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A173045
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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 = 3, read by rows.
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2
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1, 1, 1, 1, 10, 1, 1, 29, 29, 1, 1, 84, 6566, 84, 1, 1, 247, 14348916, 14348916, 247, 1, 1, 734, 282429536495, 150094635296999140, 282429536495, 734, 1, 1, 2193, 50031545098999727, 2503155504993241601315571986085883, 2503155504993241601315571986085883, 50031545098999727, 2193, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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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 = 3.
Sum_{k=0..n} T(n, k, 3) = A000295(n) + Sum_{k=0..n} 3^(n*binomial(n-2, k-1)). - G. C. Greubel, Feb 19 2021
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EXAMPLE
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Triangle begins as:
1;
1, 1;
1, 10, 1;
1, 29, 29, 1;
1, 84, 6566, 84, 1;
1, 247, 14348916, 14348916, 247, 1;
1, 734, 282429536495, 150094635296999140, 282429536495, 734, 1;
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MATHEMATICA
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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, 3], {n, 0, 9}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 19 2021 *)
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PROG
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(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, 3) for k in (0..n)] for n in (0..9)]) # G. C. Greubel, Feb 19 2021
(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, 3): k in [0..n], n in [0..9]]; // G. C. Greubel, Feb 19 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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