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 A108283 Triangle read by rows, generated from (..., 3, 2, 1). 3
 1, 1, 3, 1, 5, 6, 1, 7, 17, 10, 1, 9, 34, 49, 15, 1, 11, 57, 142, 129, 21, 1, 13, 86, 313, 547, 321, 28, 1, 15, 121, 586, 1593, 2005, 769, 36, 1, 17, 162, 985, 3711, 7737, 7108, 1793, 45, 1, 19, 209, 1534, 7465, 22461, 36409, 24604, 4097, 55, 1, 21, 262, 2257, 13539, 54121, 131836, 167481, 83653, 9217, 66 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Inverse binomial transforms of each column form the rows of A108284. Rightmost diagonal = triangular numbers, (A000217); while diagonals going to the left from (1, 3, 6, ...) are A000337 starting with 1: (1, 5, 17, 49, ...); A014915: (1, 7, 34, 142, ...); A014916: (1, 9, 57, ...); A014917: (1, 11, 86, ...). LINKS Table of n, a(n) for n=1..66. FORMULA n-th column = f(x), x = 1, 2, 3; n*x^(n-1) + (n-1)*x^(n-2) + (n-3)*x^(n-3) + ... + 1. T(n,k) = (1+ (n-k+1)^k*(n*k-k^2-1))/ (n-k)^2, n>k. - Jean-François Alcover, Sep 13 2016 EXAMPLE 4th column = 10, 49, 142, 313, ... = f(x), x = 1, 2, 3; 4x^3 + 3x^2 + 2x + 1. f(3) = 142. First few rows of the triangle: 1; 1, 3; 1, 5, 6; 1, 7, 17, 10; 1, 9, 34, 49, 15; 1, 11, 57, 142, 129, 21; ... MAPLE A108283 := proc(n, k) local x ; x := n-k+1 ; add( i*x^(i-1), i=1..k) ; end proc: seq(seq( A108283(n, k), k=1..n), n=1..10) ; # R. J. Mathar, Sep 14 2016 MATHEMATICA T[_, 1] := 1; T[n_, n_] := n (n + 1)/2; T[n_, k_] := (1 - (n - k + 1)^k*(k^2 - k*n + 1))/(n - k)^2; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 13 2016 *) CROSSREFS Cf. A059045, A108284, A000217, A000337, A014915, A014916, A014917. Sequence in context: A134083 A210551 A113445 * A208904 A344479 A209754 Adjacent sequences: A108280 A108281 A108282 * A108284 A108285 A108286 KEYWORD nonn,tabl,easy AUTHOR Gary W. Adamson, May 30 2005 EXTENSIONS More terms from Jean-François Alcover, Sep 13 2016 STATUS approved

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Last modified June 13 22:21 EDT 2024. Contains 373391 sequences. (Running on oeis4.)