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 A174719 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 3, read by rows. 3
 1, 1, 1, 1, -7, 1, 1, -51, -51, 1, 1, -239, -399, -239, 1, 1, -967, -2177, -2177, -967, 1, 1, -3639, -10191, -13831, -10191, -3639, 1, 1, -13115, -43719, -74323, -74323, -43719, -13115, 1, 1, -45919, -177119, -360799, -452639, -360799, -177119, -45919, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The row sums of this class of sequences, for varying q, is given by Sum_{k=0..n} T(n, k, q) = q^n * (n+1) + 2^n * (1 - q^n). - G. C. Greubel, Feb 09 2021 LINKS G. C. Greubel, Rows n = 0..100 of the triangle, flattened FORMULA T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q=3. Sum_{k=0..n} T(n, k, 3) = 3^n*(n+1) + 2^n*(1 - 3^n) = A027471(n+2) - A248216(n). - G. C. Greubel, Feb 09 2021 EXAMPLE Triangle begins as: 1; 1, 1; 1, -7, 1; 1, -51, -51, 1; 1, -239, -399, -239, 1; 1, -967, -2177, -2177, -967, 1; 1, -3639, -10191, -13831, -10191, -3639, 1; 1, -13115, -43719, -74323, -74323, -43719, -13115, 1; 1, -45919, -177119, -360799, -452639, -360799, -177119, -45919, 1; MATHEMATICA T[n_, k_, q_]:= 1 +(1-q^n)*(Binomial[n, k] -1); Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten PROG (Sage) def T(n, k, q): return 1 + (1-q^n)*(binomial(n, k) - 1) flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 09 2021 (Magma) T:= func< n, k, q | 1 + (1-q^n)*(Binomial(n, k) -1) >; [T(n, k, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 09 2021 CROSSREFS Cf. A000012 (q=1), A174718 (q=2), this sequence (q=3), A174720 (q=4). Cf. A027471, A248216. Sequence in context: A203389 A174689 A172345 * A176392 A015118 A340428 Adjacent sequences: A174716 A174717 A174718 * A174720 A174721 A174722 KEYWORD sign,tabl AUTHOR Roger L. Bagula, Mar 28 2010 EXTENSIONS Edited by G. C. Greubel, Feb 09 2021 STATUS approved

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Last modified April 19 08:28 EDT 2024. Contains 371782 sequences. (Running on oeis4.)