1,2

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

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.

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

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

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).

Binomial transform of A131078. - Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 17 2007

First seven rows of T are

[ 1 ]

[ 1, 2 ]

[ 1, 2, 4 ]

[ 1, 2, 4, 8 ]

[ 0, 1, 3, 7, 15 ]

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

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

(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 */

(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 */

(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 (klaus-brockhaus(AT)t-online.de), Jun 17 2007

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.

Sequence in context: A159243 A089140 A000125 this_sequence A133551 A114226 A003241

Adjacent sequences: A129958 A129959 A129960 this_sequence A129962 A129963 A129964

nonn

Paul Curtz (bpcrtz(AT)free.fr), Jun 10 2007

Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 14 2007

G.f. corrected by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 15 2009