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 A173047 Triangle T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 3, read by rows. 4
 1, 1, 1, 1, 10, 1, 1, 29, 29, 1, 1, 84, 167, 84, 1, 1, 247, 738, 738, 247, 1, 1, 734, 2930, 4393, 2930, 734, 1, 1, 2193, 10955, 21904, 21904, 10955, 2193, 1, 1, 6568, 39393, 98470, 131289, 98470, 39393, 6568, 1, 1, 19691, 137816, 413426, 689030, 689030, 413426, 137816, 19691, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The triangle sequences having the form T(n,k,q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 have the row sums Sum_{k=0..n} T(n,k,q) = 2^(n-2)*q^n + 2^n - (n-1) - (5/4)*[n=0] -(q/2)*[n=1]. - G. C. Greubel, Feb 16 2021 LINKS G. C. Greubel, Rows n = 0..100 of the triangle, flattened FORMULA T(n, k, q) = binomial(n, k) + q^n*binomial(n-2, k-1) - 1 with T(n, 0) = T(n, n) = 1 and q = 3. Sum_{k=0..n} T(n, k, 3) = (1/4)*(6^n + 2^(n+2) - 4*(n-1) - 5*[n=0] - 6*[n=1]). - G. C. Greubel, Feb 16 2021 EXAMPLE Ttiangle begins as:   1;   1,     1;   1,    10,      1;   1,    29,     29,      1;   1,    84,    167,     84,      1;   1,   247,    738,    738,    247,      1;   1,   734,   2930,   4393,   2930,    734,      1;   1,  2193,  10955,  21904,  21904,  10955,   2193,      1;   1,  6568,  39393,  98470, 131289,  98470,  39393,   6568,     1;   1, 19691, 137816, 413426, 689030, 689030, 413426, 137816, 19691, 1; MATHEMATICA T[n_, k_, q_]:= If[k==0 || k==n, 1, Binomial[n, k] +(q^n)*Binomial[n-2, k-1] -1]; Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 16 2021 *) PROG (Sage) def T(n, k, q): return 1 if (k==0 or k==n) else binomial(n, k) + q^n*binomial(n-2, k-1) -1 flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 16 2021 (Magma) T:= func< n, k, q | k eq 0 or k eq n select 1 else Binomial(n, k) + q^n*Binomial(n-2, k-1) -1 >; [T(n, k, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 16 2021 CROSSREFS Cf. A132044 (q=0), A173075 (q=1), A173046 (q=2), this sequence (q=3). Sequence in context: A146765 A190152 A154984 * A173045 A176491 A008958 Adjacent sequences:  A173044 A173045 A173046 * A173048 A173049 A173050 KEYWORD nonn,tabl AUTHOR Roger L. Bagula, Feb 08 2010 EXTENSIONS Edited by G. C. Greubel, Feb 16 2021 STATUS approved

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Last modified April 23 12:15 EDT 2021. Contains 343204 sequences. (Running on oeis4.)