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1381 sequences

• A169803: Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).
  1 · 1, 1 · 1, 2 · 1, 3, 1 … row sums 1, 2, 3, 5, 8, 13, 21, 34, …
  A169803 (−∞, 0] nonnegative
• A081579 → D+1
  1 · 2, 1 · 7, 8, 1 · 12, 33, 14, 1 … row sums 1, 3, 16, 60, 240, 896, 3328, 12096, …
  (−∞, 0] nonnegative
• basis-map coefficients of A026729 → basis A056242 → (θ+1)²
  1 · 1, 8 · 1, 20, 36 · 1, 36, 144, 128 … row sums 1, 9, 57, 309, 1530, 7134, 31864, …
  (−∞, 0] nonnegative
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
  1 · 2, 3 · 4, 12, 9 · 8, 36, 54, 27 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A013620 (−∞, 0) [−1, 0] nonnegative rational roots
• A100861 → reverse → D+1
  1 · 1, 1 · 1, 3, 1 · 0, 6, 6, 1 … row sums 1, 2, 5, 13, 38, 116, 382, 1310, …
  (−∞, 0] nonnegative
• A126216: Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A126216 (−∞, −1] nonnegative
• A007318 → (1/2+(1/2)n)θ+1+n → x ↦ x/(x+1)
  1 · 2, 5 · 3, 15, 18 · 4, 30, 72, 56 … row sums 1, 7, 36, 162, 675, 2673, 10206, 37908, …
  [−1, 0] nonnegative rational roots
• A008459 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 20, 30, 12, 1 … row sums 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  (−∞, −1] nonnegative
• A011973 → basis A007318 → D
  1 · 2 · 5, 2 · 10, 6 … row sums 0, 0, 1, 2, 7, 16, 41, 94, …
  (−∞, 0] nonnegative
• A120909 → θ+1
  3 · 3, 12 · 3, 24, 36 · 3, 36, 108, 96 … row sums 3, 15, 63, 243, 891, 3159, 10935, …
  (−∞, 0] nonnegative rational roots
• A026374 → D+1
  1 · 2, 1 · 4, 5, 1 · 5, 12, 7, 1 … row sums 1, 3, 10, 25, 75, 175, 500, 1125, …
  (−∞, 0] nonnegative
• A122431 → D
  3, 6, 3 · 1, 6, 9, 4 · 0, 2, 9, 12, 5 … row sums 12, 20, 28, 36, 44, 52, 60, 68, …
  (−∞, 0] nonnegative rational roots
• A028338 → x ↦ (−2/9)x+1 (shift +2)
  162, -27, 1 · -4374, 891, -54, 1 … row sums 136, -3536, 123760, -5445440, …
  [9/2, ∞) rational roots
• A178822 → x ↦ x+1
  1 · 12, 6 · 84, 84, 21 · 448, 672, 336, 56 … row sums 1, 18, 189, 1512, 10206, 61236, 336798, …
  (−∞, −1] nonnegative rational roots
• A130123 → reverse
  1 · 2 · 4 · 8 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, 0] nonnegative rational roots
• A023531 → (1/2)D²+2D+1 → x ↦ x+1
  1 · 3, 1 · 6, 6, 1 · 10, 18, 9, 1 … row sums 1, 4, 13, 38, 104, 272, 688, 1696, …
  (−∞, −1] nonnegative
• A334218 → x ↦ x+1
  1 · 1 · 2, 2 · 6, 9, 3 … row sums 1, 1, 4, 18, 104, 750, 6492, 65562, …
  (−∞, −1] nonnegative
• A023531 → 2D+1+2n → x ↦ x+1
  1 · 5, 3 · 9, 14, 5 · 13, 33, 27, 7 … row sums 1, 8, 28, 80, 208, 512, 1216, 2816, …
  (−∞, −1] nonnegative rational roots
• A345013: Triangle read by rows, related to clusters of type D.
  A062196 → x ↦ x+1
  1 · 4, 3 · 15, 20, 6 · 56, 105, 60, 10 … row sums 1, 7, 41, 231, 1289, 7183, 40081, …
  A345013 (−∞, −1] nonnegative
• A033282 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 11 · 1, 12, 42, 45 … row sums 1, 4, 19, 100, 562, 3304, 20071, …
  [−1, 0) nonnegative
• A111125 → D+1
  1 · 4, 1 · 10, 7, 1 · 21, 28, 10, 1 … row sums 1, 5, 18, 60, 191, 589, 1775, 5257, …
  (−∞, 0] nonnegative
• A011973 → basis A007318 → θ+1
  1 · 1 · 2, 2 · 3, 4 … row sums 1, 1, 4, 7, 18, 37, 84, 179, …
  (−∞, 0] nonnegative
• A099093 → θ+1
  1 · 0, 6 · 0, 6, 27 · 0, 0, 54, 108 … row sums 1, 6, 33, 162, 756, 3402, 14931, 64314, …
  (−∞, 0] nonnegative
• A099040 → x ↦ x−1
  1 · -2, 2 · 2, -6, 4 · 0, 8, -16, 8 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  (−1, 1]
• A094646: Triangle of generalized Stirling numbers of the first kind read by rows: T(n, k) = (-1)^(n+k)*Sum_{m=0..n} Stirling1(n, m) * binomial(m, k) * 2^(m-k), where Stirling1 is A048994.
  A028338 → x ↦ 2x−5
  1 · -2, 1 · 2, -3, 1 · 0, 2, -3, 1 … row sums 1, -1, 0, 0, 0, 0, 0, 0, …
  A094646 (−∞, 5/2] rational roots
• A274498 → θ+1
  1 · 3 · 3, 12 · 9, 36 … row sums 1, 3, 15, 45, 189, 567, 2187, 6561, …
  (−∞, 0] nonnegative rational roots
• A249248 → x ↦ x+1
  1 · 4, 3 · 11, 13, 3 · 38, 65, 31, 3 … row sums 1, 7, 27, 137, 681, 3687, 20595, …
  (−∞, −1] nonnegative
• A391045: Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.
  A172431 → ⊙ C(n,k+1) (shift -1)
  1 · 1, 1 · 1, 4, 1 · 1, 9, 10, 1 … row sums 1, 2, 6, 21, 80, 322, 1347, 5798, …
  A391045 (−∞, 0] nonnegative
• A137320 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 2, 6 · 0, 4, 20, 24 … row sums 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  [−1, 0] nonnegative rational roots
• A091044 → ⊙ C(n+2,k+1) → D
  1 · 5, 2 · 14, 28, 3 · 30, 168, 90, 4 … row sums 0, 1, 7, 45, 292, 1930, 12993, 88893, …
  (−∞, 0] nonnegative
• A062145 → reverse
  1 · 4, 1 · 10, 10, 1 · 20, 45, 18, 1 … row sums 1, 5, 21, 84, 330, 1287, 5005, 19448, …
  (−∞, 0] nonnegative
• A013610 → θ+1
  1 · 1, 6 · 1, 12, 27 · 1, 18, 81, 108 … row sums 1, 7, 40, 208, 1024, 4864, 22528, …
  (−∞, 0] nonnegative rational roots
• A081578 → D
  1 · 5, 2 · 9, 18, 3 · 13, 66, 39, 4 … row sums 0, 1, 7, 30, 122, 455, 1641, 5740, …
  (−∞, 0] nonnegative
• type_b_eulerian → γ (scale 1/4) → x ↦ x+1
  1 · 1 · 2, 1 · 6, 5 … row sums 1, 1, 3, 11, 57, 361, 2763, 24611, …
  (−∞, −1] nonnegative
• A178820 → basis A007318
  1 · 8, 4 · 40, 40, 10 · 160, 240, 120, 20 … row sums 1, 12, 90, 540, 2835, 13608, 61236, …
  (−∞, −1] nonnegative rational roots
• A145596 → D+1
  1 · 4, 2 · 11, 14, 3 · 24, 60, 32, 4 … row sums 1, 6, 28, 120, 495, 2002, 8008, 31824, …
  (−∞, 0] nonnegative
• A011973 → x ↦ x+2 (shift +1) → D
  1 · 2 · 7, 2 · 16, 6 … row sums 0, 1, 2, 9, 22, 68, 174, 475, …
  (−∞, 0] nonnegative
• A249248 → θ+1
  1 · 1, 6 · 1, 14, 9 · 1, 24, 66, 12 … row sums 1, 7, 24, 103, 419, 1815, 7970, 36277, …
  (−∞, 0] nonnegative
• basis-map coefficients of A297191, shifted → D+1
  1 · 0, 2, 1 · 0, 0, 0, 4, 1 · 0, 0, 0, 0, 0, 6, 1 … row sums 1, 3, 5, 7, 9, 11, 13, 15, …
  (−∞, 0] nonnegative rational roots
• A099040 → D
  2 · 2, 8 · 0, 16, 24 · 0, 8, 72, 64 … row sums 0, 2, 10, 40, 144, 488, 1592, 5056, …
  (−∞, 0] nonnegative
• A125662 → (θ+1)²
  1 · 3, 4 · 8, 24, 9 · 21, 100, 81, 16 … row sums 1, 7, 41, 218, 1091, 5237, 24379, …
  (−∞, 0] nonnegative
• A099040 → D+1
  1 · 2, 2 · 2, 10, 4 · 0, 16, 32, 8 … row sums 1, 4, 16, 56, 188, 608, 1920, 5952, …
  (−∞, 0] nonnegative
• chebyshev_t → x ↦ x+1
  1 · 1, 1 · 1, 4, 2 · 1, 9, 12, 4 … row sums 1, 2, 7, 26, 97, 362, 1351, 5042, …
  (−2, 0) nonnegative
• A143495 → x ↦ x/(x+1)
  1 · 3, 4 · 9, 25, 17 · 27, 118, 167, 77 … row sums 1, 7, 51, 389, 3121, 26423, 236403, …
  [−1, 0] nonnegative
• basis-map coefficients of A scaled Hermite triangle.
  0, 0, 1 · 0, 0, 0, 1 · 0, 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0] nonnegative rational roots
• A178822 → D+1
  1 · 12, 6 · 63, 84, 21 · 224, 504, 336, 56 … row sums 1, 18, 168, 1120, 6048, 28224, 118272, …
  (−∞, 0] nonnegative rational roots
• A122542 → D+1
  1 · 1, 1 · 2, 4, 1 · 2, 10, 7, 1 … row sums 1, 2, 7, 20, 57, 158, 431, 1160, …
  (−∞, 0] nonnegative
• A039756 → D+1
  1 · 2, 1 · 5, 6, 1 · 10, 35, 16, 1 … row sums 1, 3, 12, 62, 372, 2492, 18384, 147800, …
  (−∞, 0] nonnegative
• A102662 → x ↦ x+1
  1 · 4, 3 · 9, 11, 3 · 22, 38, 20, 3 … row sums 1, 7, 23, 83, 295, 1051, 3743, 13331, …
  (−∞, −1] nonnegative
• A007318 → basis A049020 → x ↦ x+1
  1 · 3, 1 · 11, 7, 1 · 47, 43, 12, 1 … row sums 1, 4, 19, 103, 622, 4117, 29521, …
  (−∞, −1] nonnegative