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A362905
Array read by antidiagonals: T(n,k) is the number of n element multisets of length k vectors over GF(2) that sum to zero.
5
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 1, 8, 5, 3, 1, 1, 1, 16, 15, 11, 3, 1, 1, 1, 32, 51, 50, 14, 4, 1, 1, 1, 64, 187, 276, 99, 24, 4, 1, 1, 1, 128, 715, 1768, 969, 232, 30, 5, 1, 1, 1, 256, 2795, 12496, 11781, 3504, 429, 45, 5, 1, 1, 1, 512, 11051, 93600, 162877, 73440, 10659, 835, 55, 6, 1
OFFSET
0,9
COMMENTS
Equivalently, T(n,k) is the number multisets with n elements drawn from {0..2^k-1} such that the bitwise-xor of all the elements gives zero.
T(n,k) is the number of equivalence classes of n X k binary matrices with an even number of 1's in each column under permutation of rows.
T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and complementation of columns.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).
FORMULA
T(n,k) = binomial(2^k+n-1, n)/2^k for odd n;
T(n,k) = (binomial(2^k+n-1, n) + (2^k-1)*binomial(2^(k-1)+n/2-1, n/2))/2^k for even n.
G.f. of column k: (1/(1-x)^(2^k) + (2^k-1)/(1-x^2)^(2^(k-1)))/2^k.
EXAMPLE
Array begins:
=========================================
n/k| 0 1 2 3 4 5 6 ...
---+-------------------------------------
0 | 1 1 1 1 1 1 1 ...
1 | 1 1 1 1 1 1 1 ...
2 | 1 2 4 8 16 32 64 ...
3 | 1 2 5 15 51 187 715 ...
4 | 1 3 11 50 276 1768 12496 ...
5 | 1 3 14 99 969 11781 162877 ...
6 | 1 4 24 232 3504 73440 1878976 ...
7 | 1 4 30 429 10659 394383 18730855 ...
...
MATHEMATICA
A362905[n_, k_]:=(Binomial[2^k+n-1, n]+If[EvenQ[n], (2^k-1)Binomial[2^(k-1)+n/2-1, n/2], 0])/2^k; Table[A362905[n-k, k], {n, 0, 15}, {k, n, 0, -1}] (* Paolo Xausa, Nov 19 2023 *)
PROG
(PARI) T(n, k)={(binomial(2^k+n-1, n) + if(n%2==0, (2^k-1)*binomial(2^(k-1)+n/2-1, n/2)))/2^k}
CROSSREFS
Columns k=0..4 are A000012, A004526(n+2), A053307, A362906, A363350.
Rows n=2..3 are A000079, A007581.
Main diagonal is A363351.
Sequence in context: A300384 A252890 A173398 * A370771 A368106 A368104
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, May 27 2023
STATUS
approved