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 A132044 Triangle T(n,k) = binomial(n, k) - 1 with T(n,0) = T(n,n) = 1, read by rows. 15
 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 1, 1, 4, 9, 9, 4, 1, 1, 5, 14, 19, 14, 5, 1, 1, 6, 20, 34, 34, 20, 6, 1, 1, 7, 27, 55, 69, 55, 27, 7, 1, 1, 8, 35, 83, 125, 125, 83, 35, 8, 1, 1, 9, 44, 119, 209, 251, 209, 119, 44, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Row sums = A132045: (1, 2, 3, 6, 13, 28, 59, ...). 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 12 2021 LINKS G. C. Greubel, Rows n = 0..100 of the triangle, flattened FORMULA T(n, k) = A007318(n,k) + A103451(n,k) - A000012(n,k), an infinite lower triangular matrix. T(n, k) = binomial(n, k) - 1, with T(n,0) = T(n,n) = 1. - Roger L. Bagula, Feb 08 2010 From G. C. Greubel, Feb 12 2021: (Start) 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 = 0. Sum_{k=0..n} T(n, k, 0) = 2^n - (n-1) - [n=0]. (End) EXAMPLE First few rows of the triangle: 1; 1, 1; 1, 1, 1; 1, 2, 2, 1; 1, 3, 5, 3, 1; 1, 4, 9, 9, 4, 1; 1, 5, 14, 19, 14, 5, 1; 1, 6, 20, 34, 34, 20, 6, 1; 1, 7, 27, 55, 69, 55, 27, 7, 1; MATHEMATICA T[n_, k_]:= If[k==0 || k==n, 1, Binomial[n, k] - 1]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* Roger L. Bagula, Feb 08 2010 *) 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, 0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 12 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, 0): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 12 2021 CROSSREFS Cf. A007318, A103451, A132045. Cf. this sequence (q=0), A173075 (q=1), A173046 (q=2), A173047 (q=3). Sequence in context: A161671 A144444 A054106 * A034327 A034254 A157103 Adjacent sequences: A132041 A132042 A132043 * A132045 A132046 A132047 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, Aug 08 2007 STATUS approved

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Last modified September 23 09:36 EDT 2023. Contains 365544 sequences. (Running on oeis4.)