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1381 sequences

• A344565 → x ↦ x/(x+1)
  3 · 15, 30 · 45, 180, 180 · 105, 630, 1260, 840 … row sums 3, 45, 405, 2835, 17010, 91854, 459270, …
  [−1, 0] nonnegative rational roots
• A007318 → θ+1 → x ↦ x+1
  1 · 3, 2 · 8, 10, 3 · 20, 36, 21, 4 … row sums 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  (−∞, −1] nonnegative rational roots
• A108838 → ⊙ C(n+1,k+1) (shift +1) → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 13, 33, 22 … row sums 1, 3, 13, 69, 417, 2763, 19609, 146793, …
  [−1, 0] nonnegative
• A274498 → (θ+1)²
  1 · 3 · 3, 24 · 9, 72 … row sums 1, 3, 27, 81, 477, 1431, 7047, 21141, …
  (−∞, 0] nonnegative
• A143683 → θ+1
  1 · 1, 2 · 1, 20, 3 · 1, 38, 57, 4 … row sums 1, 3, 24, 100, 528, 2352, 11008, 48960, …
  (−∞, 0] nonnegative
• A120909 → (θ+1)²
  3 · 3, 24 · 3, 48, 108 · 3, 72, 324, 384 … row sums 3, 27, 159, 783, 3483, 14499, 57591, …
  (−∞, 0] nonnegative
• A023531 → 2D+1+2n → reverse
  1 · 3, 2 · 5, 4 · 7, 6 … row sums 1, 5, 9, 13, 17, 21, 25, 29, …
  (−∞, 0] nonnegative rational roots
• A121574 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 13, 10 · 8, 40, 64, 33 … row sums 1, 5, 27, 145, 779, 4185, 22483, …
  [−1, 0] nonnegative
• A063967 → x ↦ x−8
  1 · -7, 1 · 42, -13, 1 · -245, 119, -19, 1 … row sums 1, -6, 30, -144, 684, -3240, 15336, …
  (3, 7]
• A030528 → x ↦ x/(x+1)
  1 · 1, 2 · 0, 2, 3 · 0, 1, 5, 5 … row sums 1, 3, 5, 11, 21, 43, 85, 171, …
  [−1, 0] nonnegative
• A249251 → basis A007318
  1 · 4, 3 · 13, 18, 6 · 43, 87, 57, 12 … row sums 1, 7, 37, 199, 1069, 5743, 30853, …
  (−∞, −1] nonnegative
• A123519 → (θ+1)²
  1, 24, 36 · 1, 48, 180, 128 · 1, 80, 540, 896, 400 … row sums 61, 357, 1917, 9717, 47249, 222633, …
  (−∞, 0] nonnegative
• A267482 → x ↦ x+1
  1 · 2, 1 · 1, 3, 1 · -1, 3, 4, 1 … row sums 1, 3, 5, 7, 9, 11, 13, 15, …
  (−3, 1)
• delannoy → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 10, 15, 4 … row sums 1, 3, 10, 30, 87, 245, 676, 1836, …
  (−∞, 0] nonnegative
• A063967 → θ+1
  1 · 1, 2 · 2, 6, 3 · 3, 14, 15, 4 … row sums 1, 3, 11, 36, 116, 364, 1124, 3424, …
  (−∞, 0] nonnegative
• A249248 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 9, 11 · 1, 15, 49, 38 … row sums 1, 5, 21, 103, 519, 2793, 15645, 91623, …
  [−1, 0] nonnegative
• A124038 → x ↦ x+1
  1 · -1, 1 · -2, 0, 1 · -1, -3, 1, 1 … row sums 1, 0, -1, -2, -3, -4, -5, -6, …
  (−3, 2)
• A143494 → x ↦ x+1
  1 · 3, 1 · 10, 7, 1 · 37, 40, 12, 1 … row sums 1, 4, 18, 90, 494, 2946, 18926, 130066, …
  (−∞, −1] nonnegative
• A110441 → reverse
  1 · 1, 3 · 1, 6, 7 · 1, 9, 23, 15 … row sums 1, 4, 14, 48, 164, 560, 1912, 6528, …
  (−∞, 0] nonnegative
• A249248 → reverse
  1 · 3, 1 · 3, 7, 1 · 3, 22, 12, 1 … row sums 1, 4, 11, 38, 131, 490, 1897, 7714, …
  (−∞, 0] nonnegative
• A334218 → θ+1
  1 · 1 · 0, 4 · 0, 6, 9 … row sums 1, 1, 4, 15, 72, 420, 2880, 22680, …
  (−∞, 0] nonnegative
• A135065: A127733 * A007318 as infinite lower triangular matrices.
  A007318 → ×(1+2n+n²)
  1 · 4, 4 · 9, 18, 9 · 16, 48, 48, 16 … row sums 1, 8, 36, 128, 400, 1152, 3136, 8192, …
  A135065 (−∞, −1] nonnegative rational roots
• A008292 → γ (scale 1/4) (shift +1) → θ+1
  1 · 2 · 4, 4 · 8, 32 … row sums 1, 2, 8, 40, 240, 1680, 13440, 120960, …
  (−∞, 0] nonnegative
• A063967 → x ↦ x−12
  1 · -11, 1 · 110, -21, 1 · -1089, 319, -31, 1 … row sums 1, -10, 90, -800, 7100, -63000, 559000, …
  (7, 11]
• A344565 → (θ+1)²
  3 · 15, 60 · 45, 360, 405 · 105, 1260, 2835, 1680 … row sums 3, 75, 810, 5880, 33600, 163296, …
  (−∞, 0] nonnegative
• A288874: Row reversed version of triangle A201637 (second-order Eulerian triangle).
  A201637 → reverse (shift +1)
  1 · 0, 1 · 0, 2, 1 · 0, 6, 8, 1 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  A288874 (−∞, 0] nonnegative
• basis-map coefficients of A026729, shifted → (θ+1)²
  1 · 0, -4 · 0, 0, 9 · 0, 0, 0, -16 … row sums 1, -4, 9, -16, 25, -36, 49, -64, …
  (−∞, 0] rational roots
• A274496 → θ+1
  1 · 2 · 2, 4 · 4, 8 … row sums 1, 2, 6, 12, 32, 64, 160, 320, …
  (−∞, 0] nonnegative rational roots
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A010054 (−∞, −1] [−1, 0] [−3−2√2, −3+2√2] nonnegative rational roots
• A026729: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
  1 · 0, 1 · 0, 1, 1 · 0, 0, 2, 1 … row sums 1, 1, 2, 3, 5, 8, 13, 21, …
  A026729 (−4, 0] nonnegative
• A391533 → x ↦ x+1
  9, 6, 1 · 18, 21, 8, 1 · 36, 60, 37, 10, 1 … row sums 16, 48, 144, 432, 1296, 3888, 11664, …
  (−∞, −1] nonnegative rational roots
• A091044 → ⊙ C(n+2,k+1) → D+1
  1 · 2, 1 · 6, 7, 1 · 15, 42, 17, 1 … row sums 1, 3, 14, 75, 438, 2702, 17324, 114291, …
  (−∞, 0] nonnegative
• A116414 → x ↦ x/(x+1)
  1 · 4, 5 · 13, 34, 22 · 40, 162, 216, 95 … row sums 1, 9, 69, 513, 3789, 27945, 206037, …
  [−1, 0) nonnegative
• A081578 → θ+1
  1 · 1, 2 · 1, 10, 3 · 1, 18, 27, 4 … row sums 1, 3, 14, 50, 183, 637, 2188, 7380, …
  (−∞, 0] nonnegative
• A007318 → basis A049020 → D+1
  1 · 3, 1 · 10, 7, 1 · 37, 40, 12, 1 … row sums 1, 4, 18, 90, 494, 2946, 18926, 130066, …
  (−∞, 0] nonnegative
• A007318 → (θ+1)² → x ↦ x+1
  1 · 5, 4 · 18, 26, 9 · 56, 114, 75, 16 … row sums 1, 9, 53, 261, 1161, 4833, 19197, …
  (−∞, −1] nonnegative
• A152828 → D+1
  1 · 1 · 1 · 2, 1 … row sums 1, 1, 1, 3, 3, 3, 8, 8, …
  (−∞, 0] nonnegative rational roots
• A082985 → θ+1
  1 · 1, 6 · 1, 10, 15 · 1, 14, 42, 28 … row sums 1, 7, 26, 85, 265, 804, 2393, 7019, …
  (−∞, 0] nonnegative
• A391533 → reverse
  1, 4, 4 · 1, 5, 8, 4 · 1, 6, 13, 12, 4 … row sums 9, 18, 36, 72, 144, 288, 576, 1152, …
  (−∞, 0] nonnegative rational roots
• A154986 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 14, 36, 24 … row sums 1, 3, 13, 75, 537, 4611, 46053, 525483, …
  [−1, 0] nonnegative
• A030528 → (θ+1)²
  1 · 1, 4 · 0, 8, 9 · 0, 4, 27, 16 … row sums 1, 5, 17, 47, 116, 266, 579, 1213, …
  (−∞, 0] nonnegative
• A039756 → D
  1 · 4, 2 · 9, 26, 3 · 16, 116, 120, 4 … row sums 0, 1, 6, 38, 256, 1844, 14296, 119160, …
  (−∞, 0] nonnegative
• A172431 → ⊙ C(n,k+1) (shift -1) → reverse
  1 · 1, 1 · 1, 4, 1 · 1, 10, 9, 1 … row sums 1, 2, 6, 21, 80, 322, 1347, 5798, …
  (−∞, 0] nonnegative
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x/(x+1)
  1 · 3, 6 · 6, 24, 24 · 10, 60, 120, 80 … row sums 1, 9, 54, 270, 1215, 5103, 20412, …
  [−1, 0] nonnegative rational roots
• A124038 → D+1
  1 · -1, 1 · -3, 0, 1 · 0, -6, 1, 1 … row sums 1, 0, -2, -4, -3, 2, 7, 6, …
  (−∞, ∞)
• A108838 → x ↦ x+1
  2 · 5, 2 · 14, 12, 2 · 42, 56, 21, 2 … row sums 2, 7, 28, 121, 550, 2591, 12536, 61921, …
  (−∞, −1] nonnegative
• A159854 → θ+1
  0, 0, 3 · 0, 0, 3, 4 · 0, 0, 3, 8, 5 … row sums 3, 7, 16, 36, 80, 176, 384, 832, …
  (−∞, 0] nonnegative rational roots
• basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
  1 · 0, 2 · -3, 3, 3 · 0, -12, 12, 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 1]
• A081577 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 16, 24, 10, 1 … row sums 1, 3, 13, 51, 205, 819, 3277, 13107, …
  (−∞, −1] nonnegative
• A169803 → D
  1 · 2 · 3, 2 · 4, 6 … row sums 0, 1, 2, 5, 10, 20, 38, 71, …
  (−∞, 0] nonnegative