A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}*Sum_{j=1..n} (cos((2*j-1)*Pi/(2*n+1)))^{k-1}.

1, 2, 1, 3, 1, 1, 4, 1, 3, 1, 5, 1, 5, 4, 1, 6, 1, 7, 4, 7, 1, 8, 1, 11, 4, 19, 16, 18, 1, 9, 1, 13, 4, 25, 16, 38, 29, 1, 10, 1, 15, 4, 31, 16, 58, 57, 47, 1, 11, 1, 17, 4, 37, 16, 78, 64, 117, 76, 1, 12, 1, 19, 4, 43, 16, 98, 64, 187, 193, 123, 1

Offset: 1

Author

L. Edson Jeffery, Jan 12 2013

Keywords

• nonn
• tabl

Comments

Antidiagonal sums: {1,3,5,9,16,26,46,78,136,...}.

Examples

			Array begins as
.1..1...1..1...1...1
.2..1...3..4...7..11
.3..1...5..4..13..16
.4..1...7..4..19..16
.5..1...9..4..25..16
.6..1..11..4..31..16
		

Crossrefs

Rows: cf. A000012, A000032, A094649, A189234, A216605, etc.

Cf. A185095, A186740.

Formula

T(n,k) = 2^{k-1}*Sum_{j=1..n} (cos((2*j-1)*Pi/(2*n+1)))^{k-1}.

Empirical g.f. for row n: F(x) = (Sum_{u=0..n-1} A122765(n,n-1-u)*x^u)/(Sum_{v=0..n} A108299(n,v)*x^v).

Empirical: odd column first differences tend to A000984 = {1, 2, 6, 20, 70, 252, ...} (central binomial coefficients).