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 A214292 #18 Mar 23 2025 08:40:11
%S A214292 0,1,-1,2,0,-2,3,2,-2,-3,4,5,0,-5,-4,5,9,5,-5,-9,-5,6,14,14,0,-14,-14,
%T A214292 -6,7,20,28,14,-14,-28,-20,-7,8,27,48,42,0,-42,-48,-27,-8,9,35,75,90,
%U A214292 42,-42,-90,-75,-35,-9,10,44,110,165,132,0,-132,-165,-110,-44,-10
%N A214292 Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n with T(n,0) = n and T(n,n) = -n.
%H A214292 Reinhard Zumkeller, <a href="/A214292/b214292.txt">Rows n=0..150 of triangle, flattened</a>
%H A214292 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A214292 T(n,k) = A007318(n+1,k+1) - A007318(n+1,k), 0<=k<=n, i.e. first differences of rows in Pascal's triangle;
%F A214292 T(n,k) = -T(n,k);
%F A214292 row sums and central terms equal 0, cf. A000004;
%F A214292 sum of positive elements of n-th row = A014495(n+1);
%F A214292 T(n,0) = n;
%F A214292 T(n,1) = A000096(n-2) for n > 1; T(n,1) = - A080956(n) for n > 0;
%F A214292 T(n,2) = A005586(n-4) for n > 3; T(n,2) = A129936(n-2);
%F A214292 T(n,3) = A005587(n-6) for n > 5;
%F A214292 T(n,4) = A005557(n-9) for n > 8;
%F A214292 T(n,5) = A064059(n-11) for n > 10;
%F A214292 T(n,6) = A064061(n-13) for n > 12;
%F A214292 T(n,7) = A124087(n) for n > 14;
%F A214292 T(n,8) = A124088(n) for n > 16;
%F A214292 T(2*n+1,n) = T(2*n+2,n) = A000108(n+1), Catalan numbers;
%F A214292 T(2*n+3,n) = A000245(n+2);
%F A214292 T(2*n+4,n) = A002057(n+1);
%F A214292 T(2*n+5,n) = A000344(n+3);
%F A214292 T(2*n+6,n) = A003517(n+3);
%F A214292 T(2*n+7,n) = A000588(n+4);
%F A214292 T(2*n+8,n) = A003518(n+4);
%F A214292 T(2*n+9,n) = A001392(n+5);
%F A214292 T(2*n+10,n) = A003519(n+5);
%F A214292 T(2*n+11,n) = A000589(n+6);
%F A214292 T(2*n+12,n) = A090749(n+6);
%F A214292 T(2*n+13,n) = A000590(n+7).
%e A214292 The triangle begins:
%e A214292 0: 0
%e A214292 1: 1 -1
%e A214292 2: 2 0 -2
%e A214292 3: 3 2 -2 -3
%e A214292 4: 4 5 0 -5 -4
%e A214292 5: 5 9 5 -5 -9 -5
%e A214292 6: 6 14 14 0 -14 -14 -6
%e A214292 7: 7 20 28 14 -14 -28 -20 -7
%e A214292 8: 8 27 48 42 0 -42 -48 -27 -8
%e A214292 9: 9 35 75 90 42 -42 -90 -75 -35 -9
%e A214292 10: 10 44 110 165 132 0 -132 -165 -110 -44 -10
%e A214292 11: 11 54 154 275 297 132 -132 -297 -275 -154 -54 -11 .
%t A214292 row[n_] := Table[Binomial[n, k], {k, 0, n}] // Differences;
%t A214292 T[n_, k_] := row[n + 1][[k + 1]];
%t A214292 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 31 2018 *)
%o A214292 (Haskell)
%o A214292 a214292 n k = a214292_tabl !! n !! k
%o A214292 a214292_row n = a214292_tabl !! n
%o A214292 a214292_tabl = map diff $ tail a007318_tabl
%o A214292 where diff row = zipWith (-) (tail row) row
%Y A214292 Cf. A007318, A000004, A000096, A000108, A000245, A000344, A000588, A000589, A000590, A001392, A002057, A003517, A003518, A003519, A005557, A005586, A005587, A008313, A014495, A064059, A064061, A080956, A090749, A097808, A112467, A124087, A124088, A129936, A259525.
%K A214292 sign,tabl
%O A214292 0,4
%A A214292 _Reinhard Zumkeller_, Jul 12 2012