Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A257613: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 4.
  1 · 4, 4 · 16, 48, 16 · 64, 416, 416, 64 … Σ 1, 8, 80, 960, 13440, 215040, 3870720, …
  (−∞, 0)nonnegative
• A130123 → basis A008277 → reverse
  1 · 2, 2 · 4, 12, 4 · 8, 48, 56, 8 … Σ 1, 4, 20, 120, 832, 6496, 56128, …
  (−∞, 0]nonnegative
• A102537 → x ↦ x+1
  1 · 4, 3 · 21, 32, 12 · 126, 290, 220, 55 … Σ 1, 7, 65, 691, 7953, 96447, 1213761, …
  (−∞, −1]nonnegative
• A172431 → reverse → (θ+1)²
  1 · 2, 4 · 3, 16, 9 · 4, 40, 54, 16 … Σ 1, 6, 28, 114, 427, 1512, 5141, 16950, …
  (−∞, 0]nonnegative
• A143491 → (θ+1)²
  6, 20, 9 · 24, 104, 81, 16 · 120, 616, 639, 224, 25 … Σ 35, 225, 1624, 13132, 118124, 1172700, …
  (−∞, 0]nonnegative
• A280580 → D+1
  1 · 2, 1 · 8, 8, 1 · 35, 60, 18, 1 … Σ 1, 3, 17, 114, 831, 6363, 50299, …
  (−∞, 0]nonnegative
• A038305 → basis A013611 → reverse
  144, 312, 169 · 1728, 5616, 6084, 2197 … Σ 625, 15625, 390625, 9765625, 244140625, …
  (−∞, 0]nonnegativerational roots
• A125092 → D+1 → D
  4 · 26, 18 · 66, 150, 48 · 124, 492, 492, 100 … Σ 0, 4, 44, 264, 1208, 4720, 16608, …
  (−∞, 0]nonnegative
• A038335 → basis A013610 → reverse
  1 · 27, 21 · 729, 1134, 441 … Σ 1, 48, 2304, 110592, 5308416, …
  (−∞, 0]nonnegativerational roots
• A038335 → x ↦ x+1 → basis A007318
  1 · 30, 9 · 900, 540, 81 · 27000, 24300, 7290, 729 … Σ 1, 39, 1521, 59319, 2313441, 90224199, …
  (−∞, −1]nonnegativerational roots
• A059473 → basis A013609 → x ↦ x/(x+1)
  1 · 4, 8 · 18, 72, 70 · 80, 480, 944, 608 … Σ 1, 12, 160, 2112, 27904, 368640, …
  [−1, 0]nonnegative
• A038221 → basis A013610 → θ+1
  1 · 6, 18 · 36, 216, 243 · 216, 1944, 4374, 2916 … Σ 1, 24, 495, 9450, 172125, 3037500, …
  (−∞, 0]nonnegativerational roots
• A013609 → Veronese 2 → (θ+1)²
  1 · 1 · 1, 16 · 1, 48 … Σ 1, 1, 17, 49, 241, 881, 3425, 12545, …
  (−∞, 0]nonnegative
• A137320 → (θ+1)²
  1 · 0, 8 · 0, 8, 36 · 0, 16, 108, 128 … Σ 1, 8, 44, 252, 1612, 11624, 94040, …
  (−∞, 0]nonnegative
• A085880 → basis A013611 → D
  4 · 32, 64 · 240, 960, 960 … Σ 0, 4, 96, 2160, 48384, 1088640, …
  (−∞, 0]nonnegativerational roots
• A069466 → D+1
  1 · 4, 2 · 30, 36, 6 · 200, 540, 240, 20 … Σ 1, 6, 72, 1000, 14700, 222264, 3415104, …
  (−∞, 0]nonnegative
• A114656 → reverse
  1 · 1, 2 · 1, 6, 4 · 1, 12, 24, 8 … Σ 1, 3, 11, 45, 197, 903, 4279, 20793, …
  (−∞, 0]nonnegative
• A071951 → (θ+1)²
  1 · 2, 4 · 4, 32, 9 · 8, 208, 180, 16 … Σ 1, 6, 45, 412, 4589, 61450, 971577, …
  (−∞, 0]nonnegative
• A130123 → Veronese 2 → D
  4 · 0, 32 · 0, 0, 192 · 0, 0, 0, 1024 … Σ 0, 0, 4, 0, 32, 0, 192, 0, …
  (−∞, 0]nonnegativerational roots
• A081579 → basis A013609 → x ↦ x/(x+1)
  1 · 2, 4 · 8, 32, 28 · 24, 144, 272, 160 … Σ 1, 6, 68, 600, 5776, 53856, 507968, …
  [−1, 0]nonnegative
• A145596 → D → D+1
  2 · 14, 6 · 60, 64, 12 · 190, 390, 180, 20 … Σ 0, 2, 20, 136, 780, 4070, 20020, 94640, …
  (−∞, 0]nonnegative
• A142459 → θ+1
  1 · 1, 2 · 1, 20, 3 · 1, 118, 177, 4 … Σ 1, 3, 24, 300, 5040, 105840, 2661120, …
  (−∞, 0]nonnegative
• A008459 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 18, 13 · 20, 90, 132, 63 … Σ 1, 5, 37, 305, 2641, 23525, 213445, …
  [−1, 0]nonnegative
• A391533 → Pₙ + Pₙ₊₁ → D
  12, 12, 3 · 20, 36, 21, 4 · 28, 76, 75, 32, 5 … Σ 27, 81, 216, 540, 1296, 3024, 6912, …
  (−∞, 0]nonnegativerational roots
• A143491 → basis A013610 → x ↦ x+1
  42, 39, 9 · 336, 438, 189, 27 … Σ 90, 990, 11880, 154440, 2162160, …
  (−∞, −1]nonnegativerational roots
• A038336 → x ↦ x/(x+1) → θ+1
  144, 1056, 1452 · 1728, 19008, 52272, 42592 … Σ 2652, 115600, 4795088, 192432384, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A026729 → basis A013611 → D+1
  1 · 5, 4 · 9, 40, 16 · 13, 108, 240, 64 … Σ 1, 9, 65, 425, 2625, 15625, 90625, …
  (−∞, 0]nonnegativerational roots
• A225470 → basis A013610 → reverse
  1 · 3, 3 · 9, 27, 18 · 27, 162, 297, 162 … Σ 1, 6, 54, 648, 9720, 174960, 3674160, …
  (−∞, 0]nonnegativerational roots
• A082985 → x ↦ x+1 → D+1
  1 · 7, 3 · 26, 25, 5 · 85, 126, 56, 7 … Σ 1, 10, 56, 274, 1276, 5798, 25936, …
  (−∞, 0]nonnegative
• A269951 → reverse
  1 · 1 · 1, 2 · 1, 5, 5 … Σ 1, 1, 3, 11, 49, 261, 1631, 11743, …
  (−∞, 0]nonnegative
• chebyshev_t → x ↦ x+1 → (θ+1)²
  1 · 1, 4 · 1, 16, 18 · 1, 36, 108, 64 … Σ 1, 5, 35, 209, 1137, 5817, 28483, …
  (−∞, 0]nonnegative
• A038319 → basis A013609 → x ↦ x/(x+1)
  256, 832, 676 · 4096, 19968, 32448, 17576 … Σ 1764, 74088, 3111696, 130691232, …
  [−1, 0]nonnegativerational roots
• A063007 → basis A013610 → reverse
  1 · 6, 3 · 54, 54, 13 · 540, 810, 396, 63 … Σ 1, 9, 121, 1809, 28401, 458649, …
  (−∞, 0]nonnegative
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
  1 · 3, 5 · 9, 30, 25 · 27, 135, 225, 125 … Σ 1, 8, 64, 512, 4096, 32768, 262144, …
  (−∞, 0)[−1, 0]nonnegative+1
• A120909 → D+1
  3 · 9, 6 · 15, 36, 12 · 21, 90, 108, 24 … Σ 3, 15, 63, 243, 891, 3159, 10935, …
  (−∞, 0]nonnegativerational roots
• A099097 → basis A013609 → reverse
  1 · 6, 3 · 36, 38, 10 · 216, 348, 186, 33 … Σ 1, 9, 84, 783, 7299, 68040, 634257, …
  (−∞, 0]nonnegative
• A286718 → basis A013610 → θ+1
  1 · 2, 6 · 10, 42, 27 · 80, 396, 405, 108 … Σ 1, 8, 79, 989, 15166, 276302, 5840200, …
  (−∞, 0]nonnegative
• A257606: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4.
  1 · 4, 4 · 16, 40, 16 · 64, 296, 296, 64 … Σ 1, 8, 72, 720, 7920, 95040, 1235520, …
  (−∞, 0)nonnegative
• A011973 → basis A008277 → θ+1
  1 · 1 · 2, 2 · 3, 4 … Σ 1, 1, 4, 7, 20, 43, 113, 267, …
  (−∞, 0]nonnegative
• A166960 → (θ+1)²
  1 · 1, 4 · 1, 24, 9 · 1, 108, 189, 16 … Σ 1, 5, 34, 314, 3832, 59084, 1119868, …
  (−∞, 0]nonnegative
• A207823 → D
  1 · 8, 2 · 46, 24, 3 · 232, 186, 48, 4 … Σ 0, 1, 10, 73, 470, 2828, 16310, 91371, …
  (−∞, 0]nonnegative
• A038320 → x ↦ x+1 → reverse
  36, 204, 289 · 216, 1836, 5202, 4913 … Σ 529, 12167, 279841, 6436343, 148035889, …
  (−∞, 0]nonnegativerational roots
• A094816 → basis A013610 → D+1
  1 · 5, 3 · 20, 33, 9 · 85, 231, 162, 27 … Σ 1, 8, 62, 505, 4433, 42248, 437338, …
  (−∞, 0]nonnegative
• A038291 → basis A013609 → (θ+1)²
  1 · 10, 8 · 100, 160, 36 · 1000, 2400, 1080, 128 … Σ 1, 18, 296, 4608, 69120, 1009152, …
  (−∞, 0]nonnegative
• A038336 → basis A013609
  484, 880, 400 · 10648, 29040, 26400, 8000 … Σ 1764, 74088, 3111696, 130691232, …
  (−∞, 0]nonnegativerational roots
• A290319 → x ↦ x+1 → θ+1
  1 · 2, 2 · 12, 16, 3 · 120, 184, 54, 4 … Σ 1, 4, 31, 362, 5661, 111024, 2619387, …
  (−∞, 0]nonnegative
• A225469 → x ↦ x+1
  1 · 4, 1 · 20, 12, 1 · 128, 124, 24, 1 … Σ 1, 5, 33, 277, 2785, 32197, 418113, …
  (−∞, −1]nonnegative
• A056939 → x ↦ x+1 → D
  1 · 6, 2 · 33, 26, 3 · 184, 232, 72, 4 … Σ 0, 1, 8, 62, 492, 4035, 34128, 296548, …
  (−∞, 0]nonnegative
• A391533 → x ↦ x+1 → x ↦ x/(x+1)
  9, 24, 16 · 18, 75, 104, 48 · 36, 204, 433, 408, 144 … Σ 49, 245, 1225, 6125, 30625, 153125, …
  [−1, 0]nonnegativerational roots
• A075497 → reverse → D+1
  1 · 3, 2 · 7, 14, 4 · 13, 68, 52, 8 … Σ 1, 5, 25, 141, 901, 6393, 49589, …
  (−∞, 0]nonnegative