Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A126216 → D → D
  2 · 18, 6 · 112, 84, 12 · 600, 720, 240, 20 … Σ 0, 0, 2, 24, 208, 1580, 11190, 75936, …
  (−∞, 0]nonnegative
• A059364 → basis A013609 → D+1
  1 · 5, 2 · 37, 38, 8 · 349, 620, 332, 48 … Σ 1, 7, 83, 1349, 27613, 679323, …
  (−∞, 0]nonnegative
• A038335 → basis A013610 → basis A007318
  1 · 48, 27 · 2304, 2592, 729 … Σ 1, 75, 5625, 421875, 31640625, …
  (−∞, −1]nonnegativerational roots
• A063007 → x ↦ x+1 → D+1
  1 · 5, 2 · 31, 30, 6 · 195, 312, 150, 20 … Σ 1, 7, 67, 677, 6941, 71435, 735295, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → (θ+1)² → reverse
  9 · 16 · 25 · 36 … Σ 9, 16, 25, 36, 49, 64, 81, 100, …
  (−∞, 0]nonnegativerational roots
• A102365 → θ+1
  1 · 1 · 1, 2 · 1, 10, 3 … Σ 1, 1, 3, 14, 86, 657, 5995, 63541, …
  (−∞, 0]nonnegative
• A056939 → θ+1 → D
  2 · 8, 6 · 20, 60, 12 · 40, 300, 240, 20 … Σ 0, 2, 14, 92, 600, 3950, 26334, 177800, …
  (−∞, 0]nonnegative
• A290316 → reverse
  1 · 6, 1 · 90, 48, 1 · 2160, 2214, 234, 1 … Σ 1, 7, 139, 4609, 214297, 12820999, …
  (−∞, 0]nonnegative
• A099093 → x ↦ x−5
  1 · -15, 3 · 210, -87, 9 · -2925, 1845, -387, 27 … Σ 1, -12, 132, -1440, 15696, -171072, …
  (11/3, 5]
• A099040 → basis A013609 → basis A007318
  1 · 6, 4 · 42, 52, 16 · 288, 528, 320, 64 … Σ 1, 10, 110, 1200, 13100, 143000, …
  (−∞, −1]nonnegative
• A196347 → basis A013611
  1 · 2, 4 · 8, 32, 32 · 48, 288, 576, 384 … Σ 1, 6, 72, 1296, 31104, 933120, …
  [−1, 0]nonnegativerational roots
• A144438 → θ+1
  1 · 1, 2 · 1, 10, 3 · 1, 28, 42, 4 … Σ 1, 3, 14, 75, 471, 3402, 27844, 254970, …
  (−∞, 0]nonnegative
• A142075 → reverse
  1 · 2, 1 · 4, 8, 1 · 8, 44, 22, 1 … Σ 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  (−∞, 0]nonnegative
• A114687 → basis A013610 → x ↦ x+1
  1 · 9, 6 · 89, 114, 36 · 945, 1764, 1080, 216 … Σ 1, 15, 239, 4005, 69917, 1261275, …
  (−∞, −1]nonnegative
• A110441 → basis A013610 → D
  3 · 24, 18 · 132, 216, 81 · 624, 1620, 1296, 324 … Σ 0, 3, 42, 429, 3864, 32523, 262374, …
  (−∞, 0]nonnegative
• A038305 → basis A013611 → (θ+1)²
  169, 1248, 1296 · 2197, 24336, 50544, 27648 … Σ 2713, 104725, 3720625, 125078125, …
  (−∞, 0]nonnegative
• A109692 → basis A013609 → D+1
  1 · 4, 2 · 28, 44, 12 · 248, 744, 632, 120 … Σ 1, 6, 84, 1744, 47328, 1578624, …
  (−∞, 0]nonnegative
• A111006 → (θ+1)²
  1 · 0, 4 · 0, 4, 18 · 0, 0, 18, 48 … Σ 1, 4, 22, 66, 214, 586, 1598, 4098, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/4) (shift +1) → x ↦ x+1 → θ+1
  1 · 2 · 6, 4 · 24, 32 … Σ 1, 2, 10, 56, 408, 3456, 34016, 378496, …
  (−∞, 0]nonnegative
• A038291 → basis A013610 → θ+1
  1 · 10, 6 · 100, 120, 27 · 1000, 1800, 810, 108 … Σ 1, 16, 247, 3718, 54925, 799708, …
  (−∞, 0]nonnegativerational roots
• A124860 → D+1
  1 · 2, 1 · 9, 12, 3 · 20, 45, 30, 5 … Σ 1, 3, 24, 100, 528, 2352, 11008, 48960, …
  (−∞, 0]nonnegativerational roots
• A134991 → D+1
  1 · 4, 3 · 11, 40, 15 · 26, 235, 420, 105 … Σ 1, 7, 66, 786, 11348, 192788, 3769328, …
  (−∞, 0]nonnegative
• A081579 → γ → basis A007318
  1 · 1 · 5, 4 · 9, 8 … Σ 1, 1, 9, 17, 89, 225, 937, 2737, …
  (−∞, −1]nonnegative
• A108838 → D → D
  4 · 30, 12 · 120, 144, 24 · 350, 840, 420, 40 … Σ 0, 0, 4, 42, 288, 1650, 8580, 42042, …
  (−∞, 0]nonnegative
• A033282 → θ+1 → (θ+1)²
  1 · 1, 16 · 1, 40, 135 · 1, 72, 567, 896 … Σ 1, 17, 176, 1536, 12251, 92363, 669856, …
  (−∞, 0]nonnegative
• A136011 → D+1
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 18, 49, 142, 437, …
  (−∞, 0]nonnegative
• A161198 → basis A013610 → reverse
  1 · 6, 3 · 36, 48, 15 · 216, 540, 426, 105 … Σ 1, 9, 99, 1287, 19305, 328185, 6235515, …
  (−∞, 0]nonnegativerational roots
• A133087 → reverse → D+1
  1 · 3, 2 · 3, 4, 1 · 5, 14, 11, 2 … Σ 1, 5, 8, 32, 48, 176, 256, 896, …
  (−∞, 0]nonnegative
• A089231 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 3, 5 · 13, 44, 37 · 73, 375, 639, 361 … Σ 1, 8, 94, 1448, 27496, 619232, …
  [−1, 0]nonnegative
• A081582 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 11, 11 · 1, 20, 54, 36 … Σ 1, 3, 23, 111, 655, 3519, 19727, …
  [−1, 0]nonnegative
• A052553 → basis A013611 → D+1
  1 · 1 · 6, 4 · 11, 8 … Σ 1, 1, 10, 19, 93, 232, 861, 2405, …
  (−∞, 0]nonnegative
• A038337 → basis A007318 → basis A007318
  1156, 748, 121 · 39304, 38148, 12342, 1331 … Σ 2025, 91125, 4100625, 184528125, …
  (−∞, −1]nonnegativerational roots
• A172431 → D → D+1
  2 · 10, 6 · 26, 44, 12 · 50, 162, 120, 20 … Σ 0, 2, 16, 82, 352, 1368, 4980, 17302, …
  (−∞, 0]nonnegative
• A034870 → Veronese 2 → D
  1 · 6, 2 · 15, 30, 3 · 28, 140, 84, 4 … Σ 0, 1, 8, 48, 256, 1280, 6144, 28672, …
  (−∞, 0]nonnegative
• A367270 → D → x ↦ x+1
  2 · 12, 6 · 60, 60, 12 · 280, 420, 180, 20 … Σ 0, 0, 2, 18, 132, 900, 5910, 37926, …
  (−∞, −1]nonnegative
• basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
  1 · 0, -1 · 0, 0, 2 · 0, 0, 0, -6 … Σ 1, -1, 2, -6, 24, -120, 720, -5040, …
  [−1, 0]rational roots
• A038319 → basis A013609
  256, 320, 100 · 4096, 7680, 4800, 1000 … Σ 676, 17576, 456976, 11881376, …
  (−∞, −1]nonnegativerational roots
• A110441 → x ↦ x/(x+1)
  1 · 3, 4 · 7, 20, 14 · 15, 68, 100, 48 … Σ 1, 7, 41, 231, 1289, 7175, 39913, …
  [−1, 0)nonnegative
• A156992 → basis A013610 → θ+1
  1 · 4, 12 · 24, 144, 162 · 192, 1728, 3888, 2592 … Σ 1, 16, 330, 8400, 255000, 9000000, …
  (−∞, 0]nonnegativerational roots
• A178820 → basis A013609 → D
  8 · 80, 80 · 480, 960, 480 · 2240, 6720, 6720, 2240 … Σ 0, 8, 160, 1920, 17920, 143360, …
  (−∞, 0]nonnegativerational roots
• A229032: Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).
  1 · 2 · 3, 4 · 4, 16 … Σ 1, 2, 7, 20, 61, 182, 547, 1640, …
  (−∞, 0)nonnegative
• A232535 → basis A007318 → x ↦ x/(x+1)
  1 · 3, 5 · 12, 38, 29 · 48, 224, 341, 169 … Σ 1, 8, 79, 782, 7741, 76628, 758539, …
  [−1, 0]nonnegative
• A038221 → basis A013609 → D
  6 · 72, 72 · 648, 1296, 648 … Σ 0, 6, 144, 2592, 41472, 622080, …
  (−∞, 0]nonnegativerational roots
• A121574 → reverse → x ↦ x+1
  1 · 3, 2 · 10, 13, 4 · 33, 64, 40, 8 … Σ 1, 5, 27, 145, 779, 4185, 22483, …
  (−∞, −1]nonnegative
• A136572 → D → θ+1
  1 · 0, 8 · 0, 0, 54 · 0, 0, 0, 384 … Σ 0, 1, 8, 54, 384, 3000, 25920, 246960, …
  (−∞, 0]nonnegativerational roots
• A290319 → basis A013611 → x ↦ x+1
  1 · 6, 4 · 60, 64, 16 · 840, 1136, 480, 64 … Σ 1, 10, 140, 2520, 55440, 1441440, …
  (−∞, −1]nonnegativerational roots
• A008292 → γ (scale 1/2) (shift +1) → θ+1 → x ↦ x+1
  1 · 1 · 3, 2 · 9, 8 … Σ 1, 1, 5, 17, 93, 513, 3477, 25569, …
  (−∞, −1]nonnegative
• A046716 → basis A013611 → D+1
  1 · 6, 4 · 25, 52, 16 · 116, 452, 368, 64 … Σ 1, 10, 93, 1000, 14801, 315886, …
  (−∞, 0]nonnegative
• A080248 → reverse
  1 · 1, 1 · 1, 4, 1 · 1, 10, 13, 1 … Σ 1, 2, 6, 25, 135, 909, 7417, 71698, …
  (−∞, 0]nonnegative
• A008955 → basis A013610 → D
  3 · 39, 72 · 660, 2826, 2916 … Σ 0, 3, 111, 6402, 567090, 72607965, …
  (−∞, 0]nonnegative