Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A142075 → x ↦ x+1
  1 · 3, 2 · 13, 16, 4 · 75, 134, 68, 8 … Σ 1, 5, 33, 285, 3081, 40005, 606033, …
  (−∞, −1]nonnegative
• A119308 → x ↦ x/(x+1)
  1 · 2, 3 · 3, 11, 9 · 4, 26, 49, 28 … Σ 1, 5, 23, 107, 509, 2473, 12235, 61463, …
  [−1, 0]nonnegative
• A225471 → basis A013610 → D
  3 · 36, 18 · 528, 432, 81 · 9600, 10080, 3240, 324 … Σ 0, 3, 54, 1041, 23244, 600447, …
  (−∞, 0]nonnegative
• A156992 → basis A013609
  1 · 4, 4 · 24, 48, 24 · 192, 576, 576, 192 … Σ 1, 8, 96, 1536, 30720, 737280, …
  (−∞, 0]nonnegativerational roots
• A147716 → basis A013609 → (θ+1)²
  1 · 15, 8 · 225, 240, 36 · 3375, 5400, 1620, 128 … Σ 1, 23, 501, 10523, 215305, 4318527, …
  (−∞, 0]nonnegative
• A132393 → reverse → (θ+1)²
  1 · 1 · 1, 4 · 1, 12, 18 … Σ 1, 1, 5, 31, 220, 1756, 15596, 152804, …
  (−∞, 0]nonnegative
• A133214 → basis A007318 → θ+1
  13, 32, 12 · 63, 228, 180, 32 … Σ 57, 503, 4577, 42135, 389625, 3608295, …
  (−∞, 0]nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.
  81, 198, 121 · 729, 2673, 3267, 1331 … Σ 400, 8000, 160000, 3200000, 64000000, …
  (−∞, 0)nonnegativerational roots
• A039756 → x ↦ x+1 → x ↦ x+1
  1 · 3, 1 · 13, 8, 1 · 79, 73, 19, 1 … Σ 1, 4, 22, 172, 1732, 20560, 277336, …
  (−∞, −1]nonnegative
• narayana → basis A013611 → x ↦ x/(x+1)
  1 · 2, 6 · 5, 30, 41 · 14, 126, 354, 306 … Σ 1, 8, 76, 800, 8992, 105728, 1284544, …
  [−1, 0]nonnegative
• A038319 → basis A013611 → D
  640, 800 · 15360, 38400, 24000 … Σ 1440, 77760, 3732480, 167961600, …
  (−∞, 0]nonnegativerational roots
• A109692 → θ+1
  1 · 1, 2 · 1, 8, 9 · 1, 18, 69, 60 … Σ 1, 3, 18, 148, 1520, 18656, 266112, …
  (−∞, 0]nonnegative
• A062196 → D+1
  1 · 4, 3 · 9, 20, 6 · 16, 75, 60, 10 … Σ 1, 7, 35, 161, 714, 3102, 13299, 56485, …
  (−∞, 0]nonnegative
• A038301 → reverse → D+1
  319, 360, 81 · 4598, 8613, 4860, 729 … Σ 760, 18800, 448000, 10400000, …
  (−∞, 0]nonnegativerational roots
• A103327 → reverse → x ↦ x+1
  1 · 4, 3 · 16, 20, 5 · 64, 112, 56, 7 … Σ 1, 7, 41, 239, 1393, 8119, 47321, …
  (−∞, −1]nonnegative
• A136124 → basis A013609 → x ↦ x+1
  1 · 5, 2 · 30, 22, 4 · 210, 214, 72, 8 … Σ 1, 7, 56, 504, 5040, 55440, 665280, …
  (−∞, −1]nonnegativerational roots
• A119900 → (θ+1)²
  1 · 0, 8 · 0, 4, 27 · 0, 0, 36, 64 … Σ 1, 8, 31, 100, 294, 812, 2144, 5472, …
  (−∞, 0]nonnegative
• A038337 → basis A007318 → reverse
  121, 506, 529 · 1331, 8349, 17457, 12167 … Σ 1156, 39304, 1336336, 45435424, …
  (−∞, 0]nonnegativerational roots
• A125662 → θ+1
  1 · 3, 2 · 8, 12, 3 · 21, 50, 27, 4 … Σ 1, 5, 23, 102, 441, 1871, 7823, 32332, …
  (−∞, 0]nonnegative
• A142459: Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).
  1 · 1, 1 · 1, 10, 1 · 1, 59, 59, 1 … Σ 1, 2, 12, 120, 1680, 30240, 665280, …
  (−∞, 0)nonnegative
• A027467 → basis A013611 → D+1
  1 · 20, 4 · 384, 160, 16 · 7168, 4608, 960, 64 … Σ 1, 24, 560, 12800, 288000, 6400000, …
  (−∞, 0]nonnegativerational roots
• A099097 → basis A013609 → D
  6 · 38, 72 · 186, 696, 648 · 814, 4544, 8424, 5184 … Σ 0, 6, 110, 1530, 18966, 220644, …
  (−∞, 0]nonnegative
• A367270 → basis A013611 → θ+1
  1 · 1 · 3, 16 · 10, 96, 144 … Σ 1, 1, 19, 250, 2979, 34206, 385646, …
  (−∞, 0]nonnegative
• A038271 → D
  5 · 70, 50 · 735, 1050, 375 … Σ 0, 5, 120, 2160, 34560, 518400, …
  (−∞, 0]nonnegativerational roots
• A038338 → basis A013609 → (θ+1)²
  576, 4608, 5184 · 13824, 165888, 373248, 221184 … Σ 10368, 774144, 53084160, 3439853568, …
  (−∞, 0]nonnegative
• A038208 → basis A007318 → θ+1
  1 · 4, 4 · 16, 32, 12 · 64, 192, 144, 32 … Σ 1, 8, 60, 432, 3024, 20736, 139968, …
  (−∞, 0]nonnegativerational roots
• A008299 → D
  3 · 10 · 25, 30 · 56, 210 … Σ 0, 0, 3, 10, 55, 266, 1414, 7862, …
  (−∞, 0]nonnegative
• A112493 → D+1
  1 · 2, 1 · 5, 10, 3 · 12, 61, 70, 15 … Σ 1, 3, 18, 158, 1808, 25448, 424784, …
  (−∞, 0]nonnegative
• A081578 → basis A013609 → basis A007318
  1 · 4, 2 · 25, 22, 4 · 136, 180, 72, 8 … Σ 1, 6, 51, 396, 3141, 24786, 195831, …
  (−∞, −1]nonnegative
• A172431 → basis A007318 → (θ+1)²
  1 · 3, 8 · 8, 40, 27 · 21, 152, 198, 64 … Σ 1, 11, 75, 435, 2339, 12027, 59963, …
  (−∞, 0]nonnegative
• A013611 → basis A007318
  1 · 5, 4 · 25, 40, 16 · 125, 300, 240, 64 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, −1]nonnegativerational roots
• A059110 → D+1
  1 · 2, 1 · 7, 6, 1 · 34, 39, 12, 1 … Σ 1, 3, 14, 86, 648, 5752, 58576, 671568, …
  (−∞, 0]nonnegative
• A038328 → basis A013611 → (θ+1)²
  196, 896, 576 · 2744, 18816, 24192, 8192 … Σ 1668, 53944, 1628176, 46893792, …
  (−∞, 0]nonnegative
• A085880 → basis A013611 → D+1
  1 · 6, 4 · 40, 96, 32 · 280, 1200, 1440, 320 … Σ 1, 10, 168, 3240, 66528, 1415232, …
  (−∞, 0]nonnegativerational roots
• A207607 → basis A013610 → θ+1
  1 · 3, 12 · 8, 54, 54 · 21, 198, 405, 216 … Σ 1, 15, 116, 840, 5851, 39669, 263654, …
  (−∞, 0]nonnegative
• A001263 → basis A013610 → D+1
  1 · 5, 3 · 20, 33, 9 · 77, 225, 162, 27 … Σ 1, 8, 62, 491, 3975, 32730, 272940, …
  (−∞, 0]nonnegative
• A038330 → basis A007318 → D+1
  384, 160, 16 · 7168, 4608, 960, 64 … Σ 560, 12800, 288000, 6400000, 140800000, …
  (−∞, 0]nonnegativerational roots
• A178820 → D
  4 · 20, 20 · 60, 120, 60 · 140, 420, 420, 140 … Σ 0, 4, 40, 240, 1120, 4480, 16128, …
  (−∞, 0]nonnegativerational roots
• A165253 → x ↦ x+1
  1 · 1 · 2, 1 · 5, 5, 1 … Σ 1, 1, 3, 11, 41, 153, 571, 2131, …
  (−5, −1)nonnegative
• A038221 → γ
  1 · 3 · 9 · 27 … Σ 1, 3, 9, 27, 81, 243, 729, 2187, …
  (−∞, −1][−1, 0][−3−2√2, −3+2√2]+2
• A132393 → θ+1 → (θ+1)²
  1 · 0, 8 · 0, 8, 27 · 0, 16, 81, 64 … Σ 1, 8, 35, 161, 854, 5248, 36966, …
  (−∞, 0]nonnegative
• A039755 → reverse → D
  1 · 4, 2 · 9, 26, 3 · 16, 116, 120, 4 … Σ 0, 1, 6, 38, 256, 1844, 14296, 119160, …
  (−∞, 0]nonnegative
• A038226 → basis A013609 → D
  352, 512 · 5808, 16896, 12288 … Σ 864, 34992, 1259712, 42515280, …
  (−∞, 0]nonnegativerational roots
• A026729 → θ+1 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 2, 5 · 0, 0, 6, 10 … Σ 1, 2, 7, 16, 41, 94, 219, 492, …
  [−1, 0]nonnegative
• A207824 → basis A013609 → reverse
  1 · 2, 6 · 4, 24, 35 · 8, 72, 212, 204 … Σ 1, 8, 63, 496, 3905, 30744, 242047, …
  (−1/2, 0]nonnegative
• A038320 → θ+1
  121, 264, 108 · 1331, 4356, 3564, 864 … Σ 493, 10115, 201433, 3925487, 75252421, …
  (−∞, 0]nonnegativerational roots
• A013622 → x ↦ x/(x+1) → D+1
  1 · 11, 8 · 57, 176, 64 · 243, 1368, 2112, 512 … Σ 1, 19, 297, 4235, 57233, 746691, …
  (−∞, 0]nonnegativerational roots
• A280580 → θ+1
  1 · 1, 2 · 2, 12, 3 · 5, 60, 45, 4 … Σ 1, 3, 17, 114, 831, 6363, 50299, …
  (−∞, 0]nonnegative
• A028421 → θ+1 → x ↦ x/(x+1)
  1 · 1, 5 · 2, 16, 23 · 6, 62, 160, 120 … Σ 1, 6, 41, 348, 3589, 43802, 618869, …
  [−1, 0]nonnegative
• A011973 → D → θ+1
  1 · 2 · 3, 4 · 4, 12 … Σ 0, 0, 1, 2, 7, 16, 38, 82, …
  (−∞, 0]nonnegative