Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A038208 → Veronese 2 → D+1
  1 · 2 · 8, 4 · 32, 24 … Σ 1, 2, 12, 56, 256, 1152, 5120, 22528, …
  (−∞, 0]nonnegative
• A116412 → basis A013610 → D+1
  1 · 7, 3 · 37, 42, 9 · 169, 342, 189, 27 … Σ 1, 10, 88, 727, 5764, 44428, 335440, …
  (−∞, 0]nonnegative
• A204420 → basis A007318
  1 · 1, 1 · 9, 12, 3 · 225, 345, 135, 15 … Σ 1, 2, 24, 720, 40320, 3628800, …
  (−∞, −1]nonnegativerational roots
• A008288 → basis A013609 → x ↦ x+1
  1 · 4, 2 · 19, 18, 4 · 88, 124, 56, 8 … Σ 1, 6, 41, 276, 1861, 12546, 84581, …
  (−∞, −1]nonnegative
• A169803 → D → θ+1
  1 · 2 · 3, 4 · 4, 12 … Σ 0, 1, 2, 7, 16, 38, 82, 173, …
  (−∞, 0]nonnegative
• A038338 → x ↦ x+1
  576, 576, 144 · 13824, 20736, 10368, 1728 … Σ 1296, 46656, 1679616, 60466176, …
  (−∞, −1]nonnegativerational roots
• A008288 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 10, 15, 4 … Σ 1, 3, 10, 30, 87, 245, 676, 1836, …
  (−∞, 0]nonnegative
• A207607 → D
  2 · 5, 4 · 9, 18, 6 · 14, 50, 39, 8 … Σ 0, 2, 9, 33, 111, 355, 1098, 3316, …
  (−∞, 0]nonnegative
• A013622 → basis A013609 → (θ+1)²
  1 · 8, 40 · 64, 640, 900 · 512, 7680, 21600, 16000 … Σ 1, 48, 1604, 45792, 1193616, 29299968, …
  (−∞, 0]nonnegative
• A038303 → basis A013610 → D
  3 · 66, 18 · 1089, 594, 81 · 15972, 13068, 3564, 324 … Σ 0, 3, 84, 1764, 32928, 576240, 9680832, …
  (−∞, 0]nonnegativerational roots
• A053122 → x ↦ x+1
  1 · -1, 1 · 0, -2, 1 · 1, 1, -3, 1 … Σ 1, 0, -1, 0, 1, 0, -1, 0, …
  (−1, 3)
• A080248 → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 26, 30, 4 … Σ 1, 3, 12, 61, 385, 2950, 26861, 285687, …
  (−∞, 0]nonnegative
• A321434 → (θ+1)²
  1 · 0, 4 · 0, 4 · 0, 4, 18 … Σ 1, 4, 4, 22, 22, 154, 154, 1306, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729 → basis A013610 → D
  3 · 6, 18 · 9, 54, 81 · 12, 108, 324, 324 … Σ 0, 3, 24, 144, 768, 3840, 18432, 86016, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A371080, shifted → x ↦ x−1
  1 · -1, 1 · -2, 1, 1 · -10, 3, 6, 1 … Σ 1, 0, 0, 0, 0, 0, 0, 0, …
  (−∞, 1]rational roots
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
  1 · 1, 4 · 1, 8, 16 · 1, 12, 48, 64 … Σ 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  (−∞, 0)[−1, 0]nonnegative+1
• A038226 → basis A007318
  121, 176, 64 · 1331, 2904, 2112, 512 … Σ 361, 6859, 130321, 2476099, 47045881, …
  (−∞, −1]nonnegativerational roots
• A038291 → θ+1
  1 · 9, 2 · 81, 36, 3 · 729, 486, 81, 4 … Σ 1, 11, 120, 1300, 14000, 150000, …
  (−∞, 0]nonnegativerational roots
• A257615 → D
  5 · 70, 50 · 715, 1430, 375 … Σ 0, 5, 120, 2520, 53760, 1209600, …
  (−∞, 0]nonnegative
• A225470 → basis A013609 → basis A013609
  1 · 5, 4 · 40, 52, 16 · 440, 732, 384, 64 … Σ 1, 9, 108, 1620, 29160, 612360, …
  (−∞, −1]nonnegativerational roots
• A082985 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 4, 7 · 11, 37, 31 · 29, 143, 234, 127 … Σ 1, 11, 79, 533, 3553, 23627, 157039, …
  [−1, 0]nonnegative
• A102365 → x ↦ x+1 → θ+1
  1 · 1 · 2, 2 · 7, 14, 3 … Σ 1, 1, 4, 24, 195, 1995, 24570, 353430, …
  (−∞, 0]nonnegative
• A013610 → basis A007318 → θ+1
  1 · 4, 6 · 16, 48, 27 · 64, 288, 324, 108 … Σ 1, 10, 91, 784, 6517, 52822, 420175, …
  (−∞, 0]nonnegativerational roots
• A081577 → basis A013611 → D+1
  1 · 6, 4 · 30, 56, 16 · 112, 416, 352, 64 … Σ 1, 10, 102, 944, 8396, 72408, 611080, …
  (−∞, 0]nonnegative
• A062196 → (θ+1)²
  1 · 1, 12 · 1, 32, 54 · 1, 60, 270, 160 … Σ 1, 13, 87, 491, 2562, 12762, 61611, …
  (−∞, 0]nonnegative
• A257618 → x ↦ x/(x+1)
  1 · 2, 4 · 4, 48, 48 · 8, 496, 1440, 960 … Σ 1, 6, 100, 2904, 117520, 6105696, …
  [−1, 0]nonnegative
• A225470 → basis A013611 → D
  4 · 36, 32 · 396, 576, 192 · 5400, 10080, 5760, 1024 … Σ 0, 4, 68, 1164, 22264, 481256, …
  (−∞, 0]nonnegative
• A075499 → x ↦ x/(x+1)
  16, 44, 29 · 64, 304, 440, 201 … Σ 89, 1009, 13457, 210105, 3747753, …
  [−1, 0]nonnegative
• A062264 → reverse → D
  1 · 12, 2 · 63, 42, 3 · 224, 336, 96, 4 … Σ 0, 1, 14, 108, 660, 3575, 18018, 86632, …
  (−∞, 0]nonnegative
• A078812 → x ↦ x+1 → θ+1
  1 · 3, 2 · 8, 12, 3 · 21, 50, 27, 4 … Σ 1, 5, 23, 102, 441, 1871, 7823, 32332, …
  (−∞, 0]nonnegative
• A027466 → basis A013609 → (θ+1)²
  1 · 8, 8 · 64, 128, 36 · 512, 1536, 864, 128 … Σ 1, 16, 228, 3040, 38800, 480000, …
  (−∞, 0]nonnegative
• A286724 → D+1
  1 · 3, 1 · 16, 10, 1 · 120, 108, 21, 1 … Σ 1, 4, 27, 250, 2917, 40896, 667999, …
  (−∞, 0]nonnegative
• A137597 → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 9, 22, 15 … Σ 1, 3, 11, 47, 227, 1215, 7107, 44959, …
  (−∞, 0]nonnegative
• A116412 → basis A007318 → reverse
  1 · 1, 4 · 1, 8, 13 · 1, 12, 42, 43 … Σ 1, 5, 22, 98, 436, 1940, 8632, 38408, …
  (−∞, 0]nonnegative
• A371259 → (θ+1)²
  1 · 36, 4 · 1764, 400, 9 · 112896, 39456, 1800, 16 … Σ 1, 40, 2173, 154168, 13853865, …
  (−∞, 0]nonnegative
• A089503 → reverse
  1 · 4, 1 · 30, 12, 1 · 336, 168, 24, 1 … Σ 1, 5, 43, 529, 8501, 169021, 4010455, …
  (−∞, 0]nonnegative
• A063967 → basis A013611 → (θ+1)²
  1 · 2, 16 · 6, 80, 144 · 16, 320, 1152, 1024 … Σ 1, 18, 230, 2512, 25052, 235224, …
  (−∞, 0]nonnegative
• A128099 → D → D+1
  2 · 4 · 14, 8 · 32, 24 … Σ 0, 0, 2, 4, 22, 56, 178, 460, …
  (−∞, 0]nonnegative
• A131689 → D+1
  1 · 1, 1 · 1, 5, 2 · 1, 13, 24, 6 … Σ 1, 2, 8, 44, 308, 2612, 25988, 296564, …
  (−∞, 0]nonnegative
• A013620 → D+1
  1 · 5, 3 · 16, 30, 9 · 44, 144, 135, 27 … Σ 1, 8, 55, 350, 2125, 12500, 71875, …
  (−∞, 0]nonnegativerational roots
• A059364 → basis A013611 → D+1
  1 · 7, 4 · 59, 108, 32 · 593, 1992, 1904, 384 … Σ 1, 11, 199, 4873, 149489, 5498659, …
  (−∞, 0]nonnegative
• A147716 → basis A013609 → x ↦ x+1
  1 · 17, 2 · 289, 68, 4 · 4913, 1734, 204, 8 … Σ 1, 19, 361, 6859, 130321, 2476099, …
  (−∞, −1]nonnegativerational roots
• A143491 → basis A013610 → x ↦ x/(x+1)
  12, 45, 42 · 60, 321, 570, 336 … Σ 99, 1287, 19305, 328185, 6235515, …
  [−1, 0]nonnegativerational roots
• A127773 → D+1
  1 · 3, 3 · 0, 12, 6 · 0, 0, 30, 10 … Σ 1, 6, 18, 40, 75, 126, 196, 288, …
  (−∞, 0]nonnegativerational roots
• A008459 → basis A013609 → (θ+1)²
  1 · 2, 8 · 6, 48, 36 · 20, 240, 432, 128 … Σ 1, 10, 90, 820, 7390, 65484, 571060, …
  (−∞, 0]nonnegative
• A099093 → x ↦ x−12
  1 · -36, 3 · 1260, -213, 9 · -44064, 11232, -954, 27 … Σ 1, -33, 1056, -33759, 1079199, …
  (32/3, 12]
• A340556: E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.
  A163936 → reverse
  1 · 0, 1 · 0, 1, 2 · 0, 1, 8, 6 … Σ 1, 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, 0]nonnegative
• A356546 → x ↦ x+1 → (θ+1)²
  1 · 4, 8 · 24, 96, 54 · 160, 960, 1080, 320 … Σ 1, 12, 174, 2520, 35910, 503496, …
  (−∞, 0]nonnegative
• A173018 → x ↦ x+1 → (θ+1)²
  1 · 1 · 2, 4 · 6, 24, 9 … Σ 1, 1, 6, 39, 310, 2935, 32186, 400855, …
  (−∞, 0]nonnegative
• A178820 → basis A013611 → θ+1
  1 · 8, 32 · 40, 320, 480 · 160, 1920, 5760, 5120 … Σ 1, 40, 840, 12960, 166320, 1886976, …
  (−∞, 0]nonnegativerational roots