Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A113953 → basis A013610 → θ+1
  1 · 1, 6 · 3, 24, 27 · 5, 66, 189, 108 … Σ 1, 7, 54, 368, 2432, 15552, 97408, …
  (−∞, 0]nonnegative
• image under θ+1 → D+1
  1 · 2, 2 · 0, 6, 3 · 0, 0, 12, 4 … Σ 1, 4, 9, 16, 25, 36, 49, 64, …
  (−∞, 0]nonnegativerational roots
• A128813 → basis A007318 → x ↦ x+1
  1 · 3, 1 · 21, 16, 3 · 273, 334, 135, 18 … Σ 1, 4, 40, 760, 23560, 1083760, …
  (−∞, −1]nonnegativerational roots
• A304236 → θ+1
  1 · 1 · 1, 6 · 1, 12 … Σ 1, 1, 7, 13, 46, 106, 301, 739, …
  (−∞, 0]nonnegative
• A143491 → D+1
  11, 7, 1 · 50, 44, 12, 1 · 274, 296, 113, 18, 1 … Σ 19, 107, 702, 5274, 44712, 422568, …
  (−∞, 0]nonnegative
• A013612 → basis A013609 → D+1
  156, 320, 100 · 1296, 4680, 4800, 1000 … Σ 576, 11776, 229376, 4325376, 79691776, …
  (−∞, 0]nonnegativerational roots
• A113214 → x ↦ x−7
  1 · -5, 1 · 28, -11, 1 · -161, 93, -17, 1 … Σ 1, -4, 18, -84, 396, -1872, 8856, …
  (3, 7]
• A026729 → basis A013610 → reverse
  1 · 3, 1 · 9, 9, 2 · 27, 45, 21, 3 … Σ 1, 4, 20, 96, 464, 2240, 10816, 52224, …
  (−∞, 0]nonnegative
• A116414 → D+1
  1 · 5, 1 · 21, 10, 1 · 82, 66, 15, 1 … Σ 1, 6, 32, 164, 819, 4012, 19363, 92352, …
  (−∞, 0]nonnegative
• A075502 → x ↦ x+1
  71, 23, 1 · 729, 430, 45, 1 · 8842, 7809, 1441, 74, 1 … Σ 95, 1205, 18167, 316341, 6201063, …
  (−∞, −1]nonnegative
• A013612 → basis A013611 → D+1
  276, 1040, 400 · 2376, 16560, 31200, 8000 … Σ 1716, 58136, 1863056, 57578976, …
  (−∞, 0]nonnegativerational roots
• A080248 → D+1
  1 · 2, 1 · 5, 6, 1 · 14, 33, 13, 1 … Σ 1, 3, 12, 61, 385, 2950, 26861, 285687, …
  (−∞, 0]nonnegative
• A028326 → basis A007318 → basis A013609
  2 · 6, 4 · 18, 24, 8 · 54, 108, 72, 16 … Σ 2, 10, 50, 250, 1250, 6250, 31250, …
  (−∞, 0]nonnegativerational roots
• A099040 → basis A007318 → x ↦ x+1
  1 · 4, 2 · 20, 18, 4 · 96, 128, 56, 8 … Σ 1, 6, 42, 288, 1980, 13608, 93528, …
  (−∞, −1]nonnegative
• A129710 → (θ+1)²
  1 · 2 · 2, 4 · 2, 12 … Σ 1, 2, 6, 14, 31, 66, 135, 270, …
  (−∞, 0]nonnegative
• A038271 → basis A013610 → reverse
  1 · 15, 12 · 225, 360, 144 · 3375, 8100, 6480, 1728 … Σ 1, 27, 729, 19683, 531441, 14348907, …
  (−∞, 0]nonnegativerational roots
• A030528 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 3, 18, 24 · 5, 45, 129, 116 … Σ 1, 7, 45, 295, 1925, 12575, 82125, …
  [−1, 0]nonnegative
• A133087 → reverse → θ+1
  1 · 1, 4 · 1, 4, 3 · 1, 8, 15, 8 … Σ 1, 5, 8, 32, 48, 176, 256, 896, …
  (−∞, 0]nonnegative
• A081577 → basis A013609 → D+1
  1 · 4, 2 · 18, 20, 4 · 64, 128, 64, 8 … Σ 1, 6, 42, 264, 1620, 9672, 56712, …
  (−∞, 0]nonnegative
• A109692: Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).
  1 · 1, 1 · 1, 4, 3 · 1, 9, 23, 15 … Σ 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  (−∞, 0)nonnegativerational roots
• A039756 → x ↦ x+1 → θ+1
  1 · 2, 2 · 6, 12, 3 · 24, 76, 48, 4 … Σ 1, 4, 21, 152, 1361, 13956, 159701, …
  (−∞, 0]nonnegative
• A130123 → basis A013610 → D+1
  1 · 8, 6 · 28, 96, 36 · 80, 504, 864, 216 … Σ 1, 14, 160, 1664, 16384, 155648, …
  (−∞, 0]nonnegativerational roots
• A104029 → D+1
  1 · 3, 1 · 8, 7, 1 · 17, 31, 12, 1 … Σ 1, 4, 16, 61, 223, 790, 2734, 9295, …
  (−∞, 0]nonnegative
• A131689 → D
  1 · 1, 4 · 1, 12, 18 · 1, 28, 108, 96 … Σ 0, 1, 5, 31, 233, 2071, 21305, 249271, …
  (−∞, 0]nonnegative
• A130123 → basis A013610 → θ+1
  1 · 2, 12 · 4, 48, 108 · 8, 144, 648, 864 … Σ 1, 14, 160, 1664, 16384, 155648, …
  (−∞, 0]nonnegativerational roots
• A103327 → basis A013609 → basis A007318
  1 · 6, 2 · 44, 32, 4 · 328, 376, 120, 8 … Σ 1, 8, 80, 832, 8704, 91136, 954368, …
  (−∞, −1]nonnegative
• A249253 → basis A007318
  1 · 4, 1 · 17, 5 · 89, 30, 1 … Σ 1, 5, 22, 120, 764, 5588, 46232, …
  (−∞, −1]nonnegative
• A038327 → basis A013609 → D+1
  1 · 15, 2 · 221, 60, 4 · 3211, 1326, 180, 8 … Σ 1, 17, 285, 4725, 77625, 1265625, …
  (−∞, 0]nonnegativerational roots
• A272866 → basis A007318 → D
  5, 2 · 50, 70, 30, 4 · 375, 900, 825, 360, 75, 6 … Σ 0, 7, 154, 2541, 37268, 512435, …
  (−∞, −1]nonnegative
• A227450 → θ+1
  2 · 2, 8 · 2, 24, 24 · 2, 56, 144, 64 … Σ 2, 10, 50, 266, 1522, 9354, 61490, …
  (−∞, 0]nonnegative
• A161198 → θ+1
  1 · 1, 4 · 3, 16, 12 · 15, 92, 108, 32 … Σ 1, 5, 31, 247, 2433, 28653, 393279, …
  (−∞, 0]nonnegative
• A038243 → reverse
  1 · 1, 5 · 1, 10, 25 · 1, 15, 75, 125 … Σ 1, 6, 36, 216, 1296, 7776, 46656, …
  (−∞, 0]nonnegativerational roots
• A062145 → x ↦ x+1 → D+1
  1 · 9, 4 · 51, 50, 10 · 252, 378, 165, 20 … Σ 1, 13, 111, 815, 5573, 36633, 234979, …
  (−∞, 0]nonnegative
• A257606 → D
  4 · 40, 32 · 296, 592, 192 · 1928, 7104, 5784, 1024 … Σ 0, 4, 72, 1080, 15840, 237600, 3706560, …
  (−∞, 0]nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
  121, 110, 25 · 1331, 1815, 825, 125 … Σ 256, 4096, 65536, 1048576, 16777216, …
  (−∞, −1]nonnegativerational roots
• A156992 → basis A008277 → (θ+1)²
  1 · 4, 8 · 24, 120, 54 · 192, 1824, 1944, 384 … Σ 1, 12, 198, 4344, 122400, 4303440, …
  (−∞, 0]nonnegative
• A003506 → x ↦ x+1 → D+1
  1 · 6, 2 · 24, 18, 3 · 80, 96, 36, 4 … Σ 0, 1, 8, 45, 216, 945, 3888, 15309, …
  (−∞, 0]nonnegativerational roots
• A132812 → basis A007318 → x ↦ x/(x+1)
  1 · 4, 6 · 15, 45, 33 · 56, 252, 372, 180 … Σ 1, 10, 93, 860, 7985, 74550, 699685, …
  [−1, 0]nonnegative
• A054142 → Veronese 2 → D+1
  1 · 1 · 2, 1 · 7, 6 … Σ 1, 1, 3, 13, 34, 102, 305, 875, …
  (−∞, 0]nonnegative
• A081580 → basis A013611 → x ↦ x+1
  1 · 6, 4 · 61, 68, 16 · 516, 872, 448, 64 … Σ 1, 10, 145, 1900, 25525, 340750, …
  (−∞, −1]nonnegative
• A286718 → basis A013611 → θ+1
  1 · 2, 8 · 10, 56, 48 · 80, 528, 720, 256 … Σ 1, 10, 114, 1584, 26352, 513216, …
  (−∞, 0]nonnegative
• A167884 → D+1
  1 · 2, 1 · 19, 20, 1 · 180, 537, 182, 1 … Σ 1, 3, 40, 900, 28080, 1113840, …
  (−∞, 0]nonnegative
• A130123 → basis A008277 → θ+1
  1 · 2, 4 · 4, 24, 12 · 8, 112, 144, 32 … Σ 1, 6, 40, 296, 2416, 21568, 208832, …
  (−∞, 0]nonnegative
• A116414 → basis A013609 → D+1
  1 · 7, 2 · 42, 28, 4 · 233, 258, 84, 8 … Σ 1, 9, 74, 583, 4461, 33416, 246289, …
  (−∞, 0]nonnegative
• A145324 → basis A013611 → x ↦ x/(x+1)
  1 · 3, 11 · 12, 92, 176 · 60, 712, 2812, 3696 … Σ 1, 14, 280, 7280, 232960, 8852480, …
  [−1, 0]nonnegativerational roots
• A137597 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 3, 4 · 11, 29, 19 · 47, 184, 239, 103 … Σ 1, 7, 59, 573, 6241, 74887, 977419, …
  [−1, 0]nonnegative
• A111125 → D → x ↦ x+1
  1 · 7, 2 · 31, 20, 3 · 115, 120, 39, 4 … Σ 0, 1, 9, 54, 278, 1323, 6003, 26380, …
  (−∞, −1]nonnegative
• A079510 → (θ+1)²
  1 · 0, 8 · 0, 12, 54 · 0, 0, 180, 384 … Σ 1, 8, 66, 564, 5215, 52380, 571228, …
  (−∞, 0]nonnegative
• A124932 → reverse
  1 · 3, 2 · 6, 9, 3 · 10, 24, 18, 4 … Σ 1, 5, 18, 56, 160, 432, 1120, 2816, …
  (−∞, 0]nonnegativerational roots
• A143495 → (θ+1)²
  1 · 3, 4 · 9, 28, 9 · 27, 148, 108, 16 … Σ 1, 7, 46, 299, 1967, 13248, 91903, …
  (−∞, 0]nonnegative