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%I #9 Mar 09 2021 02:53:23
%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 431,153,43,9,1,1,512,2584,4023,3292,1757,686,211,53,10,1,1,1024,6765,12664,11949,7168,3088,1030,281,64,11,1
%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.
%H G. C. Greubel, <a href="/A099239/b099239.txt">Antidiagonal rows n = 0..50, flattened</a>
%F T(n, k) = Sum_{j=0..n} binomial(k*n -(k-1)*(j-1), j), n, k>=0. (square array)
%F T(n, k) = Sum_{j=0..n} binomial(k + (n-1)*(j+1), n*(j+1) -1), n>0. (square array)
%F T(n, k) = Sum_{j=0..n-k} binomial(k*(n-k) - (k-1)*(j-1), j). (number triangle)
%F Rows of the square array are generated by 1/((1-x)^k-x).
%F Rows satisfy a(n) = a(n-1) - Sum_{k=1..n} (-1)^(k^binomial(n, k)) * a(n-k).
%e Rows begin
%e 1, 1, 1, 1, 1, ... A000012;
%e 1, 2, 4, 8, 16, ... 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)
%t T[n_, k_]:= Sum[Binomial[k*(n-k) - (k-1)*(j-1), j], {j,0,n-k}];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 09 2021 *)
%o (Sage)
%o def A099239(n,k): return sum( binomial(k*(n-k) -(k-1)*(j-1), j) for j in (0..n-k) )
%o flatten([[A099239(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 09 2021
%o (Magma)
%o A099239:= func< n,k | (&+[Binomial(k*(n-k) -(k-1)*(j-1), j): j in [0..n-k]]) >;
%o [A099239(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 09 2021
%Y Cf. A034856, A099240, A099241, A099242, A099253.
%Y Cf. A000079, A001906, A052529, A055991, A079675.
%K easy,nonn,tabl
%O 0,5
%A _Paul Barry_, Oct 08 2004
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