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%I #14 Jul 31 2018 02:43:18
%S 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 -6,7,20,28,14,-14,-28,-20,-7,8,27,48,42,0,-42,-48,-27,-8,9,35,75,90,
%U 42,-42,-90,-75,-35,-9,10,44,110,165,132,0,-132,-165,-110,-44,-10
%N 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.
%C 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;
%C T(n,k) = -T(n,k);
%C row sums and central terms equal 0, cf. A000004;
%C sum of positive elements of n-th row = A014495(n+1);
%C T(n,0) = n;
%C T(n,1) = A000096(n-2) for n > 1; T(n,1) = - A080956(n) for n > 0;
%C T(n,2) = A005586(n-4) for n > 3; T(n,2) = A129936(n-2);
%C T(n,3) = A005587(n-6) for n > 5;
%C T(n,4) = A005557(n-9) for n > 8;
%C T(n,5) = A064059(n-11) for n > 10;
%C T(n,6) = A064061(n-13) for n > 12;
%C T(n,7) = A124087(n) for n > 14;
%C T(n,8) = A124088(n) for n > 16;
%C T(2*n+1,n) = T(2*n+2,n) = A000108(n+1), Catalan numbers;
%C T(2*n+3,n) = A000245(n+2);
%C T(2*n+4,n) = A002057(n+1);
%C T(2*n+5,n) = A000344(n+3);
%C T(2*n+6,n) = A003517(n+3);
%C T(2*n+7,n) = A000588(n+4);
%C T(2*n+8,n) = A003518(n+4);
%C T(2*n+9,n) = A001392(n+5);
%C T(2*n+10,n) = A003519(n+5);
%C T(2*n+11,n) = A000589(n+6);
%C T(2*n+12,n) = A090749(n+6);
%C T(2*n+13,n) = A000590(n+7).
%H Reinhard Zumkeller, <a href="/A214292/b214292.txt">Rows n=0..150 of triangle, flattened</a>
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%e The triangle begins:
%e . 0: 0
%e . 1: 1 -1
%e . 2: 2 0 -2
%e . 3: 3 2 -2 -3
%e . 4: 4 5 0 -5 -4
%e . 5: 5 9 5 -5 -9 -5
%e . 6: 6 14 14 0 -14 -14 -6
%e . 7: 7 20 28 14 -14 -28 -20 -7
%e . 8: 8 27 48 42 0 -42 -48 -27 -8
%e . 9: 9 35 75 90 42 -42 -90 -75 -35 -9
%e . 10: 10 44 110 165 132 0 -132 -165 -110 -44 -10
%e . 11: 11 54 154 275 297 132 -132 -297 -275 -154 -54 -11 .
%t row[n_] := Table[Binomial[n, k], {k, 0, n}] // Differences;
%t T[n_, k_] := row[n + 1][[k + 1]];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 31 2018 *)
%o (Haskell)
%o a214292 n k = a214292_tabl !! n !! k
%o a214292_row n = a214292_tabl !! n
%o a214292_tabl = map diff $ tail a007318_tabl
%o where diff row = zipWith (-) (tail row) row
%Y 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 sign,tabl
%O 0,4
%A _Reinhard Zumkeller_, Jul 12 2012
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