A173043 Triangle T(n, k, q) = binomial(n, k) - 1 + q^(n*binomial(n-2, k-1)) with T(n, 0, q) = T(n, n, q) = 1 and q = 2, read by rows.
1, 1, 1, 1, 5, 1, 1, 10, 10, 1, 1, 19, 261, 19, 1, 1, 36, 32777, 32777, 36, 1, 1, 69, 16777230, 68719476755, 16777230, 69, 1, 1, 134, 34359738388, 1180591620717411303458, 1180591620717411303458, 34359738388, 134, 1
Offset: 0
Examples
Triangle begins as: 1; 1, 1; 1, 5, 1; 1, 10, 10, 1; 1, 19, 261, 19, 1; 1, 36, 32777, 32777, 36, 1; 1, 69, 16777230, 68719476755, 16777230, 69, 1;
Links
- G. C. Greubel, Rows n = 0..12 of the triangle, flattened
Programs
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Magma
T:= func< n,k,q | k eq 0 or k eq n select 1 else Binomial(n,k) -1 +q^(n*Binomial(n-2, k-1)) >; [T(n,k,2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 19 2021
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Mathematica
T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1 + q^(n*Binomial[n-2, k-1])]; Table[t[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Feb 19 2021 *) -
Sage
def T(n,k,q): if (k==0 or k==n): return 1 else: return binomial(n,k) -1 +q^(n*binomial(n-2, k-1)) flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 19 2021
Formula
T(n, k, q) = binomial(n, k) - 1 + q^(n*binomial(n-2, k-1)) with T(n, 0, q) = T(n, n, q) = 1 and q = 2.
Sum_{k=0..n} T(n, k, 2) = A000295(n) + Sum_{k=0..n} 2^(n*binomial(n-2, k-1)). - G. C. Greubel, Feb 19 2021
Extensions
Edited by G. C. Greubel, Feb 19 2021