A260534 Square array read by ascending antidiagonals, T(n,k) = Sum_{j=0..k} n^j*(C(k-j,j) mod 2).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 5, 1, 7, 2, 1, 1, 1, 6, 1, 13, 5, 3, 1, 1, 1, 7, 1, 21, 10, 11, 1, 1, 1, 1, 8, 1, 31, 17, 31, 1, 4, 1, 1, 1, 9, 1, 43, 26, 69, 1, 23, 3, 1, 1, 1, 10, 1, 57, 37, 131, 1, 94, 21, 5, 1, 1, 1, 11

Offset: 0

Author

Peter Luschny, Sep 20 2015

Keywords

• nonn
• tabl

Comments

A parametrization of Stern's diatomic series (which is here T(1,k)). (For other generalizations of Dijkstra's fusc function see the Luschny link.)

Examples

			Array starts:
n\k[0, 1,  2, 3,  4,  5,   6, 7,    8,    9,    10]
[0] 1, 1,  1, 1,  1,  1,   1, 1,    1,    1,     1, ...
[1] 1, 1,  2, 1,  3,  2,   3, 1,    4,    3,     5, ... [A002487]
[2] 1, 1,  3, 1,  7,  5,  11, 1,   23,   21,    59, ... [A101624]
[3] 1, 1,  4, 1, 13, 10,  31, 1,   94,   91,   355, ...
[4] 1, 1,  5, 1, 21, 17,  69, 1,  277,  273,  1349, ... [A101625]
[5] 1, 1,  6, 1, 31, 26, 131, 1,  656,  651,  3881, ...
[6] 1, 1,  7, 1, 43, 37, 223, 1, 1339, 1333,  9295, ...
[7] 1, 1,  8, 1, 57, 50, 351, 1, 2458, 2451, 19559, ...
[8] 1, 1,  9, 1, 73, 65, 521, 1, 4169, 4161, 37385, ...
-,-,-,-,A002061,A002522,A071568,-,-,A059826,-,A002523,
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10010
• Peter Luschny, Rational Trees and Binary Partitions.

Crossrefs

Cf. A002061, A002487, A002522, A002523, A059826, A071568, A101624, A101625.

Programs

Maple

T := (n,k) -> add(modp(binomial(k-j,j),2)*n^j, j=0..k):
seq(lprint(seq(T(n,k),k=0..10)),n=0..5);

Mathematica

Table[If[(n - k) == 0, 1, Sum[(n - k)^j Mod[Binomial[k - j, j], 2], {j, 0, k}]], {n, 0, 10}, {k, 0, n}] (* Michael De Vlieger, Sep 21 2015 *)

Python

def A260534_T(n,k):
    return sum(0 if ~(k-j) & j else n**j for j in range(k+1)) # Chai Wah Wu, Feb 08 2016