A144394 Triangle read by rows (n >= 4, 0 <= k <= n - 4): row n gives the coefficients in the expansion of ((x + 1)^n - (x^n + n*x^(n - 1) + n*x + 1))/x^2.
6, 10, 10, 15, 20, 15, 21, 35, 35, 21, 28, 56, 70, 56, 28, 36, 84, 126, 126, 84, 36, 45, 120, 210, 252, 210, 120, 45, 55, 165, 330, 462, 462, 330, 165, 55, 66, 220, 495, 792, 924, 792, 495, 220, 66, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91
Offset: 4
Examples
Triangle begins:
6;
10, 10;
15, 20, 15;
21, 35, 35, 21;
28, 56, 70, 56, 28;
36, 84, 126, 126, 84, 36;
45, 120, 210, 252, 210, 120, 45;
55, 165, 330, 462, 462, 330, 165, 55;
66, 220, 495, 792, 924, 792, 495, 220, 66;
78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78;
91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91;
105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105;
...
Links
- Reinhard Zumkeller, Rows n = 4..120 of triangle, flattened
- Hermann Stamm-Wilbrandt, Sum of Pascal's triangle reciprocals [Cached copy from the Wayback Machine]
- Index entries for triangles and arrays related to Pascal's triangle
Programs
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Haskell
a144394 n k = a144394_tabl !! (n-4) !! k a144394_row n = a144394_tabl !! (n-4) a144394_tabl = map (drop 2 . reverse . drop 2) $ drop 4 a007318_tabl -- Reinhard Zumkeller, Dec 24 2012
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Mathematica
p[x_, n_] = ((x + 1)^n - (x^n + n*x^(n - 1) + n*x + 1))/x^2 Table[CoefficientList[p[x, n], x], {n, 4, 15}] // Flatten -
Maxima
create_list(binomial(n, k + 2), n, 4, 20, k, 0, n - 4); /* Franck Maminirina Ramaharo, Jan 25 2019 */
Formula
T(n,k) = binomial(n, k + 2), n >= 4, 0 <= k <= n - 4.
Sum_{n >= 4, 0 <= k <= n-4} 1/T(n,k) = 3/2. - Hermann Stamm-Wilbrandt, Jul 21 2014
Extensions
Edited by Franklin T. Adams-Watters, Apr 07 2010
Comments