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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.
5

%I #8 Sep 08 2022 08:45:30

%S 1,2,4,8,15,26,42,64,94,140,232,464,1092,2744,6840,16384,37384,81296,

%T 169120,338240,654192,1232288,2280864,4194304,7761376,14635712,

%U 28384384,56768768,116566080,243472256,511907712,1073741824,2232713344

%N 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 First column is periodically 1 1 1 1 0 0 0 0 (see A131078).

%C 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 Antidiagonal sums are 1, 1, 3, 3, 6, 5, 11, ... (see A131077).

%F G.f.: x*(1-x)^4/((1-2*x)*(1-4*x+6*x^2-4*x^3+2*x^4)).

%F 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 Binomial transform of A131078. - _Klaus Brockhaus_, Jun 17 2007

%e First seven rows of T are

%e [ 1 ]

%e [ 1, 2 ]

%e [ 1, 2, 4 ]

%e [ 1, 2, 4, 8 ]

%e [ 0, 1, 3, 7, 15 ]

%e [ 0, 0, 1, 4, 11, 26 ]

%e [ 0, 0, 0, 1, 5, 16, 42 ].

%o (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 (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 (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 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 nonn

%O 1,2

%A _Paul Curtz_, Jun 10 2007

%E Edited and extended by _Klaus Brockhaus_, Jun 14 2007

%E G.f. corrected by _Klaus Brockhaus_, Oct 15 2009