A099239 Square array read by antidiagonals associated with sections of 1/(1-x-x^k).
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, 144, 129, 69, 26, 7, 1, 1, 128, 377, 406, 250, 106, 34, 8, 1, 1, 256, 987, 1278, 907, 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
Offset: 0
Examples
Rows begin 1, 1, 1, 1, 1, ... A000012; 1, 2, 4, 8, 16, ... 1-section of 1/(1-x-x) A000079; 1, 3, 8, 21, 55, .... bisection of 1/(1-x-x^2) A001906; 1, 4, 13, 41, 129, ... trisection of 1/(1-x-x^3) A052529; (essentially) 1, 5, 19, 69, 250, ... quadrisection of 1/(1-x-x^4) A055991; 1, 6, 26, 106, 431, ... quintisection of 1/(1-x-x^5) A079675; (essentially)
Links
- G. C. Greubel, Antidiagonal rows n = 0..50, flattened
Crossrefs
Programs
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Magma
A099239:= func< n,k | (&+[Binomial(k*(n-k) -(k-1)*(j-1), j): j in [0..n-k]]) >; [A099239(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021
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Mathematica
T[n_, k_]:= Sum[Binomial[k*(n-k) - (k-1)*(j-1), j], {j,0,n-k}]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 09 2021 *) -
Sage
def A099239(n,k): return sum( binomial(k*(n-k) -(k-1)*(j-1), j) for j in (0..n-k) ) flatten([[A099239(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021
Formula
T(n, k) = Sum_{j=0..n} binomial(k*n -(k-1)*(j-1), j), n, k>=0. (square array)
T(n, k) = Sum_{j=0..n} binomial(k + (n-1)*(j+1), n*(j+1) -1), n>0. (square array)
T(n, k) = Sum_{j=0..n-k} binomial(k*(n-k) - (k-1)*(j-1), j). (number triangle)
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^binomial(n, k)) * a(n-k).
Comments