Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A173018 → x ↦ x+1 → x ↦ x+1
  1 · 1 · 3, 1 · 13, 8, 1 … Σ 1, 1, 4, 22, 160, 1456, 15904, 202672, …
  (−∞, −1]nonnegative
• A152828 → basis A007318 → basis A007318
  1 · 1 · 1 · 3, 1 … Σ 1, 1, 1, 4, 4, 4, 16, 16, …
  (−∞, −1]nonnegativerational roots
• A038279 → basis A013611 → (θ+1)²
  1 · 9, 16 · 81, 288, 144 · 729, 3888, 3888, 1024 … Σ 1, 25, 513, 9529, 166465, 2787993, …
  (−∞, 0]nonnegative
• A225471 → basis A013611 → (θ+1)²
  1 · 4, 16 · 32, 192, 144 · 384, 2816, 3456, 1024 … Σ 1, 20, 368, 7680, 185344, 5132288, …
  (−∞, 0]nonnegative
• A088996 → basis A013609 → D
  2 · 10, 16 · 68, 200, 144 · 596, 2488, 3408, 1536 … Σ 0, 2, 26, 412, 8028, 187182, 5095206, …
  (−∞, 0]nonnegative
• A103633 → basis A013609 → reverse
  1 · 2, 1 · 4, 6, 2 · 8, 16, 10, 2 … Σ 1, 3, 12, 36, 144, 432, 1728, 5184, …
  (−∞, 0]nonnegativerational roots
• A207824 → basis A013610 → θ+1
  1 · 6, 6 · 35, 72, 27 · 204, 636, 486, 108 … Σ 1, 12, 134, 1434, 14905, 151668, …
  (−∞, 0]nonnegative
• A156319 → θ+1
  1 · 2, 2 · 0, 4, 3 · 0, 0, 6, 4 … Σ 1, 4, 7, 10, 13, 16, 19, 22, …
  (−∞, 0]nonnegativerational roots
• A038324 → (θ+1)²
  1 · 11, 40 · 121, 880, 900 … Σ 1, 51, 1901, 61551, 1835001, 51778251, …
  (−∞, 0]nonnegative
• A110441 → basis A013609 → D+1
  1 · 6, 2 · 30, 24, 4 · 136, 184, 72, 8 … Σ 1, 8, 58, 400, 2668, 17376, 111160, …
  (−∞, 0]nonnegative
• A099093 → θ+1 → D+1
  1 · 6, 6 · 6, 60, 27 · 0, 108, 378, 108 … Σ 1, 12, 93, 594, 3402, 18144, 92043, …
  (−∞, 0]nonnegative
• A129710 → θ+1
  1 · 2 · 2, 2 · 2, 6 … Σ 1, 2, 4, 8, 15, 28, 51, 92, …
  (−∞, 0]nonnegative
• A122542 → θ+1 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 4, 7 · 0, 4, 20, 20 … Σ 1, 2, 11, 44, 181, 726, 2879, 11296, …
  [−1, 0]nonnegative
• A125092 → θ+1
  1 · 1, 8 · 1, 16, 27 · 1, 24, 81, 64 … Σ 1, 9, 44, 170, 576, 1792, 5248, 14688, …
  (−∞, 0]nonnegative
• A137320 → Veronese 2 → D
  4 · 12 · 44, 32 · 200, 320 … Σ 0, 0, 4, 12, 76, 520, 4008, 34608, …
  (−∞, 0]nonnegative
• A038321 → basis A007318 → D+1
  576, 350, 49 · 12636, 12096, 3675, 343 … Σ 975, 28750, 828125, 23437500, …
  (−∞, 0]nonnegativerational roots
• A081578 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 7, 35, 37 · 20, 150, 348, 245 … Σ 1, 7, 79, 763, 7711, 76867, 769399, …
  [−1, 0]nonnegative
• basis-map coefficients of A026729, shifted → D → D
  2 · 0, -6 · 0, 0, 12 · 0, 0, 0, -20 … Σ 0, 0, 2, -6, 12, -20, 30, -42, …
  (−∞, 0]rational roots
• Hadamard-factor parent of A155495 → x ↦ x/(x+1)
  1, 8, 8 · 1, 18, 48, 32 · 1, 32, 160, 256, 128 … Σ 17, 99, 577, 3363, 19601, 114243, …
  [−1, 0]nonnegative
• A059473 → basis A013610 → x ↦ x+1
  1 · 10, 6 · 108, 126, 36 · 1160, 2016, 1152, 216 … Σ 1, 16, 270, 4544, 76484, 1287360, …
  (−∞, −1]nonnegative
• A038277 → x ↦ x+1 → (θ+1)²
  1 · 18, 44 · 324, 1584, 1089 … Σ 1, 62, 2997, 128702, 5147761, …
  (−∞, 0]nonnegative
• A367270 → basis A013611 → (θ+1)²
  1 · 1 · 3, 32 · 10, 192, 432 … Σ 1, 1, 35, 634, 9411, 128286, 1668686, …
  (−∞, 0]nonnegative
• A127648 → (θ+1)² → x ↦ x+1
  1 · 8, 8 · 27, 54, 27 · 64, 192, 192, 64 … Σ 0, 1, 16, 108, 512, 2000, 6912, 21952, …
  (−∞, −1]nonnegativerational roots
• A081577 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 10, 24, 16 … Σ 1, 3, 13, 51, 205, 819, 3277, 13107, …
  [−1, 0]nonnegative
• A330510: Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <=…
  2 · 3, 3 · 4, 10, 4 · 5, 22, 22, 5 … Σ 2, 6, 18, 54, 162, 486, 1458, 4374, …
  (−∞, 0)nonnegative
• A114656 → basis A007318 → (θ+1)²
  1 · 3, 4 · 11, 32, 9 · 45, 204, 135, 16 … Σ 1, 7, 52, 400, 3136, 24864, 198528, …
  (−∞, 0]nonnegative
• A131689 → θ+1 → D+1
  1 · 2, 2 · 2, 14, 6 · 2, 38, 90, 24 … Σ 1, 4, 22, 154, 1306, 12994, 148282, …
  (−∞, 0]nonnegative
• A013609 → basis A013610
  1 · 3, 6 · 9, 36, 36 · 27, 162, 324, 216 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  [−1, 0]nonnegativerational roots
• A143494 → θ+1 → x ↦ x+1
  1 · 4, 2 · 17, 16, 3 · 77, 104, 39, 4 … Σ 1, 6, 36, 224, 1464, 10088, 73288, …
  (−∞, −1]nonnegative
• A152828 → D
  1 · 1 · 1 · 2, 2 … Σ 0, 0, 0, 1, 1, 1, 4, 4, …
  (−∞, 0]nonnegativerational roots
• A249251 → basis A007318 → x ↦ x+1
  1 · 7, 3 · 37, 30, 6 · 199, 237, 93, 12 … Σ 1, 10, 73, 541, 4006, 29665, 219673, …
  (−∞, −1]nonnegative
• A038321 → basis A013609
  324, 504, 196 · 5832, 13608, 10584, 2744 … Σ 1024, 32768, 1048576, 33554432, …
  (−∞, 0]nonnegativerational roots
• A178821 → basis A013610
  1 · 10, 15 · 60, 180, 135 · 280, 1260, 1890, 945 … Σ 1, 25, 375, 4375, 43750, 393750, …
  [−1, 0]nonnegativerational roots
• A008277 → θ+1 → reverse → x ↦ x/(x+1)
  1 · 2, 3 · 3, 12, 10 · 4, 30, 62, 37 … Σ 1, 5, 25, 133, 761, 4677, 30745, …
  [−1, 0]nonnegative
• A136668 → D+1
  -2, 4, 2 · -11, -11, 24, 8 · 8, -148, -74, 192, 48 … Σ 4, 10, 26, 14, -1306, -29954, -594842, …
  (−∞, ∞)
• A038301 → basis A013611 → D+1
  2160, 5632, 1936 · 60800, 285120, 371712, 85184 … Σ 9728, 802816, 62914560, 4764729344, …
  (−∞, 0]nonnegativerational roots
• A038305 → x ↦ x+1 → (θ+1)²
  169, 312, 81 · 2197, 6084, 3159, 432 … Σ 562, 11872, 240640, 4734976, 91095040, …
  (−∞, 0]nonnegative
• A161198 → (θ+1)²
  1 · 1, 8 · 3, 32, 36 · 15, 184, 324, 128 … Σ 1, 9, 71, 651, 7057, 89249, 1295639, …
  (−∞, 0]nonnegative
• A152842 → reverse
  1 · 3, 1 · 3, 4, 1 · 9, 15, 7, 1 … Σ 1, 4, 8, 32, 64, 256, 512, 2048, …
  (−∞, 0]nonnegativerational roots
• A281000 → D
  1 · 10, 2 · 70, 42, 3 · 420, 504, 108, 4 … Σ 0, 1, 12, 115, 1036, 9135, 79904, …
  (−∞, 0]nonnegative
• A358735 → basis A007318 → D
  1 · 8, 4 · 60, 68, 18 · 484, 884, 516, 96 … Σ 0, 1, 12, 146, 1980, 30179, 514264, …
  (−∞, 0]nonnegative
• A038284 → reverse
  36, 96, 64 · 216, 864, 1152, 512 … Σ 196, 2744, 38416, 537824, 7529536, …
  (−∞, 0]nonnegativerational roots
• A269944 → reverse
  1 · 1 · 1, 1 · 1, 5, 4 … Σ 1, 1, 2, 10, 100, 1700, 44200, 1635400, …
  (−∞, 0]nonnegativerational roots
• A038284 → basis A007318 → D+1
  364, 240, 36 · 6272, 6552, 2160, 216 … Σ 640, 15200, 352000, 8000000, 179200000, …
  (−∞, 0]nonnegativerational roots
• A028421 → D
  2 · 6, 6 · 22, 36, 12 · 100, 210, 120, 20 … Σ 0, 2, 12, 70, 450, 3248, 26264, 236248, …
  (−∞, 0]nonnegative
• A008277 → θ+1 → reverse → D
  1 · 6, 2 · 18, 28, 3 · 40, 150, 90, 4 … Σ 0, 1, 8, 49, 284, 1658, 9974, 62375, …
  (−∞, 0]nonnegative
• A113953 → x ↦ x−5
  1 · -5, 1 · 15, -8, 1 · -25, 35, -11, 1 … Σ 1, -4, 8, 0, -64, 256, -512, 0, …
  (−3, 5]
• A143685 → basis A007318 → x ↦ x+1
  1 · 3, 1 · 27, 15, 1 · 135, 117, 27, 1 … Σ 1, 4, 43, 280, 2281, 16684, 128323, …
  (−∞, −1]nonnegative
• A137320 → basis A013610 → θ+1
  1 · 2, 12 · 6, 60, 108 · 24, 312, 972, 864 … Σ 1, 14, 174, 2172, 28212, 386064, …
  (−∞, 0]nonnegative
• A003506 → Veronese 2 (j=1) → θ+1
  2 · 6 · 12, 8 · 20, 40 … Σ 0, 0, 2, 6, 20, 60, 168, 448, …
  (−∞, 0]nonnegative