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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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS G. C. Greubel, Rows n = 0..12 of the triangle, flattened 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 Adjacent sequences:  A173040 A173041 A173042 * A173044 A173045 A173046 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Feb 08 2010 EXTENSIONS Edited by G. C. Greubel, Feb 19 2021 STATUS approved

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Last modified May 6 06:00 EDT 2021. Contains 343580 sequences. (Running on oeis4.)