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A129961 Main diagonal of triangular array T: T(j,1) = 1 for ((j-1) mod 8) < 4, else 0; T(j,k) = T(j-1,k-1)+T(j,k-1) for 2 <= k <= j.

%I A129961 #8 Sep 08 2022 08:45:30
%S A129961 1,2,4,8,15,26,42,64,94,140,232,464,1092,2744,6840,16384,37384,81296,
%T A129961 169120,338240,654192,1232288,2280864,4194304,7761376,14635712,
%U A129961 28384384,56768768,116566080,243472256,511907712,1073741824,2232713344
%N A129961 Main diagonal of triangular array T: T(j,1) = 1 for ((j-1) mod 8) < 4, else 0; T(j,k) = T(j-1,k-1)+T(j,k-1) for 2 <= k <= j.
%C A129961 First column is periodically 1 1 1 1 0 0 0 0 (see A131078).
%C A129961 First subdiagonal is 1, 2, 4, 7, 11, 16, 22, ... (see A131075); next subdiagonals are 1, 2, 3, 4, 5, 6, 8, 16, 46, 140, ..., 1, 1, 1, 1, 1, 2, 8, 30, 94, 256, ..., 0, 0, 0, 0, 1, 6, 22, 64, 162, 372, ..., 0, 0, 0, 1, 5, 16, 42, 98, 210, 420, ...., 0, 0, 1, 4, 11, 26, 56, 112, 210, 372, ..., 0, 1, 3, 7, 15, 30, 56, 98, 162, 256, ...,1, 2, 4, 8, 15, 26, 42, 64, 94, 140, ... . Main diagonal and eighth subdiagonal agree; generally j-th subdiagonal equals (j+8)-th subdiagonal.
%C A129961 Antidiagonal sums are 1, 1, 3, 3, 6, 5, 11, ... (see A131077).
%F A129961 G.f.: x*(1-x)^4/((1-2*x)*(1-4*x+6*x^2-4*x^3+2*x^4)).
%F A129961 a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 8, a(5) = 15; for n > 5, a(n) = 6*a(n-1)-14*a(n-2)+16*a(n-3)-10*a(n-4)+4*a(n-5).
%F A129961 Binomial transform of A131078. - _Klaus Brockhaus_, Jun 17 2007
%e A129961 First seven rows of T are
%e A129961 [ 1 ]
%e A129961 [ 1, 2 ]
%e A129961 [ 1, 2, 4 ]
%e A129961 [ 1, 2, 4, 8 ]
%e A129961 [ 0, 1, 3, 7, 15 ]
%e A129961 [ 0, 0, 1, 4, 11, 26 ]
%e A129961 [ 0, 0, 0, 1, 5, 16, 42 ].
%o A129961 (PARI) {m=33; v=concat([1, 2, 4, 8, 15], vector(m-5)); for(n=6, m, v[n]=6*v[n-1]-14*v[n-2]+16*v[n-3]-10*v[n-4]+4*v[n-5]); v} \\ _Klaus Brockhaus_, Jun 14 2007
%o A129961 (Magma) m:=33; M:=ZeroMatrix(IntegerRing(), m, m); for j:=1 to m do if (j-1) mod 8 lt 4 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[n, n]: n in [1..m] ]; // _Klaus Brockhaus_, Jun 14 2007
%o A129961 (Magma) m:=33; S:=[ [1, 1, 1, 1, 0, 0, 0, 0][(n-1) mod 8 + 1]: n in [1..m] ]; [ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; // _Klaus Brockhaus_, Jun 17 2007
%Y A129961 Cf. A129339, A131074 (T read by rows), A131075 (first subdiagonal of T), A131076 (row sums of T), A131077 (antidiagonal sums of T). First through sixth column of T are in A131078, A131079, A131080, A131081, A131082, A131083 resp.
%K A129961 nonn
%O A129961 1,2
%A A129961 _Paul Curtz_, Jun 10 2007
%E A129961 Edited and extended by _Klaus Brockhaus_, Jun 14 2007
%E A129961 G.f. corrected by _Klaus Brockhaus_, Oct 15 2009