This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A144394 #24 Sep 10 2022 08:37:49
%S A144394 6,10,10,15,20,15,21,35,35,21,28,56,70,56,28,36,84,126,126,84,36,45,
%T A144394 120,210,252,210,120,45,55,165,330,462,462,330,165,55,66,220,495,792,
%U A144394 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
%N 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.
%C A144394 Interior of Pascal's triangle, stripping out the initial 1, n and final n, 1 in each row.
%H A144394 Reinhard Zumkeller, <a href="/A144394/b144394.txt">Rows n = 4..120 of triangle, flattened</a>
%H A144394 Hermann Stamm-Wilbrandt, <a href="https://web.archive.org/web/20171109040115/https://www.ibm.com/developerworks/community/blogs/HermannSW/entry/sum_of_pascal_s_triangle_reciprocals10?lang=en">Sum of Pascal's triangle reciprocals</a> [Cached copy from the Wayback Machine]
%H A144394 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A144394 T(n,k) = binomial(n, k + 2), n >= 4, 0 <= k <= n - 4.
%F A144394 Sum_{n >= 4, 0 <= k <= n-4} 1/T(n,k) = 3/2. - _Hermann Stamm-Wilbrandt_, Jul 21 2014
%e A144394 Triangle begins:
%e A144394 6;
%e A144394 10, 10;
%e A144394 15, 20, 15;
%e A144394 21, 35, 35, 21;
%e A144394 28, 56, 70, 56, 28;
%e A144394 36, 84, 126, 126, 84, 36;
%e A144394 45, 120, 210, 252, 210, 120, 45;
%e A144394 55, 165, 330, 462, 462, 330, 165, 55;
%e A144394 66, 220, 495, 792, 924, 792, 495, 220, 66;
%e A144394 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78;
%e A144394 91, 364, 1001, 2002, 3003, 3432, 3003, 2002, 1001, 364, 91;
%e A144394 105, 455, 1365, 3003, 5005, 6435, 6435, 5005, 3003, 1365, 455, 105;
%e A144394 ...
%t A144394 p[x_, n_] = ((x + 1)^n - (x^n + n*x^(n - 1) + n*x + 1))/x^2
%t A144394 Table[CoefficientList[p[x, n], x], {n, 4, 15}] // Flatten
%o A144394 (Haskell)
%o A144394 a144394 n k = a144394_tabl !! (n-4) !! k
%o A144394 a144394_row n = a144394_tabl !! (n-4)
%o A144394 a144394_tabl = map (drop 2 . reverse . drop 2) $ drop 4 a007318_tabl
%o A144394 -- _Reinhard Zumkeller_, Dec 24 2012
%o A144394 (Maxima) create_list(binomial(n, k + 2), n, 4, 20, k, 0, n - 4); /* _Franck Maminirina Ramaharo_, Jan 25 2019 */
%Y A144394 Cf. A007318, A052515 (row sums), A024746 (sorted), A144393.
%K A144394 nonn,easy,tabl
%O A144394 4,1
%A A144394 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 02 2008
%E A144394 Edited by _Franklin T. Adams-Watters_, Apr 07 2010