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A099239 Square array read by antidiagonals associated with sections of 1/(1-x-x^k). 4


%S 1,1,1,1,2,1,1,4,3,1,1,8,8,4,1,1,16,21,13,5,1,1,32,55,41,19,6,1,1,64,

%T 144,129,69,26,7,1,1,128,377,406,250,106,34,8,1,1,256,987,1278,907,

%U 431,153,43,9,1,1,512,2584,4023,3292,1757,686,211,53,10,1,1,1024,6765,12664

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

%C Rows include A099242, A099253. Columns include A034856. Main diagonal is A099240. Sums of antidiagonals are A099241.

%F Square array T(n, k)=sum{j=0..n, binomial(k*n-(k-1)(j-1), j)}, n, k>=0. Also, T(n, k)=sum{j=0..n, binomial(k+(n-1)(j+1), n(j+1)-1}, n>0. As a number triangle read by row, this is T(n, k)=sum{j=0..n-k, binomial(k(n-k)-(k-1)(j-1)}. Rows of the square array are generated by 1/((1-x)^k-x). Rows satisfy a(n)=a(n-1)-sum{k=1..n, (-1)^k^C(n, k)a(n-k)}.

%e Rows begin

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

%e 1,2,4,8,16,32,... 1-section of 1/(1-x-x) A000079

%e 1,3,8,21,55,..... bisection of 1/(1-x-x^2) A001906

%e 1,4,13,41,129,... trisection of 1/(1-x-x^3) A052529 (essentially)

%e 1,5,19,69,250,... quadrisection of 1/(1-x-x^4) A055991

%e 1,6,26,106,431,.. quintisection of 1/(1-x-x^5) A079675 (essentially)

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Oct 08 2004

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Last modified May 27 06:24 EDT 2019. Contains 323599 sequences. (Running on oeis4.)