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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.
2
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
0,5
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
EXAMPLE
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;
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 *)
PROG
(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
(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
CROSSREFS
Cf. A132044 (q=0), A007318 (q=1), this sequence (q=2), A173045 (q=3).
Cf. A000295.
Sequence in context: A188461 A188474 A173046 * A082046 A132787 A181370
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Feb 08 2010
EXTENSIONS
Edited by G. C. Greubel, Feb 19 2021
STATUS
approved