Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A286718 → basis A013609 → reverse
  1 · 2, 2 · 4, 14, 10 · 8, 60, 132, 80 … Σ 1, 4, 28, 280, 3640, 58240, 1106560, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A026729, shifted → basis A013610 → θ+1
  1 · -1, -6 · 1, 12, 27 · -1, -18, -81, -108 … Σ 1, -7, 40, -208, 1024, -4864, 22528, …
  (−∞, 0]rational roots
• A030528 → basis A013611 → reverse
  1 · 4, 2 · 16, 16, 3 · 64, 96, 40, 5 … Σ 1, 6, 35, 205, 1200, 7025, 41125, …
  (−∞, 0]nonnegative
• A110441 → D
  1 · 6, 2 · 23, 18, 3 · 72, 96, 36, 4 … Σ 0, 1, 8, 44, 208, 908, 3776, 15200, …
  (−∞, 0]nonnegative
• A008957 → D
  1 · 5, 2 · 14, 42, 3 · 30, 294, 255, 4 … Σ 0, 1, 7, 59, 583, 6902, 97298, 1604561, …
  (−∞, 0]nonnegative
• A116414 → basis A013611 → reverse
  1 · 4, 5 · 16, 40, 22 · 64, 240, 276, 95 … Σ 1, 9, 78, 675, 5841, 50544, 437373, …
  (−∞, 0]nonnegative
• A136124 → x ↦ x+1 → basis A007318
  1 · 4, 1 · 20, 9, 1 · 120, 74, 15, 1 … Σ 1, 5, 30, 210, 1680, 15120, 151200, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A026729, shifted → basis A013609 → D
  -2 · 4, 8 · -6, -24, -24 · 8, 48, 96, 64 … Σ 0, -2, 12, -54, 216, -810, 2916, …
  (−∞, 0]rational roots
• A131431 → D+1 → reverse
  1 · 4, 4 · 7, 14 · 10, 30 … Σ 1, 8, 21, 40, 65, 96, 133, 176, …
  (−∞, 0]nonnegativerational roots
• A178822 → x ↦ x+1 → θ+1
  1 · 12, 12 · 84, 168, 63 · 448, 1344, 1008, 224 … Σ 1, 24, 315, 3024, 23814, 163296, …
  (−∞, 0]nonnegativerational roots
• A207824 → basis A013609 → basis A007318
  1 · 8, 2 · 63, 32, 4 · 496, 380, 96, 8 … Σ 1, 10, 99, 980, 9701, 96030, 950599, …
  (−5, −3)nonnegative
• A039755 → x ↦ x+1 → D
  1 · 6, 2 · 34, 24, 3 · 208, 224, 60, 4 … Σ 0, 1, 8, 61, 496, 4361, 41400, 422437, …
  (−∞, 0]nonnegative
• A131431 → D → θ+1
  4 · 0, 28 · 0, 0, 90 · 0, 0, 0, 208 … Σ 0, 4, 28, 90, 208, 400, 684, 1078, …
  (−∞, 0]nonnegativerational roots
• A114687 → basis A013611 → θ+1
  1 · 3, 16 · 11, 112, 192 · 45, 672, 2304, 2048 … Σ 1, 19, 315, 5069, 81381, 1313271, …
  (−∞, 0]nonnegative
• A207823 → basis A013611 → θ+1
  1 · 5, 8 · 24, 80, 48 · 115, 584, 720, 256 … Σ 1, 13, 152, 1675, 17767, 183504, …
  (−∞, 0]nonnegative
• A257614: Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*…
  1 · 2, 2 · 4, 28, 4 · 8, 244, 244, 8 … Σ 1, 4, 36, 504, 9576, 229824, 6664896, …
  (−∞, 0)nonnegative
• delannoy → Veronese 2 → x ↦ x+1
  1 · 1 · 2, 1 · 6, 5 … Σ 1, 1, 3, 11, 31, 87, 255, 743, …
  (−∞, −1]nonnegative
• A038330 → basis A007318
  256, 128, 16 · 4096, 3072, 768, 64 … Σ 400, 8000, 160000, 3200000, 64000000, …
  (−∞, −1]nonnegativerational roots
• A001263 → basis A013609 → x ↦ x/(x+1)
  1 · 2, 4 · 5, 20, 19 · 14, 84, 162, 100 … Σ 1, 6, 44, 360, 3152, 28896, 273856, …
  [−1, 0]nonnegative
• A091044 → ⊙ C(n+2,k+1) → x ↦ x+1
  1 · 2, 1 · 7, 7, 1 · 30, 45, 17, 1 … Σ 1, 3, 15, 93, 653, 4983, 40339, 341337, …
  (−∞, −1]nonnegative
• A166972 → (θ+1)²
  1 · 1, 4 · 1, 40, 9 · 1, 332, 369, 16 … Σ 1, 5, 50, 718, 14640, 394356, 13383372, …
  (−∞, 0]nonnegative
• A063967 → basis A013611 → x ↦ x+1
  1 · 6, 4 · 42, 52, 16 · 288, 528, 320, 64 … Σ 1, 10, 110, 1200, 13100, 143000, …
  (−∞, −1]nonnegative
• A038337 → basis A013609 → D
  1012, 968 · 34914, 66792, 31944 … Σ 1980, 133650, 8019000, 451068750, …
  (−∞, 0]nonnegativerational roots
• A136532 → x ↦ x+1
  -20, -8, 4 · -80, 20, 35, -5 · -192, 456, 174, -84, 6 … Σ -24, -30, 360, 4410, 33120, 169290, …
  (−∞, ∞)
• A143491 → basis A007318 → θ+1
  12, 14, 3 · 60, 94, 36, 4 · 360, 684, 357, 72, 5 … Σ 29, 194, 1478, 12664, 120696, 1267440, …
  (−∞, 0]nonnegative
• A091042 → Veronese 2 → θ+1
  1 · 1 · 1, 10 · 1, 70 … Σ 1, 1, 11, 71, 280, 1156, 5344, 23856, …
  (−∞, 0]nonnegative
• A059473 → basis A013609 → (θ+1)²
  1 · 4, 16 · 18, 144, 144 · 80, 960, 2016, 1024 … Σ 1, 20, 306, 4080, 50196, 585072, …
  (−∞, 0]nonnegative
• A013622 → basis A013611 → basis A007318
  1 · 28, 20 · 784, 1120, 400 … Σ 1, 48, 2304, 110592, 5308416, …
  (−∞, −1]nonnegativerational roots
• A131689 → x ↦ x+1
  1 · 1, 1 · 3, 5, 2 · 13, 31, 24, 6 … Σ 1, 2, 10, 74, 730, 9002, 133210, …
  (−2, −1]nonnegative
• A038277 → x ↦ x/(x+1) → (θ+1)²
  1 · 7, 72 · 49, 1008, 2916 · 343, 10584, 61236, 93312 … Σ 1, 79, 3973, 165475, 6195625, …
  (−∞, 0]nonnegative
• A130123 → (θ+1)² → (θ+1)²
  1 · 0, 32 · 0, 0, 324 · 0, 0, 0, 2048 … Σ 1, 32, 324, 2048, 10000, 41472, 153664, …
  (−∞, 0]nonnegativerational roots
• A099040 → (θ+1)²
  1 · 0, 8 · 0, 8, 36 · 0, 0, 72, 128 … Σ 1, 8, 44, 200, 820, 3136, 11424, 40128, …
  (−∞, 0]nonnegative
• A159854 → basis A013610 → D
  6, 18 · 15, 72, 81 · 36, 234, 486, 324 … Σ 24, 168, 1080, 6600, 39000, 225000, …
  (−∞, 0]nonnegativerational roots
• A038320 → reverse
  36, 132, 121 · 216, 1188, 2178, 1331 … Σ 289, 4913, 83521, 1419857, 24137569, …
  (−∞, 0]nonnegativerational roots
• A114656 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 6 · 11, 46, 44 · 45, 288, 576, 360 … Σ 1, 9, 101, 1269, 17081, 240849, …
  [−1, 0]nonnegative
• A038328 → x ↦ x+1
  196, 56, 4 · 2744, 1176, 168, 8 … Σ 256, 4096, 65536, 1048576, 16777216, …
  (−∞, −1]nonnegativerational roots
• A144440: Triangle T(n,k) by rows: T(n, k) = (3*n-3*k+1)*T(n-1, k-1) +(3*k-2)*T(n-1, k) + 3*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 11, 1 · 1, 54, 54, 1 … Σ 1, 2, 13, 110, 1249, 17816, 306619, …
  (−∞, 0)nonnegative
• A130123 → D → D
  8 · 0, 48 · 0, 0, 192 · 0, 0, 0, 640 … Σ 0, 0, 8, 48, 192, 640, 1920, 5376, …
  (−∞, 0]nonnegativerational roots
• A062264 → reverse
  1 · 5, 1 · 15, 12, 1 · 35, 63, 21, 1 … Σ 1, 6, 28, 120, 495, 2002, 8008, 31824, …
  (−∞, 0]nonnegative
• A038337 → basis A013609 → θ+1
  529, 2024, 1452 · 12167, 69828, 100188, 42592 … Σ 4005, 224775, 12119625, 635596875, …
  (−∞, 0]nonnegativerational roots
• A245962 → D
  2 · 3 · 8, 4 · 15, 10 … Σ 0, 0, 2, 3, 12, 25, 66, 147, …
  (−∞, 0]nonnegative
• A113413 → θ+1
  1 · 2, 2 · 2, 8, 3 · 2, 16, 18, 4 … Σ 1, 4, 13, 40, 117, 332, 921, 2512, …
  (−∞, 0]nonnegative
• A038277 → x ↦ x/(x+1) → reverse
  1 · 18, 7 · 324, 252, 49 · 5832, 6804, 2646, 343 … Σ 1, 25, 625, 15625, 390625, 9765625, …
  (−∞, 0]nonnegativerational roots
• chebyshev_u → D+1
  1 · 2, 2 · -1, 8, 4 · -4, -4, 24, 8 … Σ 1, 4, 11, 24, 45, 76, 119, 176, …
  (−∞, ∞)
• A034870 → basis A013610 → D+1
  1 · 16, 30, 9 · 112, 528, 864, 540, 81 … Σ 1, 55, 2125, 71875, 2265625, 68359375, …
  (−∞, 0]nonnegativerational roots
• A125662 → D
  1 · 6, 2 · 25, 18, 3 · 90, 102, 36, 4 … Σ 0, 1, 8, 46, 232, 1091, 4912, 21468, …
  (−∞, 0]nonnegative
• A147716 → D
  1 · 28, 2 · 588, 84, 3 · 10976, 2352, 168, 4 … Σ 0, 1, 30, 675, 13500, 253125, 4556250, …
  (−∞, 0]nonnegativerational roots
• A114655 → θ+1 → (θ+1)²
  2 · 4, 16 · 8, 96, 54 · 16, 384, 648, 128 … Σ 2, 20, 158, 1176, 8442, 59036, 404630, …
  (−∞, 0]nonnegative
• A038320 → x ↦ x+1 → D
  204, 72 · 5202, 3672, 648 … Σ 276, 9522, 292008, 8395230, 231708348, …
  (−∞, 0]nonnegativerational roots
• A178821 → basis A013609 → θ+1
  1 · 10, 20 · 60, 240, 180 · 280, 1680, 2520, 1120 … Σ 1, 30, 480, 5600, 53760, 451584, …
  (−∞, 0]nonnegativerational roots