%I #8 Sep 08 2022 08:45:30
%S 1,1,3,2,5,3,9,6,16,11,27,22,50,50,101,114,215,255,471,552,1024,1145,
%T 2169,2290,4460,4460,8921,8556,17477,16383,33861,31674,65536,62255,
%U 127791,124510,252302,252302,504605,514446,1019051,1048575,2067627
%N Antidiagonal sums of triangular array T: T(j,1) = 1 for ((j-1) mod 6) < 3, else 0; T(j,k) = T(j-1,k-1) + T(j-1,k) for 2 <= k <= j.
%H Michael De Vlieger, <a href="/A131025/b131025.txt">Table of n, a(n) for n = 1..6644</a>
%H Scott M. Bailey and Donald M. Larson, <a href="https://arxiv.org/abs/2107.01316">The A(1)-module structure of the homology of Brown-Gitler spectra</a>, arXiv:2107.01316 [math.AT], 2021.
%F G.f.: (1-3*x^2+2*x^4+2*x^6-2*x^8+x^9)/((1-x)*(1+x)*(1-x+x^2)*(1-2*x^2)*(1-3*x^2+3*x^4)).
%e For first seven rows of T see A131022 or A129339.
%t CoefficientList[Series[(1 - 3 x^2 + 2 x^4 + 2 x^6 - 2 x^8 + x^9)/((1 - x)*(1 + x)*(1 - x + x^2)*(1 - 2 x^2)*(1 - 3 x^2 + 3 x^4)), {x, 0, 42}], x] (* _Michael De Vlieger_, Oct 26 2021 *)
%o (PARI) {m=43; M=matrix(m, m); for(j=1, m, M[j, 1]=if((j-1)%6<3, 1, 0)); for(k=2, m, for(j=k, m, M[j, k]=M[j-1, k-1]+M[j, k-1])); for(j=1, m, print1(sum(k=1, (j+1)\2, M[j-k+1, k]), ","))}
%o (Magma) m:=43; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 6 lt 3 then M[j, 1]:=1; end if; end for; for k:=2 to m do for j:=k to m do M[j, k]:=M[j-1, k-1]+M[j, k-1]; end for; end for; [ &+[ M[j-k+1, k]: k in [1..(j+1) div 2] ]: j in [1..m] ];
%Y Cf. A131022 (T read by rows), A129339 (main diagonal of T), A131023 (first subdiagonal of T), A131024 (row sums of T). First through sixth column of T are in A088911, A131026, A131027, A131028, A131029, A131030 resp.
%K nonn
%O 1,3
%A _Klaus Brockhaus_, following a suggestion of _Paul Curtz_, Jun 10 2007