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A129961
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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.
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5
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1, 2, 4, 8, 15, 26, 42, 64, 94, 140, 232, 464, 1092, 2744, 6840, 16384, 37384, 81296, 169120, 338240, 654192, 1232288, 2280864, 4194304, 7761376, 14635712, 28384384, 56768768, 116566080, 243472256, 511907712, 1073741824, 2232713344
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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FORMULA
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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).
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EXAMPLE
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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 ].
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PROG
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(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, Jun 17 2007
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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