Proof methods / preserver_image

Image under a real-rootedness preserver

1,387 sequences

• A143685 → D
  1 · 11, 2 · 21, 42, 3 · 31, 282, 93, 4 … Σ 0, 1, 13, 66, 410, 2015, 10371, 49588, …
  (−∞, 0]nonnegative
• A110441 → (θ+1)²
  1 · 3, 4 · 7, 24, 9 · 15, 92, 81, 16 … Σ 1, 7, 40, 204, 968, 4372, 19048, 80736, …
  (−∞, 0]nonnegative
• A027465: Cube of lower triangular normalized binomial matrix.
  1 · 3, 1 · 9, 6, 1 · 27, 27, 9, 1 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  (−∞, −1]nonnegativerational roots
• A085880 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 8, 8 · 5, 30, 60, 40 … Σ 1, 3, 18, 135, 1134, 10206, 96228, …
  [−1, 0]nonnegativerational roots
• A135838 → D
  2 · 4, 4 · 12, 24, 12 · 16, 48, 48, 16 … Σ 0, 2, 8, 48, 128, 640, 1536, 7168, …
  (−∞, 0]nonnegativerational roots
• A081577 → D+1
  1 · 2, 1 · 5, 6, 1 · 8, 21, 10, 1 … Σ 1, 3, 12, 40, 132, 420, 1312, 4032, …
  (−∞, 0]nonnegative
• A272866 → θ+1
  1 · 1, 6, 3 · 1, 12, 33, 24, 5 … Σ 1, 10, 75, 500, 3125, 18750, 109375, …
  (−∞, 0]nonnegative
• A104684 → θ+1
  1 · 2, 2 · 6, 12, 3 · 20, 60, 36, 4 … Σ 1, 4, 21, 120, 705, 4188, 24997, …
  (−∞, 0]nonnegative
• A049020 → D
  1 · 3, 2 · 10, 12, 3 · 37, 62, 30, 4 … Σ 0, 1, 5, 25, 133, 761, 4677, 30745, …
  (−∞, −1]nonnegative
• A038208 → x ↦ x/(x+1)
  1 · 2, 4 · 4, 16, 16 · 8, 48, 96, 64 … Σ 1, 6, 36, 216, 1296, 7776, 46656, …
  [−1, 0]nonnegativerational roots
• A059340: Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.
  A008278 → x ↦ x+1 (shift +1)
  1 · 2, 1 · 5, 5, 1 · 15, 23, 10, 1 … Σ 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  (−∞, −1]nonnegative
• A127773 → θ+1
  1 · 0, 6 · 0, 0, 18 · 0, 0, 0, 40 … Σ 1, 6, 18, 40, 75, 126, 196, 288, …
  (−∞, 0]nonnegativerational roots
• A136572 → D
  1 · 0, 4 · 0, 0, 18 · 0, 0, 0, 96 … Σ 0, 1, 4, 18, 96, 600, 4320, 35280, …
  (−∞, 0]nonnegativerational roots
• A318772 → x ↦ (1/243)x → basis A007318
  1 · 3 · 9 · 27 … Σ 1, 3, 9, 27, 83, 255, 783, 2403, …
  (−∞, −1]nonnegative
• A391403 → (θ+1)²
  1 · 1 · 1, 8 · 1, 24 … Σ 1, 1, 9, 25, 103, 351, 1251, 4299, …
  (−∞, 0]nonnegative
• A133080 → D+1
  1 · 2, 1 · 0, 2, 1 · 0, 2, 4, 1 … Σ 1, 3, 3, 7, 5, 11, 7, 15, …
  (−∞, 0]nonnegative
• A030528 → D+1
  1 · 2, 1 · 2, 4, 1 · 1, 7, 6, 1 … Σ 1, 3, 7, 15, 30, 58, 109, 201, …
  (−∞, 0]nonnegative
• A118394 → basis A007318
  1 · 1 · 1 · 7, 6 … Σ 1, 1, 1, 13, 49, 121, 1681, 10501, …
  (−∞, −1]nonnegative
• A136124: Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1;…
  1 · 2, 1 · 6, 5, 1 · 24, 26, 9, 1 … Σ 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  (−∞, −1]nonnegativerational roots
• A114687 → θ+1
  1 · 1, 4 · 1, 12, 12 · 1, 24, 72, 32 … Σ 1, 5, 25, 129, 681, 3653, 19825, …
  (−∞, 0]nonnegative
• A056242 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 10 · 1, 12, 37, 34 … Σ 1, 4, 18, 84, 396, 1872, 8856, 41904, …
  [−1, 0]nonnegative
• A113953 → x ↦ x−12
  1 · -12, 1 · 120, -22, 1 · -1152, 336, -32, 1 … Σ 1, -11, 99, -847, 7139, -59895, 501787, …
  (4, 12]
• A216973 → D+1
  1 · 4, 2 · 9, 12, 3 · 16, 36, 24, 4 … Σ 0, 1, 6, 24, 80, 240, 672, 1792, …
  (−∞, 0]nonnegativerational roots
• A178822 → θ+1
  1 · 6, 12 · 21, 84, 63 · 56, 336, 504, 224 … Σ 1, 18, 168, 1120, 6048, 28224, 118272, …
  (−∞, 0]nonnegativerational roots
• A135838 → θ+1
  1 · 2, 4 · 2, 8, 6 · 4, 24, 36, 16 … Σ 1, 6, 16, 80, 192, 896, 2048, 9216, …
  (−∞, 0]nonnegativerational roots
• A229032 → D+1
  1 · 2 · 7, 4 · 20, 16 … Σ 1, 2, 11, 36, 133, 454, 1551, 5192, …
  (−∞, 0]nonnegative
• A111596: The matrix inverse of the unsigned Lah numbers A271703.
  A059110 → x ↦ −x−1
  1 · 0, 1 · 0, -2, 1 · 0, 6, -6, 1 … Σ 1, 1, -1, 1, 1, -19, 151, -1091, …
  [−1, ∞)
• A054142 → D
  1 · 3, 2 · 5, 12, 3 · 7, 30, 30, 4 … Σ 0, 1, 5, 20, 71, 235, 744, 2285, …
  (−∞, 0]nonnegative
• A114655 → D
  2 · 12, 4 · 48, 48, 6 · 160, 320, 120, 8 … Σ 0, 2, 16, 102, 608, 3530, 20256, …
  (−∞, 0]nonnegative
• A099097 → x ↦ x−5
  1 · -15, 3 · 220, -89, 9 · -3225, 1965, -399, 27 … Σ 1, -12, 140, -1632, 19024, -221760, …
  (41/9, 5]
• A063007 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 8, 13 · 1, 15, 57, 63 … Σ 1, 4, 22, 136, 886, 5944, 40636, …
  [−1, 0)nonnegative
• A116414 → reverse
  1 · 1, 4 · 1, 8, 13 · 1, 12, 42, 40 … Σ 1, 5, 22, 95, 409, 1760, 7573, 32585, …
  (−∞, 0]nonnegative
• A136572 → D+1
  1 · 1, 1 · 0, 4, 2 · 0, 0, 18, 6 … Σ 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, 0]nonnegativerational roots
• A122431 → D+1
  4, 9, 6, 1 · 1, 7, 12, 7, 1 · 0, 2, 10, 15, 8, 1 … Σ 20, 28, 36, 44, 52, 60, 68, 76, …
  (−∞, 0]nonnegative
• A099097 → x ↦ x−4
  1 · -12, 3 · 140, -71, 9 · -1632, 1248, -318, 27 … Σ 1, -9, 78, -675, 5841, -50544, 437373, …
  (32/9, 4]
• basis-map coefficients of A112227 → D+1
  0, 2, 1 · 0, 0, 3, 1 · 0, 0, 0, 4, 1 … Σ 3, 4, 5, 6, 7, 8, 9, 10, …
  (−∞, 0]nonnegativerational roots
• A227450 → D+1
  2 · 6, 4 · 14, 28, 8 · 30, 124, 96, 16 … Σ 2, 10, 50, 266, 1522, 9354, 61490, …
  (−∞, 0]nonnegative
• A266213 → θ+1
  1 · 1 · 1, 4 · 1, 8, 6 … Σ 1, 1, 5, 15, 45, 129, 361, 991, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729, shifted → basis A007318
  1 · -1, -1 · 1, 2, 1 · -1, -3, -3, -1 … Σ 1, -2, 4, -8, 16, -32, 64, -128, …
  (−∞, −1]rational roots
• A249252 → reverse
  1 · 4, 1 · 8, 8, 1 · 16, 32, 12, 1 … Σ 1, 5, 17, 61, 217, 773, 2753, 9805, …
  (−∞, 0]nonnegative
• A103633 → θ+1
  1 · 0, 2 · 0, 2, 3 · 0, 0, 3, 4 … Σ 1, 2, 5, 7, 16, 20, 44, 52, …
  (−∞, 0]nonnegativerational roots
• A059344 → D+1
  1 · 1 · 3, 2 · 7, 6 … Σ 1, 1, 5, 13, 61, 221, 1081, 4705, …
  (−∞, 0]nonnegative
• A051338: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 5.
  1 · -6, 1 · 42, -13, 1 · -336, 146, -21, 1 … Σ 1, -5, 30, -210, 1680, -15120, 151200, …
  [11/2, ∞)rational roots
• A048993 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 1, 5, 5 … Σ 1, 1, 3, 11, 49, 257, 1539, 10299, …
  [−1, 0]nonnegative
• A091044 → x ↦ x/(x+1)
  1 · 2, 4 · 3, 16, 16 · 4, 40, 96, 64 … Σ 1, 6, 35, 204, 1189, 6930, 40391, …
  [−1, 0]nonnegative
• A063967 → x ↦ x−3
  1 · -2, 1 · 2, -3, 1 · 0, 4, -4, 1 … Σ 1, -1, 0, 1, -1, 0, 1, -1, …
  (−2, 2]
• A066325 → D
  1 · 0, 2 · -3, 0, 3 · 0, -12, 0, 4 … Σ 0, 1, 2, 0, -8, -10, 36, 112, …
  (−∞, ∞)
• A133087 → reverse
  1 · 1, 2 · 1, 2, 1 · 1, 4, 5, 2 … Σ 1, 3, 4, 12, 16, 48, 64, 192, …
  (−∞, 0]nonnegativerational roots
• A033282 → (θ+1)²
  1 · 1, 8 · 1, 20, 45 · 1, 36, 189, 224 … Σ 1, 9, 66, 450, 2955, 18963, 119812, …
  (−∞, 0]nonnegative
• A114687 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 8, 11 · 1, 15, 51, 45 … Σ 1, 4, 20, 112, 672, 4224, 27456, …
  [−1, 0]nonnegative