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

0,4

Comments

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).

Links

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Index entries for triangles and arrays related to Pascal's triangle

Example

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  .

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 *)

Prog

(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

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.

Sequence in context: A220455 A208295 A285721 * A212184 A371571 A033769

Adjacent sequences: A214289 A214290 A214291 * A214293 A214294 A214295

Keyword

sign,tabl

Author

Reinhard Zumkeller, Jul 12 2012

Status

approved