Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A225470 → basis A013611 → θ+1
  1 · 3, 8 · 18, 72, 48 · 162, 792, 864, 256 … Σ 1, 11, 138, 2074, 36824, 757896, …
  (−∞, 0]nonnegative
• A272867 → x ↦ x/(x+1)
  1 · 1, 6, 6 · 1, 12, 48, 72, 36 … Σ 1, 13, 169, 2197, 28561, 371293, …
  [−1, 0]nonnegative
• A376827: T(n, k) = binomial(n, k)*hypergeom([(1 - n)/2, -n/2], [1], 4).
  1 · 1, 1 · 3, 6, 3 · 7, 21, 21, 7 … Σ 1, 2, 12, 56, 304, 1632, 9024, 50304, …
  (−∞, 0)nonnegativerational roots
• A136011 → x ↦ x+1
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 19, 55, 171, 567, …
  (−∞, −1]nonnegative
• A035607 → reverse
  1 · 2, 1 · 2, 4, 1 · 2, 8, 6, 1 … Σ 1, 3, 7, 17, 41, 99, 239, 577, …
  (−∞, 0]nonnegative
• A375853 → D+1
  2 · 16, 8 · 76, 96, 20 · 256, 648, 336, 40 … Σ 2, 24, 192, 1280, 7680, 43008, 229376, …
  (−∞, 0]nonnegative
• A013610 → D → D
  18 · 54, 162 · 108, 648, 972 · 180, 1620, 4860, 4860 … Σ 0, 0, 18, 216, 1728, 11520, 69120, …
  (−∞, 0]nonnegativerational roots
• A166344 → D+1
  7, 8, 1 · 16, 45, 18, 1 · 35, 214, 192, 38, 1 … Σ 16, 80, 480, 3360, 26880, 241920, …
  (−∞, 0]nonnegative
• A135065 → x ↦ x+1
  1 · 8, 4 · 36, 36, 9 · 128, 192, 96, 16 … Σ 1, 12, 81, 432, 2025, 8748, 35721, …
  (−∞, −1]nonnegativerational roots
• A013610 → basis A013611
  1 · 4, 12 · 16, 96, 144 · 64, 576, 1728, 1728 … Σ 1, 16, 256, 4096, 65536, 1048576, …
  [−1, 0]nonnegativerational roots
• A059344 → basis A007318 → x ↦ x+1
  1 · 1 · 5, 2 · 13, 6 … Σ 1, 1, 7, 19, 145, 601, 4951, 26587, …
  (−∞, −1]nonnegative
• A064861 → reverse → θ+1
  2, 6, 3 · 4, 16, 15, 4 · 4, 24, 39, 24, 5 … Σ 11, 39, 96, 324, 756, 2484, 5616, …
  (−∞, 0]nonnegative
• A269945 → x ↦ x+1
  1 · 1, 1 · 2, 3, 1 · 7, 14, 8, 1 … Σ 1, 2, 6, 30, 214, 2030, 24486, 362622, …
  (−∞, −1]nonnegative
• A093375 → x ↦ x/(x+1)
  1 · 2, 3 · 3, 10, 8 · 4, 21, 36, 20 … Σ 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  [−1, 0]nonnegativerational roots
• A185410 → x ↦ x+1
  1 · 1 · 3, 2 · 15, 18, 4 … Σ 1, 1, 5, 37, 377, 4921, 78365, 1473277, …
  (−∞, −1]nonnegative
• A116414 → reverse → D+1
  1 · 5, 4 · 9, 34, 13 · 13, 96, 162, 40 … Σ 1, 9, 56, 311, 1635, 8308, 41221, …
  (−∞, 0]nonnegative
• A038325 → (θ+1)²
  1 · 11, 44 · 121, 968, 1089 … Σ 1, 55, 2178, 74536, 2342560, 69574032, …
  (−∞, 0]nonnegative
• A159854 → x ↦ x+1 → x ↦ x/(x+1)
  1, 4, 4 · 2, 11, 20, 12 · 4, 28, 73, 84, 36 … Σ 9, 45, 225, 1125, 5625, 28125, 140625, …
  [−1, 0]nonnegativerational roots
• A103327 → basis A013611 → reverse
  1 · 4, 4 · 16, 48, 16 · 64, 384, 320, 64 … Σ 1, 8, 80, 832, 8704, 91136, 954368, …
  (−∞, 0]nonnegative
• A128100 → basis A013610 → D+1
  1 · 1 · 6, 3 · 11, 6 … Σ 1, 1, 9, 17, 80, 198, 721, 1999, …
  (−∞, 0]nonnegative
• A008459 → D+1
  1 · 2, 1 · 5, 6, 1 · 10, 27, 12, 1 … Σ 1, 3, 12, 50, 210, 882, 3696, 15444, …
  (−∞, 0]nonnegative
• A116414: Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).
  1 · 4, 1 · 13, 8, 1 · 40, 42, 12, 1 … Σ 1, 5, 22, 95, 409, 1760, 7573, 32585, …
  (−8, 0)nonnegative
• A257627: Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 3.
  1 · 3, 3 · 9, 60, 9 · 27, 753, 753, 27 … Σ 1, 6, 78, 1560, 42120, 1432080, …
  (−∞, 0)nonnegative
• A125662 → basis A013610 → reverse
  1 · 3, 4 · 9, 24, 15 · 27, 108, 138, 56 … Σ 1, 7, 48, 329, 2255, 15456, 105937, …
  (−∞, 0]nonnegative
• A038333 → x ↦ x+1 → (θ+1)²
  361, 1064, 441 · 6859, 30324, 25137, 5488 … Σ 1866, 67808, 2330848, 77088336, …
  (−∞, 0]nonnegative
• A062145 → D → D
  20 · 90, 120 · 252, 840, 420 · 560, 3360, 4200, 1120 … Σ 0, 0, 20, 210, 1512, 9240, 51480, …
  (−∞, 0]nonnegative
• A103406 → D
  2 · 4, 6 · 6, 18, 12 · 8, 36, 48, 20 … Σ 0, 2, 10, 36, 112, 320, 864, 2240, …
  (−∞, 0]nonnegativerational roots
• A125662 → reverse → θ+1
  1 · 1, 6 · 1, 12, 24 · 1, 18, 75, 84 … Σ 1, 7, 37, 178, 813, 3589, 15465, 65444, …
  (−∞, 0]nonnegative
• A013620 → basis A007318 → (θ+1)²
  1 · 5, 12 · 25, 120, 81 · 125, 900, 1215, 432 … Σ 1, 17, 226, 2672, 29440, 309248, …
  (−∞, 0]nonnegative
• A059364 → basis A013611 → x ↦ x+1
  1 · 7, 4 · 91, 108, 32 · 1729, 3144, 1904, 384 … Σ 1, 11, 231, 7161, 293601, 14973651, …
  (−∞, −1]nonnegativerational roots
• A272866 → basis A013609 → reverse
  1 · 4, 10, 5 · 16, 80, 140, 100, 25 … Σ 1, 19, 361, 6859, 130321, 2476099, …
  (−∞, 0]nonnegative
• A257613 → x ↦ x+1
  1 · 8, 4 · 80, 80, 16 · 960, 1440, 608, 64 … Σ 1, 12, 176, 3072, 62336, 1443072, …
  (−∞, −1]nonnegative
• A357613 → D+1
  1 · 5, 2 · 35, 32, 6 · 252, 378, 165, 20 … Σ 1, 7, 73, 815, 9281, 106375, 1221305, …
  (−∞, 0]nonnegative
• A143495 → x ↦ x+1 → D
  1 · 9, 2 · 64, 30, 3 · 427, 314, 66, 4 … Σ 0, 1, 11, 97, 811, 6761, 57363, 499969, …
  (−∞, 0]nonnegative
• A038334 → x ↦ x+1
  400, 320, 64 · 8000, 9600, 3840, 512 … Σ 784, 21952, 614656, 17210368, …
  (−∞, −1]nonnegativerational roots
• A207606 → basis A013610 → reverse
  18, 33, 13 · 54, 153, 132, 34 … Σ 64, 373, 2174, 12671, 73852, 430441, …
  (−∞, 0]nonnegative
• A081578 → basis A013609 → (θ+1)²
  1 · 2, 8 · 7, 56, 36 · 20, 240, 432, 128 … Σ 1, 10, 99, 820, 6421, 47646, 341431, …
  (−∞, 0]nonnegative
• A094638 → θ+1 → D+1
  1 · 3, 2 · 7, 18, 6 · 13, 78, 105, 24 … Σ 1, 5, 31, 220, 1756, 15596, 152804, …
  (−∞, 0]nonnegative
• A038291 → basis A013609 → basis A007318
  1 · 12, 2 · 144, 48, 4 · 1728, 864, 144, 8 … Σ 1, 14, 196, 2744, 38416, 537824, …
  (−∞, −1]nonnegativerational roots
• A125662 → x ↦ x/(x+1)
  1 · 3, 4 · 8, 22, 15 · 21, 88, 122, 56 … Σ 1, 7, 45, 287, 1829, 11655, 74269, …
  [−1, −1/2)nonnegative
• A121574 → basis A013609 → θ+1
  1 · 3, 4 · 10, 28, 12 · 33, 140, 132, 32 … Σ 1, 7, 50, 337, 2201, 14044, 88061, …
  (−∞, 0]nonnegative
• A143494 → D
  1 · 5, 2 · 19, 18, 3 · 65, 110, 42, 4 … Σ 0, 1, 7, 40, 221, 1241, 7218, 43807, …
  (−∞, 0]nonnegative
• A013610 → basis A013610 → θ+1
  1 · 4, 18 · 16, 144, 243 · 64, 864, 2916, 2916 … Σ 1, 22, 403, 6760, 107653, 1656538, …
  (−∞, 0]nonnegativerational roots
• A038338 → x ↦ x+1 → D
  576, 288 · 20736, 20736, 5184 … Σ 864, 46656, 2239488, 100776960, …
  (−∞, 0]nonnegativerational roots
• A391533 → Veronese 2 → D+1
  5, 1 · 9, 5 · 17, 15, 1 · 29, 39, 7 … Σ 6, 14, 33, 75, 168, 372, 816, 1776, …
  (−∞, 0]nonnegative
• A113214 → θ+1 → D
  2 · 6, 6 · 4, 24, 12 · 0, 30, 60, 20 … Σ 0, 2, 12, 40, 110, 270, 616, 1336, …
  (−∞, 0]nonnegative
• basis-map coefficients of A145596 → D
  2 · 2, 6 · -8, 16, 12 · -10, -30, 60, 20 … Σ 0, 2, 8, 20, 40, 70, 112, 168, …
  (−∞, ∞)
• A008288 → Veronese 2 → x ↦ x+1
  1 · 1 · 2, 1 · 6, 5 … Σ 1, 1, 3, 11, 31, 87, 255, 743, …
  (−∞, −1]nonnegative
• A130850 → D
  1 · 3, 2 · 12, 14, 3 · 60, 100, 45, 4 … Σ 0, 1, 5, 29, 209, 1809, 18269, 210853, …
  (−∞, −1]nonnegative
• A143491 → basis A008277 → x ↦ x/(x+1)
  12, 32, 21 · 60, 240, 315, 136 … Σ 65, 751, 10369, 166391, 3040897, …
  [−1, 0]nonnegative