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 A174718 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 2, read by rows. 3
 1, 1, 1, 1, -2, 1, 1, -13, -13, 1, 1, -44, -74, -44, 1, 1, -123, -278, -278, -123, 1, 1, -314, -881, -1196, -881, -314, 1, 1, -761, -2539, -4317, -4317, -2539, -761, 1, 1, -1784, -6884, -14024, -17594, -14024, -6884, -1784, 1, 1, -4087, -17884, -42412, -63874, -63874, -42412, -17884, -4087, 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=2. Sum_{k=0..n} T(n, k, 2) = 2^n *(n + 2 - 2^n) = A001787(n+1) - A020522(n). - G. C. Greubel, Feb 09 2021 EXAMPLE Triangle begins as: 1; 1, 1; 1, -2, 1; 1, -13, -13, 1; 1, -44, -74, -44, 1; 1, -123, -278, -278, -123, 1; 1, -314, -881, -1196, -881, -314, 1; 1, -761, -2539, -4317, -4317, -2539, -761, 1; 1, -1784, -6884, -14024, -17594, -14024, -6884, -1784, 1; 1, -4087, -17884, -42412, -63874, -63874, -42412, -17884, -4087, 1; MATHEMATICA T[n_, k_, q_]:= 1 +(1-q^n)*(Binomial[n, k] -1); Table[T[n, k, 2], {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, 2) 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, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 09 2021 CROSSREFS Cf. A000012 (q=1), this sequence (q=2), A174719 (q=3), A174720 (q=4). Cf. A001787, A020522. Sequence in context: A186430 A173889 A156885 * A176291 A337514 A054505 Adjacent sequences: A174715 A174716 A174717 * A174719 A174720 A174721 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 02:04 EDT 2024. Contains 371782 sequences. (Running on oeis4.)