cp's OEIS Frontend

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.

%I A362905 #18 Nov 19 2023 15:56:50
%S A362905 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,
%T A362905 50,14,4,1,1,1,64,187,276,99,24,4,1,1,1,128,715,1768,969,232,30,5,1,1,
%U A362905 1,256,2795,12496,11781,3504,429,45,5,1,1,1,512,11051,93600,162877,73440,10659,835,55,6,1
%N 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.
%C A362905 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.
%C A362905 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.
%C A362905 T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and complementation of columns.
%H A362905 Andrew Howroyd, <a href="/A362905/b362905.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%F A362905 T(n,k) = binomial(2^k+n-1, n)/2^k for odd n;
%F A362905 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.
%F A362905 G.f. of column k: (1/(1-x)^(2^k) + (2^k-1)/(1-x^2)^(2^(k-1)))/2^k.
%e A362905 Array begins:
%e A362905 =========================================
%e A362905 n/k| 0 1  2   3     4      5        6 ...
%e A362905 ---+-------------------------------------
%e A362905 0  | 1 1  1   1     1      1        1 ...
%e A362905 1  | 1 1  1   1     1      1        1 ...
%e A362905 2  | 1 2  4   8    16     32       64 ...
%e A362905 3  | 1 2  5  15    51    187      715 ...
%e A362905 4  | 1 3 11  50   276   1768    12496 ...
%e A362905 5  | 1 3 14  99   969  11781   162877 ...
%e A362905 6  | 1 4 24 232  3504  73440  1878976 ...
%e A362905 7  | 1 4 30 429 10659 394383 18730855 ...
%e A362905   ...
%t A362905 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 *)
%o A362905 (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}
%Y A362905 Columns k=0..4 are A000012, A004526(n+2), A053307, A362906, A363350.
%Y A362905 Rows n=2..3 are A000079, A007581.
%Y A362905 Main diagonal is A363351.
%Y A362905 Cf. A054724, A340030, A340312, A363349.
%K A362905 nonn,tabl
%O A362905 0,9
%A A362905 _Andrew Howroyd_, May 27 2023