Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A122542 → D → x ↦ x+1
  1 · 4, 2 · 13, 14, 3 · 40, 64, 30, 4 … Σ 0, 1, 6, 30, 138, 602, 2538, 10442, …
  (−∞, −1]nonnegative
• basis-map coefficients of A112227 → (θ+1)² → basis A007318
  9, 18, 9 · 16, 48, 48, 16 · 25, 100, 150, 100, 25 … Σ 36, 128, 400, 1152, 3136, 8192, 20736, …
  (−∞, −1]nonnegativerational roots
• A088729 → x ↦ x/(x+1)
  13, 35, 23 · 75, 304, 401, 173 … Σ 71, 953, 15897, 316543, 7318943, …
  [−1, 0]nonnegative
• A038330 → x ↦ x/(x+1)
  144, 384, 256 · 1728, 6912, 9216, 4096 … Σ 784, 21952, 614656, 17210368, …
  [−1, 0]nonnegativerational roots
• A113413 → basis A013611 → D
  4 · 24, 32 · 92, 288, 192 · 304, 1536, 2304, 1024 … Σ 0, 4, 56, 572, 5168, 43684, 354120, …
  (−∞, 0]nonnegative
• A039755 → D+1
  1 · 2, 1 · 5, 6, 1 · 14, 31, 12, 1 … Σ 1, 3, 12, 58, 324, 2044, 14320, 109960, …
  (−∞, 0]nonnegative
• A003506 → x ↦ x+1 → (θ+1)²
  1 · 4, 8 · 12, 48, 27 · 32, 192, 216, 64 … Σ 0, 1, 12, 87, 504, 2565, 11988, 52731, …
  (−∞, 0]nonnegative
• A136572 → basis A013610 → basis A007318
  1 · 4, 3 · 32, 48, 18 · 384, 864, 648, 162 … Σ 1, 7, 98, 2058, 57624, 2016840, …
  (−∞, −1]nonnegativerational roots
• A322944 → reverse
  1 · 1, 1 · 1, 6, 2 · 1, 15, 38, 6 … Σ 1, 2, 9, 60, 513, 5286, 63417, 865824, …
  (−∞, 0]nonnegative
• A133080 → D → D
  2 · 2, 6 · 0, 0, 12 · 0, 0, 12, 20 … Σ 0, 0, 2, 8, 12, 32, 30, 72, …
  (−∞, 0]nonnegativerational roots
• A232535 → basis A013610 → θ+1
  1 · 3, 12 · 12, 84, 81 · 48, 480, 999, 432 … Σ 1, 15, 177, 1959, 20913, 217743, …
  (−∞, 0]nonnegative
• delannoy → basis A013611 → (θ+1)²
  1 · 2, 16 · 5, 80, 144 · 12, 288, 1152, 1024 … Σ 1, 18, 229, 2476, 24381, 225654, …
  (−∞, 0]nonnegative
• A216973: Exponential Riordan array [x*exp(x),x].
  1 · 2, 2 · 3, 6, 3 · 4, 12, 12, 4 … Σ 0, 1, 4, 12, 32, 80, 192, 448, …
  (−∞, −1]nonnegativerational roots
• A124860 → x ↦ x+1
  1 · 2, 1 · 12, 12, 3 · 40, 60, 30, 5 … Σ 1, 3, 27, 135, 891, 5103, 31347, …
  (−∞, −1]nonnegativerational roots
• A124860 → θ+1
  1 · 1, 2 · 3, 12, 9 · 5, 30, 45, 20 … Σ 1, 3, 24, 100, 528, 2352, 11008, 48960, …
  (−∞, 0]nonnegativerational roots
• image under θ+1 → θ+1 → D+1 → (θ+1)²
  1 · 4, 16 · 0, 72, 81 · 0, 0, 432, 256 … Σ 1, 20, 153, 688, 2225, 5796, 12985, …
  (−∞, 0]nonnegativerational roots
• A113953 → basis A013611 → x ↦ x/(x+1)
  1 · 1, 5 · 3, 22, 35 · 5, 59, 215, 225 … Σ 1, 6, 60, 504, 4464, 38880, 340416, …
  [−1, 0]nonnegative
• A038320 → x ↦ x+1 → basis A013609
  529, 552, 144 · 12167, 19044, 9936, 1728 … Σ 1225, 42875, 1500625, 52521875, …
  (−∞, −1]nonnegativerational roots
• A038220 → Veronese 2 (j=1) → (θ+1)²
  2 · 12 · 54, 32 · 216, 384 … Σ 0, 2, 12, 86, 600, 3978, 25380, 157406, …
  (−∞, 0]nonnegative
• A038255 → basis A013610 → basis A013611
  1 · 10, 12 · 100, 240, 144 · 1000, 3600, 4320, 1728 … Σ 1, 22, 484, 10648, 234256, 5153632, …
  (−∞, 0]nonnegativerational roots
• A152198 → Veronese 2 → x ↦ x+1
  1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 3, 3, 7, 7, …
  (−∞, −1]nonnegative
• A257610 → x ↦ x+1
  1 · 4, 2 · 28, 28, 4 · 280, 420, 156, 8 … Σ 1, 6, 60, 864, 16200, 373896, 10248120, …
  (−∞, −1]nonnegative
• A207823: Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).
  1 · 4, 1 · 15, 8, 1 · 56, 46, 12, 1 … Σ 1, 5, 24, 115, 551, 2640, 12649, 60605, …
  (−6, −2)nonnegative
• A038284 → (θ+1)²
  64, 384, 324 · 512, 4608, 7776, 3456 … Σ 772, 16352, 320656, 5970944, 107103808, …
  (−∞, 0]nonnegative
• A126351 → D
  2 · 5, 8 · 9, 38, 24 · 14, 110, 195, 64 … Σ 0, 2, 13, 71, 383, 2129, 12360, 75242, …
  (−∞, 0]nonnegative
• A081580 → basis A007318 → D
  1 · 9, 2 · 42, 32, 3 · 202, 248, 69, 4 … Σ 0, 1, 11, 77, 523, 3225, 19383, 112693, …
  (−∞, 0]nonnegative
• A038319 → basis A013611 → (θ+1)²
  256, 2560, 3600 · 4096, 61440, 172800, 128000 … Σ 6416, 366336, 19097856, 937598976, …
  (−∞, 0]nonnegative
• A038279 → D
  1 · 16, 2 · 192, 48, 3 · 2048, 768, 96, 4 … Σ 0, 1, 18, 243, 2916, 32805, 354294, …
  (−∞, 0]nonnegativerational roots
• A207824 → basis A013611 → (θ+1)²
  1 · 6, 16 · 35, 192, 144 · 204, 1696, 2592, 1024 … Σ 1, 22, 371, 5516, 76085, 997826, …
  (−∞, 0]nonnegative
• A038226 → basis A013611 → basis A007318
  1849, 2752, 1024 · 79507, 177504, 132096, 32768 … Σ 5625, 421875, 31640625, 2373046875, …
  (−∞, −1]nonnegativerational roots
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
  3, 4, 1 · 12, 27, 18, 3 · 30, 96, 108, 48, 6 … Σ 0, 0, 8, 60, 288, 1120, 3840, 12096, …
  (−∞, 0]nonnegativerational roots
• A054142 → reverse → D+1
  1 · 2, 1 · 4, 5, 1 · 7, 16, 8, 1 … Σ 1, 3, 10, 32, 99, 299, 887, 2595, …
  (−∞, 0]nonnegative
• A102662 → D
  3 · 5, 6 · 7, 22, 9 · 9, 46, 51, 12 … Σ 0, 3, 11, 38, 118, 349, 997, 2780, …
  (−∞, 0]nonnegative
• A261711 → θ+1
  1 · 4 · 15, 2 · 56, 16 … Σ 1, 4, 17, 72, 304, 1280, 5376, 22528, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → basis A013609 → D
  -2 · 8, 8 · -24, -48, -24 · 64, 192, 192, 64 … Σ 0, -2, 16, -96, 512, -2560, 12288, …
  (−∞, 0]rational roots
• A137338 → D+1
  1 · 0, 1 · -3, -1, 1 · 9, -6, -3, 1 … Σ 1, 1, -3, 1, 13, -61, 223, -867, …
  (−∞, ∞)
• A059364 → θ+1
  1 · 2, 2 · 6, 14, 6 · 24, 92, 87, 24 … Σ 1, 4, 26, 227, 2463, 31818, 476004, …
  (−∞, 0]nonnegative
• A125662 → reverse → D
  3 · 6, 16 · 9, 50, 63 · 12, 102, 270, 220 … Σ 0, 3, 22, 122, 604, 2809, 12554, 54580, …
  (−∞, 0]nonnegative
• A059364 → D+1
  1 · 3, 1 · 13, 11, 2 · 70, 104, 47, 6 … Σ 1, 4, 26, 227, 2463, 31818, 476004, …
  (−∞, −1]nonnegative
• Veronese parent of A109447 → Veronese 2 (j=1) → D+1
  1 · 2 · 4, 3 · 8, 4 … Σ 1, 2, 7, 12, 36, 64, 176, 320, …
  (−∞, 0]nonnegative
• A330798 → θ+1
  1 · 2, 4 · 6, 30, 27 · 20, 168, 336, 192 … Σ 1, 6, 63, 716, 8285, 96210, 1116395, …
  (−∞, 0]nonnegative
• A130534 → basis A013610 → reverse
  1 · 3, 2 · 9, 15, 6 · 27, 81, 78, 24 … Σ 1, 5, 30, 210, 1680, 15120, 151200, …
  (−∞, 0]nonnegativerational roots
• A059110 → reverse
  1 · 1, 1 · 1, 4, 3 · 1, 9, 21, 13 … Σ 1, 2, 8, 44, 304, 2512, 24064, 261536, …
  (−∞, 0]nonnegative
• A013618 → (θ+1)²
  1, 88, 1089 · 1, 132, 3267, 21296 … Σ 1178, 24696, 457920, 7852032, …
  (−∞, 0]nonnegative
• A144698 → θ+1
  1 · 1, 8 · 1, 26, 48 · 1, 64, 339, 256 … Σ 1, 9, 75, 660, 6300, 65520, 740880, …
  (−∞, 0]nonnegative
• basis-map coefficients of A371080, shifted → (θ+1)²
  1 · 0, 4 · 0, 12, 9 · 0, 72, 81, 16 … Σ 1, 4, 21, 169, 1852, 25752, 434380, …
  (−∞, 0]nonnegative
• A120910 → Veronese 2 (j=1) → D+1
  3 · 12 · 39, 3 · 120, 24 … Σ 0, 3, 12, 42, 144, 489, 1644, 5472, …
  (−∞, 0]nonnegative
• A116414 → basis A007318
  1 · 5, 1 · 22, 10, 1 · 95, 69, 15, 1 … Σ 1, 6, 33, 180, 981, 5346, 29133, …
  (−9, −1)nonnegative
• A113214 → x ↦ x−2
  1 · 0, 1 · -2, -1, 1 · 4, -2, -2, 1 … Σ 1, 1, -2, 1, 1, -2, 1, 1, …
  (−2, 2]
• A013620 → basis A008288 → θ+1
  1 · 5, 6 · 25, 78, 27 · 125, 666, 567, 108 … Σ 1, 11, 130, 1466, 15991, 170357, …
  (−∞, 0]nonnegative