Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A209414 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 18, 28, 12, 1 … Σ 1, 3, 13, 59, 269, 1227, 5597, 25531, …
  (−∞, −1]nonnegative
• A257615 → D+1
  1 · 10, 5 · 95, 120, 25 · 840, 2145, 1090, 125 … Σ 1, 15, 240, 4200, 80640, 1693440, …
  (−∞, 0]nonnegative
• A272867 → basis A013611 → D+1
  1 · 30, 56, 16 · 324, 1824, 3072, 1792, 256 … Σ 1, 102, 7268, 452824, 26280720, …
  (−∞, 0]nonnegative
• A059364 → basis A013611 → θ+1
  1 · 3, 8 · 15, 88, 96 · 105, 976, 2256, 1536 … Σ 1, 11, 199, 4873, 149489, 5498659, …
  (−∞, 0]nonnegative
• A038243 → D
  1 · 10, 2 · 75, 30, 3 · 500, 300, 60, 4 … Σ 0, 1, 12, 108, 864, 6480, 46656, …
  (−∞, −1]nonnegativerational roots
• A128099: Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= …
  1 · 1 · 1, 2 · 1, 4 … Σ 1, 1, 3, 5, 11, 21, 43, 85, …
  (−∞, 0)nonnegative
• A169803 → basis A013609 → D+1
  1 · 4, 2 · 7, 4 · 15, 18, 4 … Σ 1, 6, 11, 37, 84, 233, 565, 1458, …
  (−∞, 0]nonnegative
• A207607 → basis A013609 → (θ+1)²
  1 · 3, 16 · 8, 72, 72 · 21, 264, 540, 256 … Σ 1, 19, 152, 1081, 7167, 45288, 276277, …
  (−∞, 0]nonnegative
• A038337 → θ+1
  144, 528, 363 · 1728, 9504, 13068, 5324 … Σ 1035, 29624, 815189, 21827598, …
  (−∞, 0]nonnegativerational roots
• A136124 → basis A013610 → (θ+1)²
  1 · 3, 12 · 12, 84, 81 · 60, 564, 972, 432 … Σ 1, 15, 177, 2028, 23904, 295956, …
  (−∞, 0]nonnegative
• A165253 → D+1
  1 · 1 · 2, 1 · 4, 5, 1 … Σ 1, 1, 3, 10, 32, 99, 299, 887, …
  (−∞, 0]nonnegative
• A159854 → basis A013611 → θ+1
  1, 16, 48 · 2, 40, 192, 256 · 4, 96, 624, 1536, 1280 … Σ 65, 490, 3540, 24840, 170640, 1153440, …
  (−∞, 0]nonnegative
• A173018 → reverse (shift +1) → reverse → (θ+1)²
  1 · 1 · 1, 4 · 1, 16, 9 … Σ 1, 1, 5, 26, 160, 1140, 9240, 84000, …
  (−∞, 0]nonnegative
• A099040 → basis A013611 → basis A007318
  1 · 10, 8 · 110, 168, 64 · 1200, 2720, 2048, 512 … Σ 1, 18, 342, 6480, 122796, 2326968, …
  (−∞, −1]nonnegative
• A166341 → D+1
  11, 12, 1 · 24, 69, 26, 1 · 51, 326, 288, 54, 1 … Σ 24, 120, 720, 5040, 40320, 362880, …
  (−∞, 0]nonnegative
• A081579 → basis A013610 → D+1
  1 · 5, 3 · 32, 42, 9 · 132, 360, 207, 27 … Σ 1, 8, 83, 726, 6305, 52804, 434767, …
  (−∞, 0]nonnegative
• A188440 → basis A007318 → D
  2 · 2 · 12, 8 · 12, 8 … Σ 0, 0, 2, 2, 20, 20, 150, 150, …
  (−∞, 0]nonnegativerational roots
• A063007 → (θ+1)²
  1 · 1, 8 · 1, 24, 54 · 1, 48, 270, 320 … Σ 1, 9, 79, 639, 4881, 35793, 254647, …
  (−∞, 0]nonnegative
• chebyshev_u → x ↦ x+1 → θ+1
  1 · 2, 4 · 3, 16, 12 · 4, 40, 72, 32 … Σ 1, 6, 31, 148, 673, 2962, 12735, 53800, …
  (−∞, 0]nonnegative
• A330381 → x ↦ x+1
  1 · 3, 2 · 9, 11, 3 · 27, 47, 25, 4 … Σ 1, 5, 23, 103, 457, 2021, 8927, 39415, …
  (−∞, −1]nonnegative
• A169803 → basis A013611 → D
  4 · 8 · 20, 32 · 40, 96 … Σ 0, 4, 8, 52, 136, 560, 1624, 5628, …
  (−∞, 0]nonnegative
• A075499 → θ+1
  16, 24, 3 · 64, 224, 72, 4 · 256, 1920, 1200, 160, 5 … Σ 43, 364, 3541, 38642, 465599, 6128472, …
  (−∞, 0]nonnegative
• A136124 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 6 · 12, 45, 42 · 60, 321, 570, 336 … Σ 1, 9, 99, 1287, 19305, 328185, 6235515, …
  [−1, 0]nonnegativerational roots
• A038334 → basis A013609 → D
  640, 512 · 19200, 30720, 12288 … Σ 1152, 62208, 2985984, 134369280, …
  (−∞, 0]nonnegativerational roots
• A085880 → D
  1 · 4, 4 · 15, 30, 15 · 56, 168, 168, 56 … Σ 0, 1, 8, 60, 448, 3360, 25344, 192192, …
  (−∞, 0]nonnegativerational roots
• A026729 → Veronese 2 → x ↦ x+1
  1 · 1, 1 · 2, 2 · 2, 3, 1 … Σ 1, 0, 2, 4, 6, 16, 32, 64, …
  (−∞, −1]nonnegative
• A038303 → reverse
  1 · 1, 10 · 1, 20, 100 · 1, 30, 300, 1000 … Σ 1, 11, 121, 1331, 14641, 161051, …
  (−∞, 0]nonnegativerational roots
• A081579 → basis A013609 → D
  2 · 16, 8 · 72, 112, 24 · 320, 800, 480, 64 … Σ 0, 2, 24, 208, 1664, 12384, 88960, …
  (−∞, 0]nonnegative
• A156365 → (θ+1)²
  1, 32, 36 · 1, 88, 396, 128 · 1, 208, 2376, 3328, 400 … Σ 69, 613, 6313, 73937, 970861, 14131677, …
  (−∞, 0]nonnegative
• A131689 → θ+1 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 2, 8 · 0, 2, 22, 44 … Σ 1, 2, 10, 68, 592, 6272, 78160, …
  [−1, 0]nonnegative
• A056939 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 18, 13 · 22, 99, 145, 69 … Σ 1, 5, 37, 335, 3457, 39125, 474385, …
  [−1, 0]nonnegative
• A028338 → basis A008277 → x ↦ x/(x+1)
  1 · 2, 3 · 8, 23, 16 · 48, 201, 273, 121 … Σ 1, 5, 47, 643, 11513, 254381, 6680455, …
  [−1, 0]nonnegative
• A331333 → x ↦ x/(x+1)
  1 · 1, 3 · 2, 12, 14 · 6, 54, 126, 86 … Σ 1, 4, 28, 272, 3344, 49472, 852928, …
  [−1, 0]nonnegative
• A125662 → reverse → x ↦ x+1
  1 · 4, 3 · 15, 22, 8 · 56, 122, 88, 21 … Σ 1, 7, 45, 287, 1829, 11655, 74269, …
  (−∞, −1]nonnegative
• A091044 → D → θ+1
  2 · 10, 12 · 28, 112, 36 · 60, 504, 540, 80 … Σ 0, 2, 22, 176, 1184, 7168, 40448, …
  (−∞, 0]nonnegative
• A008955 → basis A007318 → (θ+1)²
  1 · 2, 4 · 10, 52, 36 · 100, 880, 1413, 576 … Σ 1, 6, 98, 2969, 143145, 10039770, …
  (−∞, 0]nonnegative
• A038208 → basis A013610 → x ↦ x/(x+1)
  1 · 4, 10 · 16, 80, 100 · 64, 480, 1200, 1000 … Σ 1, 14, 196, 2744, 38416, 537824, …
  [−1, 0]nonnegativerational roots
• A249248 → reverse → D
  1 · 7, 2 · 22, 24, 3 · 40, 138, 54, 4 … Σ 0, 1, 9, 49, 236, 1125, 5309, 25435, …
  (−∞, 0]nonnegative
• A144696 → D
  2 · 7, 8 · 18, 66, 24 · 41, 342, 393, 64 … Σ 0, 2, 15, 108, 840, 7200, 68040, …
  (−∞, 0]nonnegative
• A075501 → x ↦ x/(x+1)
  36, 90, 55 · 216, 900, 1188, 505 … Σ 181, 2809, 50377, 1056709, 25529149, …
  [−1, 0]nonnegative
• A133214 → basis A007318 → D
  16, 8 · 114, 120, 24 · 768, 1248, 576, 64 … Σ 24, 258, 2656, 26610, 261720, 2540370, …
  (−∞, 0]nonnegative
• A145324 → basis A007318 → D+1
  1 · 5, 2 · 29, 29, 6 · 193, 329, 170, 24 … Σ 1, 7, 64, 716, 9451, 143653, 2470110, …
  (−∞, 0]nonnegative
• A038330 → basis A007318 → D
  128, 32 · 3072, 1536, 192 · 65536, 49152, 12288, 1024 … Σ 160, 4800, 128000, 3200000, 76800000, …
  (−∞, 0]nonnegativerational roots
• A290316 → x ↦ x/(x+1)
  1 · 1, 7 · 1, 50, 139 · 1, 237, 2685, 4609 … Σ 1, 8, 190, 7532, 418168, 29853224, …
  [−1, 0]nonnegative
• A114687 → basis A013611 → D
  8 · 56, 128 · 336, 1536, 1536 … Σ 0, 8, 184, 3408, 59280, 1008120, …
  (−∞, 0]nonnegative
• basis-map coefficients of A030528, shifted → basis A013610 → θ+1
  1 · -1, -6 · 1, 12, 27 · -1, -18, -81, -108 … Σ 1, -7, 40, -208, 1024, -4864, 22528, …
  (−∞, 0]rational roots
• A113953 → basis A013609 → θ+1
  1 · 1, 4 · 3, 16, 12 · 5, 44, 84, 32 … Σ 1, 5, 31, 165, 867, 4401, 21951, …
  (−∞, 0]nonnegative
• A128099 → basis A013611 → θ+1
  1 · 1 · 3, 16 · 5, 32 … Σ 1, 1, 19, 37, 315, 853, 5051, 16309, …
  (−∞, 0]nonnegative
• A173018 → reverse (shift +1) → θ+1 → x ↦ x+1
  1 · 2, 2 · 5, 8, 3 · 18, 38, 24, 4 … Σ 1, 4, 16, 84, 568, 4700, 45904, 516292, …
  (−∞, −1]nonnegative
• A094816 → x ↦ x+1 → θ+1
  1 · 2, 2 · 5, 10, 3 · 16, 46, 27, 4 … Σ 1, 4, 18, 93, 553, 3760, 28946, 249697, …
  (−∞, 0]nonnegative