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Square array read by antidiagonals associated to sections of 1/(1-x-x^k).
8

%I #19 Mar 26 2023 10:25:36

%S 1,1,1,1,1,1,1,1,2,1,1,1,3,3,1,1,1,4,6,5,1,1,1,5,10,13,8,1,1,1,6,15,

%T 26,28,13,1,1,1,7,21,45,69,60,21,1,1,1,8,28,71,140,181,129,34,1,1,1,9,

%U 36,105,251,431,476,277,55,1,1,1,10,45,148,413,882,1326,1252,595,89,1

%N Square array read by antidiagonals associated to sections of 1/(1-x-x^k).

%H Seiichi Manyama, <a href="/A099233/b099233.txt">Antidiagonals n = 0..139, flattened</a>

%F Square array T(n, k) = Sum_{j=0..n} binomial(k(n-j), j).

%F Rows are generated by 1/(1-x(1+x)^k) and satisfy a(n) = Sum_{k=0..n} binomial(n, k)a(n-k-1).

%e Rows begin

%e 1, 1, 1, 1, 1, 1, ...

%e 1, 1, 2, 3, 5, 8, ...

%e 1, 1, 3, 6, 13, 28, ...

%e 1, 1, 4, 10, 26, 69, ...

%e 1, 1, 5, 15, 45, 140, ...

%e Row 1 is the 0-section of 1/(1-x-x) (A000079);

%e Row 2 is the 1-section of 1/(1-x-x^2) (A000045);

%e Row 3 is the 2-section of 1/(1-x-x^3) (A000930);

%e Row 4 is the 3-section of 1/(1-x-x^4) (A003269);

%e etc.

%Y Sums of antidiagonals are A099236.

%Y Columns include A000217, A008778.

%Y Rows include A000045, A002478, A099234, A099235.

%Y Main diagonal gives A099237.

%Y Cf. A099238.

%K easy,nonn,tabl

%O 0,9

%A _Paul Barry_, Oct 08 2004