Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A120910 → basis A013610
  3 · 9, 9 · 27, 54, 27 · 81, 243, 243, 81 … Σ 3, 18, 108, 648, 3888, 23328, 139968, …
  (−∞, 0]nonnegativerational roots
• A048854 → x ↦ x+1
  1 · 3, 1 · 25, 14, 1 · 331, 243, 33, 1 … Σ 1, 4, 40, 608, 12224, 303872, 8969728, …
  (−∞, −1]nonnegative
• A159854 → basis A013611 → D+1
  9, 40, 16 · 22, 148, 256, 64 … Σ 65, 490, 3540, 24840, 170640, 1153440, …
  (−∞, 0]nonnegative
• A054142 → Veronese 2 → θ+1
  1 · 1 · 1, 2 · 1, 12 … Σ 1, 1, 3, 13, 34, 102, 305, 875, …
  (−∞, 0]nonnegative
• A178820 → Veronese 2 → basis A007318
  1 · 4 · 20, 10 · 80, 60 … Σ 1, 4, 30, 140, 595, 2296, 8316, 28680, …
  (−∞, −1]nonnegative
• A159854 → basis A008277 → θ+1
  1, 6, 3 · 2, 20, 21, 4 · 4, 64, 114, 48, 5 … Σ 10, 47, 235, 1248, 7023, 41767, 261792, …
  (−∞, 0]nonnegative
• A124932 → basis A007318 → (θ+1)²
  1 · 5, 12 · 18, 84, 54 · 56, 384, 486, 160 … Σ 1, 17, 156, 1086, 6435, 34263, 168966, …
  (−∞, 0]nonnegative
• A154227 → θ+1
  1 · 1, 2 · 1, 16, 3 · 1, 38, 57, 4 … Σ 1, 3, 20, 100, 690, 4550, 36160, …
  (−∞, 0]nonnegative
• A257608 → x ↦ x+1
  1 · 2, 1 · 22, 22, 1 · 440, 660, 222, 1 … Σ 1, 3, 45, 1323, 55485, 3042603, …
  (−∞, −1]nonnegative
• A001263 → basis A013610 → x ↦ x+1
  1 · 5, 3 · 29, 33, 9 · 185, 306, 162, 27 … Σ 1, 8, 71, 680, 6882, 72528, 788019, …
  (−∞, −1]nonnegative
• A027465 → Veronese 2 → θ+1
  1 · 3 · 9, 2 · 27, 18 … Σ 1, 3, 11, 45, 192, 828, 3568, 15312, …
  (−∞, 0]nonnegative
• A038325 → basis A013611
  1 · 22, 44 · 484, 1936, 1936 … Σ 1, 66, 4356, 287496, 18974736, …
  [−1, 0]nonnegativerational roots
• A121574 → basis A013609 → reverse
  1 · 2, 3 · 4, 14, 10 · 8, 44, 70, 33 … Σ 1, 5, 28, 155, 859, 4760, 26377, …
  (−∞, 0]nonnegative
• A038319 → basis A013610 → D
  480, 450 · 11520, 21600, 10125 … Σ 930, 43245, 1787460, 69264075, …
  (−∞, 0]nonnegativerational roots
• A081579 → basis A013610 → D
  3 · 24, 18 · 108, 252, 81 · 480, 1800, 1620, 324 … Σ 0, 3, 42, 441, 4224, 37839, 326646, …
  (−∞, 0]nonnegative
• A085881 → D
  1 · 6, 6 · 45, 90, 45 · 420, 1260, 1260, 420 … Σ 0, 1, 12, 180, 3360, 75600, 1995840, …
  (−∞, 0]nonnegativerational roots
• A038231 → D+1
  1 · 5, 1 · 24, 10, 1 · 112, 72, 15, 1 … Σ 1, 6, 35, 200, 1125, 6250, 34375, …
  (−∞, 0]nonnegativerational roots
• A100861 → basis A013610 → D
  3 · 9 · 36, 54 · 120, 270 … Σ 0, 0, 3, 9, 90, 390, 3285, 17703, …
  (−∞, 0]nonnegative
• A128100 → basis A007318 → D+1
  1 · 1 · 4, 1 · 7, 2 … Σ 1, 1, 5, 9, 28, 62, 165, 391, …
  (−∞, 0]nonnegative
• A136124 → basis A008277 → D+1
  1 · 4, 1 · 20, 10, 1 · 120, 90, 18, 1 … Σ 1, 5, 31, 229, 1961, 19081, 207775, …
  (−∞, 0]nonnegative
• A132812: Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
  1 · 2, 2 · 3, 9, 3 · 4, 24, 24, 4 … Σ 1, 4, 15, 56, 210, 792, 3003, 11440, …
  (−∞, 0)nonnegative
• A038321 → basis A013610 → D+1
  1080, 1638, 441 · 26244, 68040, 51597, 9261 … Σ 3159, 155142, 7296237, 333135504, …
  (−∞, 0]nonnegativerational roots
• A201637 → x ↦ x+1
  1 · 1 · 3, 2 · 15, 20, 6 … Σ 1, 1, 5, 41, 469, 6889, 123605, …
  (−∞, −1]nonnegative
• A156365 → x ↦ x+1
  13, 16, 4 · 75, 134, 68, 8 · 541, 1268, 984, 272, 16 … Σ 33, 285, 3081, 40005, 606033, 10491885, …
  (−∞, −1]nonnegative
• A038314 → x ↦ x+1 → x ↦ x/(x+1)
  484, 1496, 1156 · 10648, 49368, 76296, 39304 … Σ 3136, 175616, 9834496, 550731776, …
  [−1, 0]nonnegativerational roots
• image under θ+1 → θ+1 → D+1 → θ+1
  1 · 4, 8 · 0, 36, 27 · 0, 0, 144, 64 … Σ 1, 12, 63, 208, 525, 1116, 2107, 3648, …
  (−∞, 0]nonnegativerational roots
• A154602 → reverse
  1 · 1, 1 · 1, 4, 3 · 1, 9, 19, 11 … Σ 1, 2, 8, 40, 240, 1664, 12992, 112256, …
  (−∞, 0]nonnegative
• A103327 → x ↦ x+1 → reverse
  1 · 1, 4 · 1, 12, 16 · 1, 24, 80, 64 … Σ 1, 5, 29, 169, 985, 5741, 33461, …
  (−∞, 0]nonnegative
• A356546 → basis A013611 → D+1
  1 · 12, 8 · 120, 288, 96 · 1120, 4800, 5760, 1280 … Σ 1, 20, 504, 12960, 332640, 8491392, …
  (−∞, 0]nonnegativerational roots
• A054142 → D+1
  1 · 2, 1 · 4, 5, 1 · 6, 17, 9, 1 … Σ 1, 3, 10, 33, 105, 324, 977, 2895, …
  (−∞, 0]nonnegative
• A178822 → basis A013609
  1 · 12, 12 · 84, 168, 84 · 448, 1344, 1344, 448 … Σ 1, 24, 336, 3584, 32256, 258048, …
  (−∞, 0]nonnegativerational roots
• A094638 → D+1 → reverse
  1 · 1, 2 · 2, 7, 4 · 6, 29, 28, 7 … Σ 1, 3, 13, 70, 446, 3276, 27252, 253296, …
  (−∞, 0]nonnegative
• A038208 → basis A007318 → D
  2 · 16, 8 · 96, 96, 24 · 512, 768, 384, 64 … Σ 0, 2, 24, 216, 1728, 12960, 93312, …
  (−∞, −1]nonnegativerational roots
• A116414 → basis A013609 → D
  2 · 20, 8 · 138, 120, 24 · 820, 1128, 480, 64 … Σ 0, 2, 28, 282, 2492, 20536, 162036, …
  (−∞, 0]nonnegative
• A008459 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 6, 30, 33 · 20, 150, 348, 245 … Σ 1, 7, 69, 763, 8881, 106407, 1298949, …
  [−1, 0]nonnegative
• A013610 → Veronese 2 → θ+1
  1 · 1 · 1, 18 · 1, 54 … Σ 1, 1, 19, 55, 352, 1396, 6832, 29296, …
  (−∞, 0]nonnegative
• A257615 → x ↦ x+1
  1 · 10, 5 · 120, 120, 25 · 1680, 2520, 1090, 125 … Σ 1, 15, 265, 5415, 125905, 3285375, …
  (−∞, −1]nonnegative
• basis-map coefficients of A112227 → basis A013609 → x ↦ x+1
  9, 12, 4 · 27, 54, 36, 8 · 81, 216, 216, 96, 16 … Σ 25, 125, 625, 3125, 15625, 78125, …
  (−∞, −1]nonnegativerational roots
• A038231 → D → (θ+1)²
  1 · 8, 8 · 48, 96, 27 · 256, 768, 432, 64 … Σ 0, 1, 16, 171, 1520, 12125, 90000, …
  (−∞, 0]nonnegative
• A088729 → θ+1
  13, 18, 3 · 75, 158, 54, 4 · 541, 1530, 795, 120, 5 … Σ 34, 291, 2991, 35806, 488559, 7476265, …
  (−∞, 0]nonnegative
• A100861 → basis A007318 → x ↦ x+1
  1 · 1 · 3, 1 · 7, 3 … Σ 1, 1, 4, 10, 46, 166, 856, 3844, …
  (−∞, −1]nonnegative
• A159854 → x ↦ x+1 → θ+1
  1, 4, 3 · 2, 10, 12, 4 · 4, 24, 39, 24, 5 … Σ 8, 28, 96, 324, 1080, 3564, 11664, …
  (−∞, 0]nonnegative
• A122848 → basis A013609 → D+1
  1 · 3, 2 · 8, 14, 4 · 22, 66, 48, 8 … Σ 1, 5, 26, 144, 846, 5238, 34020, …
  (−∞, 0]nonnegative
• A062264 → x ↦ x/(x+1)
  1 · 1, 6 · 1, 14, 28 · 1, 24, 108, 120 … Σ 1, 7, 43, 253, 1462, 8378, 47818, …
  [−1, 0]nonnegative
• A086915 → D
  4 · 24, 16 · 144, 192, 48 · 960, 1920, 960, 128 … Σ 0, 4, 40, 384, 3968, 44960, 558528, …
  (−∞, 0]nonnegative
• A091042 → Veronese 2 → D+1
  1 · 1 · 6, 5 · 36, 35 … Σ 1, 1, 11, 71, 280, 1156, 5344, 23856, …
  (−∞, 0]nonnegative
• narayana → basis A013611 → reverse
  1 · 4, 2 · 16, 20, 5 · 64, 144, 84, 14 … Σ 1, 6, 41, 306, 2426, 20076, 171481, …
  (−∞, 0]nonnegative
• A132774 → Veronese 2 → θ+1
  1 · 2 · 0, 10 · 0, 12 … Σ 1, 2, 10, 12, 27, 30, 52, 56, …
  (−∞, 0]nonnegativerational roots
• A094816 → basis A013611 → D+1
  1 · 6, 4 · 25, 52, 16 · 108, 380, 336, 64 … Σ 1, 10, 93, 888, 8977, 97102, 1127021, …
  (−∞, 0]nonnegative
• A038271 → basis A013609 → (θ+1)²
  1 · 12, 40 · 144, 960, 900 … Σ 1, 52, 2004, 67408, 2092816, 61588032, …
  (−∞, 0]nonnegative