Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A100861 → basis A013611 → θ+1
  1 · 1 · 2, 8 · 4, 24 … Σ 1, 1, 10, 28, 250, 1066, 9436, 51640, …
  (−∞, 0]nonnegative
• A102413 → D
  1 · 4, 2 · 6, 12, 3 · 8, 32, 24, 4 … Σ 0, 1, 6, 21, 68, 205, 594, 1673, …
  (−∞, 0]nonnegative
• A052553 → basis A008277 → D
  1 · 2 · 6, 2 · 13, 6 … Σ 0, 0, 1, 2, 8, 19, 57, 144, …
  (−∞, 0]nonnegative
• A038329 → basis A013610 → D+1
  495, 432, 81 · 9450, 13365, 5832, 729 … Σ 1008, 29376, 829440, 22892544, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A026729 → D
  1 · 0, 2 · 0, 0, 3 · 0, 0, 0, 4 … Σ 0, 1, 2, 3, 4, 5, 6, 7, …
  (−∞, 0]nonnegativerational roots
• A059473 → γ → x ↦ x+1
  1 · 2 · 6, 2 · 16, 8 … Σ 1, 2, 8, 24, 80, 256, 832, 2688, …
  (−∞, −1]nonnegative
• A099093 → θ+1 → (θ+1)²
  1 · 0, 24 · 0, 24, 243 · 0, 0, 486, 1728 … Σ 1, 24, 267, 2214, 15552, 98172, 574965, …
  (−∞, 0]nonnegative
• A154986 → θ+1 → reverse
  1 · 2, 1 · 3, 8, 1 · 4, 33, 22, 1 … Σ 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  (−∞, 0]nonnegative
• A008955 → (θ+1)²
  1 · 1, 4 · 1, 20, 36 · 1, 56, 441, 576 … Σ 1, 5, 57, 1074, 30098, 1177048, …
  (−∞, 0]nonnegative
• A209414 → reverse
  1 · 1, 1 · 1, 4, 1 · 1, 9, 7, 1 … Σ 1, 2, 6, 18, 54, 162, 486, 1458, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → basis A013609 → D+1
  1 · -4, -2 · 12, 16, 4 · -32, -72, -48, -8 … Σ 1, -6, 32, -160, 768, -3584, 16384, …
  (−∞, 0]rational roots
• A137320 → Veronese 2 → (θ+1)²
  1 · 0, 16 · 0, 48 · 0, 176, 144 … Σ 1, 0, 16, 48, 320, 2240, 17648, 155568, …
  (−∞, 0]nonnegative
• chebyshev_t → x ↦ x+1 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 7 · 1, 12, 33, 26 … Σ 1, 3, 14, 72, 376, 1968, 10304, 53952, …
  [−1, 0)nonnegative
• A078812 → x ↦ x+1 → reverse
  1 · 1, 3 · 1, 6, 8 · 1, 9, 25, 21 … Σ 1, 4, 15, 56, 209, 780, 2911, 10864, …
  (−1, 0]nonnegative
• A008955 → basis A013609 → reverse
  1 · 2, 2 · 16, 26, 10 · 288, 628, 440, 100 … Σ 1, 4, 52, 1456, 71344, 5422144, …
  (−∞, 0]nonnegativerational roots
• A145596 → x ↦ x+1 → D
  2 · 14, 6 · 72, 64, 12 · 330, 450, 180, 20 … Σ 0, 2, 20, 148, 980, 6150, 37464, …
  (−∞, −1]nonnegative
• A249250 → basis A013609 → D+1
  1 · 10, 6 · 34, 48, 12 · 124, 288, 168, 24 … Σ 1, 16, 94, 604, 3652, 21712, 126712, …
  (−∞, 0]nonnegative
• A013610 → basis A008277
  1 · 4, 3 · 16, 33, 9 · 64, 279, 189, 27 … Σ 1, 7, 58, 559, 6127, 74920, 1007299, …
  (−∞, 0]nonnegative
• A028326 → basis A013611 → D
  8 · 32, 64 · 96, 384, 384 · 256, 1536, 3072, 2048 … Σ 0, 8, 96, 864, 6912, 51840, 373248, …
  (−∞, 0]nonnegativerational roots
• A033282 → basis A013609 → θ+1
  1 · 3, 8 · 11, 60, 60 · 45, 372, 756, 448 … Σ 1, 11, 131, 1621, 20525, 263927, …
  (−∞, 0]nonnegative
• A008278 → x ↦ x+1 → D
  1 · 5, 2 · 23, 20, 3 · 109, 152, 57, 4 … Σ 0, 1, 7, 46, 322, 2427, 19645, 170236, …
  (−∞, −1]nonnegative
• A028326 → basis A007318 → θ+1
  2 · 4, 4 · 8, 16, 6 · 16, 48, 36, 8 … Σ 2, 8, 30, 108, 378, 1296, 4374, 14580, …
  (−∞, 0]nonnegativerational roots
• Veronese parent of A109447 → Veronese 2 (j=1) → D
  3 · 4 · 10, 10 · 20, 12 … Σ 0, 0, 3, 4, 20, 32, 112, 192, …
  (−∞, 0]nonnegative
• A201730 → D
  1 · 6 · 26, 2 · 100, 20 … Σ 0, 0, 1, 6, 28, 120, 492, 1960, …
  (−∞, 0]nonnegative
• delannoy → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 5, 25, 29 · 12, 90, 216, 165 … Σ 1, 7, 59, 483, 3971, 32627, 268099, …
  [−1, 0]nonnegative
• A225470 → x ↦ x+1 → θ+1
  1 · 3, 2 · 18, 18, 3 · 162, 198, 54, 4 … Σ 1, 5, 39, 418, 5714, 95064, 1864456, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → D → D
  2 · 0, 0, 12 · 0, 0, 0, 0, 30 · 0, 0, 0, 0, 0, 0, 56 … Σ 0, 2, 12, 30, 56, 90, 132, 182, …
  (−∞, 0]nonnegativerational roots
• A126351 → reverse → (θ+1)²
  1 · 2, 4 · 4, 20, 9 · 8, 76, 81, 16 … Σ 1, 6, 33, 181, 1020, 5977, 36589, …
  (−∞, 0]nonnegative
• A113413 → (θ+1)²
  1 · 2, 4 · 2, 16, 9 · 2, 32, 54, 16 … Σ 1, 6, 27, 104, 365, 1206, 3815, 11680, …
  (−∞, 0]nonnegative
• A257611 → D
  3 · 30, 18 · 213, 426, 81 · 1308, 5964, 3924, 324 … Σ 0, 3, 48, 720, 11520, 201600, 3870720, …
  (−∞, 0]nonnegative
• A114687 → basis A007318 → D+1
  1 · 5, 2 · 25, 22, 4 · 129, 180, 72, 8 … Σ 1, 7, 51, 389, 3069, 24795, 203775, …
  (−∞, 0]nonnegative
• A049403 → basis A007318
  4, 5, 1 · 10, 18, 9, 1 · 26, 64, 51, 14, 1 … Σ 10, 38, 156, 692, 3256, 16200, 84496, …
  (−∞, −1]nonnegative
• A249251 → D+1
  1 · 4, 3 · 7, 18, 6 · 10, 51, 57, 12 … Σ 1, 7, 31, 130, 520, 2017, 7651, 28537, …
  (−∞, 0]nonnegative
• A159854 → basis A008277 → reverse
  1, 3, 1 · 1, 7, 10, 2 · 1, 12, 38, 32, 4 … Σ 5, 20, 87, 409, 2066, 11155, 64077, …
  (−∞, 0]nonnegative
• A225470 → basis A008277 → (θ+1)²
  1 · 3, 4 · 18, 40, 9 · 162, 472, 189, 16 … Σ 1, 7, 67, 839, 13037, 242151, 5234287, …
  (−∞, 0]nonnegative
• A201730 → θ+1
  1 · 2 · 5, 2 · 14, 12 … Σ 1, 2, 7, 26, 96, 352, 1284, 4664, …
  (−∞, 0]nonnegative
• A056242 → reverse → x ↦ x/(x+1)
  1 · 2, 3 · 4, 13, 10 · 8, 40, 65, 34 … Σ 1, 5, 27, 147, 801, 4365, 23787, …
  [−1, 0]nonnegative
• A053122 → D
  1 · -4, 2 · 10, -12, 3 · -20, 42, -24, 4 … Σ 0, 1, -2, 1, 2, -4, 2, 3, …
  (−∞, ∞)
• A161198 → basis A007318 → basis A008277
  1 · 5, 2 · 35, 28, 4 · 315, 378, 108, 8 … Σ 1, 7, 67, 809, 11753, 199127, 3848539, …
  (−∞, 0]nonnegative
• A156366 → x ↦ x+1
  22, 30, 9 · 160, 312, 180, 27 … Σ 61, 679, 9445, 158095, 3088765, …
  (−∞, −1]nonnegative
• A059473 → basis A013610 → θ+1
  1 · 4, 12 · 18, 108, 108 · 80, 720, 1512, 864 … Σ 1, 16, 234, 3176, 41240, 519552, …
  (−∞, 0]nonnegative
• A257624 → θ+1
  1 · 5, 10 · 25, 160, 75 · 125, 1830, 2745, 500 … Σ 1, 15, 260, 5200, 118560, 3043040, …
  (−∞, 0]nonnegative
• A038291 → basis A013610 → D+1
  1 · 13, 3 · 160, 78, 9 · 1900, 1440, 351, 27 … Σ 1, 16, 247, 3718, 54925, 799708, …
  (−∞, 0]nonnegativerational roots
• A249250: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 3 · 1, 6, 3 · 1, 9, 15, 3 … Σ 1, 4, 10, 28, 76, 208, 568, 1552, …
  (−∞, 0)nonnegative
• A139548 → basis A013609 → (θ+1)²
  1 · 8, 64, 72 · 64, 1024, 3456, 4096, 1600 … Σ 1, 144, 10240, 573440, 28311552, …
  (−∞, 0]nonnegative
• A257619 → D
  2 · 44, 8 · 564, 1128, 24 · 6436, 45120, 19308, 64 … Σ 0, 2, 52, 1716, 70928, 3546400, …
  (−∞, 0]nonnegative
• A130123 → θ+1
  1 · 0, 4 · 0, 0, 12 · 0, 0, 0, 32 … Σ 1, 4, 12, 32, 80, 192, 448, 1024, …
  [−1, 0]nonnegativerational roots
• A038319 → x ↦ x+1 → basis A007318
  441, 210, 25 · 9261, 6615, 1575, 125 … Σ 676, 17576, 456976, 11881376, …
  (−∞, −1]nonnegativerational roots
• A120910 → basis A013611 → x ↦ x+1
  3 · 21, 12 · 147, 168, 48 · 1029, 1764, 1008, 192 … Σ 3, 33, 363, 3993, 43923, 483153, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A026729, shifted → θ+1
  1 · 0, -2 · 0, 0, 3 · 0, 0, 0, -4 … Σ 1, -2, 3, -4, 5, -6, 7, -8, …
  [−1, 0]rational roots