.
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..3240

Crossrefs

Removing the leftmost column of A349933 generates this sequence.

Cf. A273975.

Programs

PARI

T(n,k) = polcoef(sum(i=0, n-1, x^i)^(2*k), k*(n-1)); \\ Michel Marcus, Jan 23 2024

Formula

T(n,k) = A273975(2*k, n, (n-1)*k). - Andrey Zabolotskiy, Jan 23 2024