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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
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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 A033769 A074660 Adjacent sequences:  A214289 A214290 A214291 * A214293 A214294 A214295 KEYWORD sign,tabl AUTHOR Reinhard Zumkeller, Jul 12 2012 STATUS approved

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Last modified December 6 04:43 EST 2019. Contains 329784 sequences. (Running on oeis4.)