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 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. 32

%I

%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 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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Last modified January 19 06:37 EST 2020. Contains 331033 sequences. (Running on oeis4.)