cp's OEIS Frontend

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

%I A099239 #9 Mar 09 2021 02:53:23
%S A099239 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 A099239 144,129,69,26,7,1,1,128,377,406,250,106,34,8,1,1,256,987,1278,907,
%U A099239 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 A099239 Square array read by antidiagonals associated with sections of 1/(1-x-x^k).
%C A099239 Rows include A099242, A099253. Columns include A034856. Main diagonal is A099240. Sums of antidiagonals are A099241.
%H A099239 G. C. Greubel, <a href="/A099239/b099239.txt">Antidiagonal rows n = 0..50, flattened</a>
%F A099239 T(n, k) = Sum_{j=0..n} binomial(k*n -(k-1)*(j-1), j), n, k>=0. (square array)
%F A099239 T(n, k) = Sum_{j=0..n} binomial(k + (n-1)*(j+1), n*(j+1) -1), n>0. (square array)
%F A099239 T(n, k) = Sum_{j=0..n-k} binomial(k*(n-k) - (k-1)*(j-1), j). (number triangle)
%F A099239 Rows of the square array are generated by 1/((1-x)^k-x).
%F A099239 Rows satisfy a(n) = a(n-1) - Sum_{k=1..n} (-1)^(k^binomial(n, k)) * a(n-k).
%e A099239 Rows begin
%e A099239   1, 1,  1,   1,   1, ...                               A000012;
%e A099239   1, 2,  4,   8,  16, ...      1-section of 1/(1-x-x)   A000079;
%e A099239   1, 3,  8,  21,  55, ....     bisection of 1/(1-x-x^2) A001906;
%e A099239   1, 4, 13,  41, 129, ...     trisection of 1/(1-x-x^3) A052529; (essentially)
%e A099239   1, 5, 19,  69, 250, ...  quadrisection of 1/(1-x-x^4) A055991;
%e A099239   1, 6, 26, 106, 431, ...  quintisection of 1/(1-x-x^5) A079675; (essentially)
%t A099239 T[n_, k_]:= Sum[Binomial[k*(n-k) - (k-1)*(j-1), j], {j,0,n-k}];
%t A099239 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 09 2021 *)
%o A099239 (Sage)
%o A099239 def A099239(n,k): return sum( binomial(k*(n-k) -(k-1)*(j-1), j) for j in (0..n-k) )
%o A099239 flatten([[A099239(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 09 2021
%o A099239 (Magma)
%o A099239 A099239:= func< n,k | (&+[Binomial(k*(n-k) -(k-1)*(j-1), j): j in [0..n-k]]) >;
%o A099239 [A099239(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 09 2021
%Y A099239 Cf. A034856, A099240, A099241, A099242, A099253.
%Y A099239 Cf. A000079, A001906, A052529, A055991, A079675.
%K A099239 easy,nonn,tabl
%O A099239 0,5
%A A099239 _Paul Barry_, Oct 08 2004