Search: id:A129961
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%I A129961
%S A129961 1,2,4,8,15,26,42,64,94,140,232,464,1092,2744,6840,16384,37384,81296,
%T A129961 169120,338240,654192,1232288,2280864,4194304,7761376,14635712,28384384,
%U A129961 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 (klaus-brockhaus(AT)t-online.de), 
               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 (klaus-brockhaus(AT)t-online.de), 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.
%Y A129961 Sequence in context: A159243 A089140 A000125 this_sequence A133551 A114226 
               A003241
%Y A129961 Adjacent sequences: A129958 A129959 A129960 this_sequence A129962 A129963 
               A129964
%K A129961 nonn
%O A129961 1,2
%A A129961 Paul Curtz (bpcrtz(AT)free.fr), Jun 10 2007
%E A129961 Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), 
               Jun 14 2007
%E A129961 G.f. corrected by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 
               15 2009

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