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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 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 A033769 A333636 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 February 21 04:08 EST 2024. Contains 370219 sequences. (Running on oeis4.)