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.

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, -6, 7, 20, 28, 14, -14, -28, -20, -7, 8, 27, 48, 42, 0, -42, -48, -27, -8, 9, 35, 75, 90, 42, -42, -90, -75, -35, -9, 10, 44, 110, 165, 132, 0, -132, -165, -110, -44, -10

Offset: 0

Author

Reinhard Zumkeller, Jul 12 2012

Keywords

• sign
• tabl

Examples

			The triangle begins:
    0:                              0
    1:                            1   -1
    2:                          2   0   -2
    3:                       3    2   -2   -3
    4:                     4    5   0   -5   -4
    5:                  5    9    5   -5   -9   -5
    6:                6   14   14   0  -14  -14   -6
    7:             7   20   28   14  -14  -28  -20   -7
    8:           8   27   48   42   0  -42  -48  -27   -8
    9:        9   35   75   90   42  -42  -90  -75  -35   -9
   10:     10   44  110  165  132   0 -132 -165 -110  -44  -10
   11:  11   54  154  275  297  132 -132 -297 -275 -154  -54  -11  .
		

Links

• Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

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.

Programs

Haskell

a214292 n k = a214292_tabl !! n !! k
a214292_row n = a214292_tabl !! n
a214292_tabl = map diff $ tail a007318_tabl
   where diff row = zipWith (-) (tail row) row

Mathematica

row[n_] := Table[Binomial[n, k], {k, 0, n}] // Differences;
T[n_, k_] := row[n + 1][[k + 1]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 31 2018 *)

Formula

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;

T(n,k) = -T(n,k);

row sums and central terms equal 0, cf. A000004;

sum of positive elements of n-th row = A014495(n+1);

T(n,0) = n;

T(n,1) = A000096(n-2) for n > 1; T(n,1) = - A080956(n) for n > 0;

T(n,2) = A005586(n-4) for n > 3; T(n,2) = A129936(n-2);

T(n,3) = A005587(n-6) for n > 5;

T(n,4) = A005557(n-9) for n > 8;

T(n,5) = A064059(n-11) for n > 10;

T(n,6) = A064061(n-13) for n > 12;

T(n,7) = A124087(n) for n > 14;

T(n,8) = A124088(n) for n > 16;

T(2*n+1,n) = T(2*n+2,n) = A000108(n+1), Catalan numbers;

T(2*n+3,n) = A000245(n+2);

T(2*n+4,n) = A002057(n+1);

T(2*n+5,n) = A000344(n+3);

T(2*n+6,n) = A003517(n+3);

T(2*n+7,n) = A000588(n+4);

T(2*n+8,n) = A003518(n+4);

T(2*n+9,n) = A001392(n+5);

T(2*n+10,n) = A003519(n+5);

T(2*n+11,n) = A000589(n+6);

T(2*n+12,n) = A090749(n+6);

T(2*n+13,n) = A000590(n+7).