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

4,1

Comments

Interior of Pascal's triangle, stripping out the initial 1, n and final n, 1 in each row.

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

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

Example

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;
  ...

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

Prog

(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
(Maxima) create_list(binomial(n, k + 2), n, 4, 20, k, 0, n - 4); /* Franck Maminirina Ramaharo, Jan 25 2019 */

Crossrefs

Cf. A007318, A052515 (row sums), A024746 (sorted), A144393.

Sequence in context: A175612 A024746 A111093 * A250742 A315122 A087873

Adjacent sequences: A144391 A144392 A144393 * A144395 A144396 A144397

Keyword

nonn,easy,tabl

Author

Roger L. Bagula and Gary W. Adamson, Oct 02 2008

Extensions

Edited by Franklin T. Adams-Watters, Apr 07 2010

Status

approved