Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A130123 → basis A013611 → (θ+1)²
  1 · 2, 32 · 4, 128, 576 · 8, 384, 3456, 8192 … Σ 1, 34, 708, 12040, 182800, 2580000, …
  (−∞, 0]nonnegative
• A102547 → x ↦ (1/27)x → basis A007318
  1 · 3 · 9 · 28, 1 … Σ 1, 3, 9, 29, 93, 297, 949, 3033, …
  (−∞, −1]nonnegative
• A257625 → θ+1
  1 · 3, 6 · 9, 108, 27 · 27, 1242, 1863, 108 … Σ 1, 9, 144, 3240, 93312, 3265920, …
  (−∞, 0]nonnegative
• A027467 → basis A013611 → (θ+1)²
  1 · 16, 16 · 256, 512, 144 · 4096, 12288, 6912, 1024 … Σ 1, 32, 912, 24320, 620800, 15360000, …
  (−∞, 0]nonnegative
• A112486 → D+1
  1 · 2, 1 · 7, 11, 3 · 32, 96, 80, 15 … Σ 1, 3, 21, 223, 3137, 54715, 1136909, …
  (−∞, 0]nonnegative
• A038335 → x ↦ x+1 → D
  9 · 378, 162 · 11907, 10206, 2187 … Σ 0, 9, 540, 24300, 972000, 36450000, …
  (−∞, 0]nonnegativerational roots
• A391533 → θ+1
  4, 8, 3 · 4, 16, 15, 4 · 4, 24, 39, 24, 5 … Σ 15, 39, 96, 228, 528, 1200, 2688, 5952, …
  (−∞, 0]nonnegative
• A356546 → basis A013611 → reverse
  1 · 8, 4 · 96, 96, 24 · 1280, 1920, 960, 160 … Σ 1, 12, 216, 4320, 90720, 1959552, …
  (−∞, −1]nonnegativerational roots
• A111125 → θ+1
  1 · 3, 2 · 5, 10, 3 · 7, 28, 21, 4 … Σ 1, 5, 18, 60, 191, 589, 1775, 5257, …
  (−∞, 0]nonnegative
• A196347: Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
  1 · 1, 1 · 2, 4, 2 · 6, 18, 18, 6 … Σ 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  (−∞, −1]nonnegativerational roots
• A357367 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 6, 18 · 0, 24, 168, 264 … Σ 1, 2, 24, 456, 12000, 404640, 16652160, …
  [−1, 0]nonnegative
• A207606 → basis A013611 → reverse
  32, 44, 13 · 128, 272, 176, 34 … Σ 89, 610, 4181, 28657, 196418, 1346269, …
  (−∞, 0]nonnegative
• A086645 → D → θ+1
  1 · 6, 4 · 15, 60, 9 · 28, 280, 252, 16 … Σ 0, 1, 10, 84, 576, 3520, 19968, 107520, …
  (−∞, 0]nonnegative
• A038335 → basis A013609 → θ+1
  1 · 21, 36 · 441, 1512, 972 … Σ 1, 57, 2925, 141453, 6584409, …
  (−∞, 0]nonnegativerational roots
• A063967 → x ↦ x−15
  1 · -14, 1 · 182, -27, 1 · -2352, 532, -40, 1 … Σ 1, -13, 156, -1859, 22139, -263640, …
  (10, 14]
• A038221 → basis A013611 → x ↦ x/(x+1)
  1 · 6, 18 · 36, 216, 324 · 216, 1944, 5832, 5832 … Σ 1, 24, 576, 13824, 331776, 7962624, …
  [−1, 0]nonnegativerational roots
• A013612 → D+1
  11, 60, 25 · 16, 165, 450, 125 … Σ 96, 756, 5616, 40176, 279936, 1912896, …
  (−∞, 0]nonnegativerational roots
• type_b_eulerian → (θ+1)²
  1 · 1, 4 · 1, 24, 9 · 1, 92, 207, 16 … Σ 1, 5, 34, 316, 3616, 48960, 764160, …
  (−∞, 0]nonnegative
• A113214 → D+1
  1 · 3, 1 · 3, 5, 1 · 2, 10, 7, 1 … Σ 1, 4, 9, 20, 40, 78, 147, 272, …
  (−∞, 0]nonnegative
• A119308 → θ+1
  1 · 2, 2 · 3, 10, 3 · 4, 28, 27, 4 … Σ 1, 4, 16, 63, 246, 957, 3718, 14443, …
  (−∞, 0]nonnegative
• A130123 → D+1
  1 · 2, 2 · 0, 8, 4 · 0, 0, 24, 8 … Σ 1, 4, 12, 32, 80, 192, 448, 1024, …
  (−∞, 0]nonnegativerational roots
• A088617 → D+1
  1 · 2, 1 · 4, 7, 2 · 7, 26, 25, 5 … Σ 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  (−∞, 0]nonnegative
• A063967 → x ↦ x−6
  1 · -5, 1 · 20, -9, 1 · -75, 55, -13, 1 … Σ 1, -4, 12, -32, 80, -192, 448, -1024, …
  (1, 5]
• A130123 → Veronese 2 (j=1) → D
  8 · 0, 64 · 0, 0, 384 · 0, 0, 0, 2048 … Σ 0, 0, 0, 8, 0, 64, 0, 384, …
  (−∞, 0]nonnegativerational roots
• A038220 → D+1
  1 · 5, 2 · 21, 20, 4 · 81, 126, 60, 8 … Σ 1, 7, 45, 275, 1625, 9375, 53125, …
  (−∞, 0]nonnegativerational roots
• A008288 → basis A007318 → D
  1 · 5, 2 · 18, 16, 3 · 58, 80, 33, 4 … Σ 0, 1, 7, 37, 175, 777, 3315, 13757, …
  (−∞, 0]nonnegative
• A166973 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 9, 9 · 1, 46, 107, 63 … Σ 1, 3, 19, 217, 3121, 51443, 967379, …
  [−1, 0]nonnegative
• A027465 → Veronese 2 (j=1) → (θ+1)²
  1 · 6 · 27, 4 · 108, 48 … Σ 0, 1, 6, 31, 156, 774, 3780, 18160, …
  (−∞, 0]nonnegative
• A026729 → basis A013610 → D+1
  1 · 4, 3 · 11, 27, 9 · 24, 111, 126, 27 … Σ 1, 7, 47, 288, 1688, 9584, 53200, …
  (−∞, 0]nonnegative
• Veronese parent of A109446 → D+1
  1, 1 · 2, 1 · 1, 5, 5, 1 · 4, 9, 6, 1 … Σ 2, 3, 12, 20, 64, 112, 320, 576, …
  (−∞, 0]nonnegative
• A274496 → Veronese 2 (j=1) → x ↦ x+1
  2 · 4 · 8 · 16 … Σ 0, 0, 2, 4, 8, 16, 40, 80, …
  (−∞, −1]nonnegative
• A086645 → basis A013610 → θ+1
  1 · 2, 6 · 8, 48, 27 · 32, 288, 486, 108 … Σ 1, 8, 83, 914, 9845, 103340, 1062887, …
  (−∞, 0]nonnegative
• A060920 → reverse → D+1
  1 · 3, 2 · 6, 15, 5 · 10, 49, 59, 13 … Σ 1, 5, 26, 131, 636, 2997, 13806, 62503, …
  (−∞, 0]nonnegative
• A003506 → D → x ↦ x+1
  2 · 12, 6 · 48, 48, 12 · 160, 240, 120, 20 … Σ 0, 0, 2, 18, 108, 540, 2430, 10206, …
  (−∞, −1]nonnegativerational roots
• A081578 → basis A013609 → reverse
  1 · 2, 2 · 4, 14, 7 · 8, 48, 60, 20 … Σ 1, 4, 25, 136, 769, 4300, 24121, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A008277 → reverse
  1 · 1, 3, 1 · 1, 10, 25, 15, 1 … Σ 1, 5, 52, 877, 21147, 678570, 27644437, …
  (−∞, 0]nonnegative
• A103327 → basis A013611 → x ↦ x/(x+1)
  1 · 4, 8 · 16, 80, 80 · 64, 512, 1216, 832 … Σ 1, 12, 176, 2624, 39168, 584704, …
  [−1, 0]nonnegative
• A113413 → D+1
  1 · 3, 1 · 6, 6, 1 · 10, 20, 9, 1 … Σ 1, 4, 13, 40, 117, 332, 921, 2512, …
  (−∞, 0]nonnegative
• A075497 → (θ+1)²
  1 · 2, 4 · 4, 24, 9 · 8, 112, 108, 16 … Σ 1, 6, 37, 244, 1741, 13442, 111833, …
  (−∞, 0]nonnegative
• A028326 → Veronese 2 (j=1) → (θ+1)²
  2 · 4 · 6, 8 · 8, 32 … Σ 0, 2, 4, 14, 40, 108, 280, 704, …
  (−∞, 0]nonnegative
• A290315 → D
  2 · 16, 24 · 66, 568, 360 · 224, 5744, 17856, 6720 … Σ 0, 2, 40, 994, 30544, 1127330, …
  (−∞, 0]nonnegative
• A123160 → D+1
  1 · 2, 1 · 5, 10, 3 · 10, 45, 48, 10 … Σ 1, 3, 18, 113, 708, 4407, 27254, …
  (−∞, 0]nonnegative
• A038336 → x ↦ x/(x+1) → x ↦ x/(x+1)
  144, 816, 1156 · 1728, 14688, 41616, 39304 … Σ 2116, 97336, 4477456, 205962976, …
  [−1, 0]nonnegativerational roots
• A030528 → x ↦ x−7
  1 · -6, 1 · 35, -12, 1 · -203, 106, -18, 1 … Σ 1, -5, 24, -114, 540, -2556, 12096, …
  (3, 7]
• Hadamard-factor parent of A155495 → θ+1
  1, 12, 3 · 1, 30, 45, 4 · 1, 56, 210, 112, 5 … Σ 16, 80, 384, 1792, 8192, 36864, 163840, …
  (−∞, 0]nonnegative
• A038314 → D
  240, 288 · 3600, 8640, 5184 … Σ 528, 17424, 511104, 14055360, …
  (−∞, 0]nonnegativerational roots
• A123519 → D+1
  7, 14, 4 · 13, 52, 44, 8 · 21, 140, 228, 120, 16 … Σ 25, 117, 525, 2289, 9773, 41065, …
  (−∞, 0]nonnegative
• A114687 → basis A013609 → D+1
  1 · 7, 4 · 39, 60, 16 · 213, 552, 384, 64 … Σ 1, 11, 115, 1213, 13013, 141727, …
  (−∞, 0]nonnegative
• A232535 → basis A007318
  1 · 3, 2 · 12, 14, 3 · 48, 80, 37, 4 … Σ 1, 5, 29, 169, 985, 5741, 33461, …
  (−∞, −1]nonnegative
• A038327 → basis A013611 → reverse
  1 · 4, 13 · 16, 104, 169 · 64, 624, 2028, 2197 … Σ 1, 17, 289, 4913, 83521, 1419857, …
  (−∞, 0]nonnegativerational roots