Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A154227 → D
  1 · 8, 2 · 19, 38, 3 · 35, 316, 105, 4 … Σ 0, 1, 10, 60, 460, 3250, 27120, 227080, …
  (−∞, 0]nonnegative
• A069466 → basis A007318 → x ↦ x/(x+1)
  1 · 4, 6 · 36, 108, 78 · 400, 1800, 2640, 1260 … Σ 1, 10, 222, 6100, 184870, 5928300, …
  [−1, 0]nonnegative
• A144443: Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 20, 1 · 1, 159, 159, 1 … Σ 1, 2, 22, 320, 6532, 171752, 5535256, …
  (−∞, 0)nonnegative
• A290319 → basis A013610 → basis A007318
  1 · 5, 3 · 45, 42, 9 · 585, 681, 243, 27 … Σ 1, 8, 96, 1536, 30720, 737280, …
  (−∞, −1]nonnegativerational roots
• A116412 → x ↦ x/(x+1)
  1 · 3, 4 · 6, 18, 13 · 12, 57, 87, 43 … Σ 1, 7, 37, 199, 1069, 5743, 30853, …
  [−1, 0]nonnegative
• A121574 → reverse
  1 · 1, 2 · 1, 5, 4 · 1, 8, 16, 8 … Σ 1, 3, 10, 33, 109, 360, 1189, 3927, …
  (−∞, 0]nonnegative
• A132393 → D+1 → reverse
  1 · 1, 1 · 1, 3, 1 · 1, 6, 8, 2 … Σ 1, 2, 5, 17, 74, 394, 2484, 18108, …
  (−∞, 0]nonnegative
• A008459 → basis A013610 → reverse
  1 · 3, 2 · 9, 18, 6 · 27, 108, 90, 20 … Σ 1, 5, 33, 245, 1921, 15525, 127905, …
  (−∞, 0]nonnegative
• A130534 → Veronese 2 → θ+1
  1 · 1 · 2, 2 · 6, 12 … Σ 1, 1, 4, 18, 97, 615, 4497, 37296, …
  (−∞, 0]nonnegative
• A027466 → basis A013610 → D
  3 · 48, 18 · 576, 432, 81 · 6144, 6912, 2592, 324 … Σ 0, 3, 66, 1089, 15972, 219615, 2898918, …
  (−∞, 0]nonnegativerational roots
• A001263 → θ+1 → (θ+1)²
  1 · 1, 8 · 1, 24, 27 · 1, 48, 162, 64 … Σ 1, 9, 52, 275, 1386, 6762, 32208, …
  (−∞, 0]nonnegative
• A152198 → x ↦ x+1 → x ↦ x+1
  1 · 1 · 3, 1 · 3, 1 … Σ 1, 1, 4, 4, 16, 16, 64, 64, …
  (−∞, −1]nonnegativerational roots
• A185410 → D
  2 · 10, 8 · 36, 120, 24 · 116, 1032, 888, 64 … Σ 0, 0, 2, 18, 180, 2100, 28350, 436590, …
  (−∞, 0]nonnegative
• A367270 → x ↦ x+1 → θ+1
  1 · 1 · 3, 4 · 10, 24, 9 … Σ 1, 1, 7, 43, 261, 1581, 9563, 57751, …
  (−∞, 0]nonnegative
• A091044 → basis A013611 → basis A007318
  1 · 12, 8 · 128, 160, 48 · 1344, 2432, 1408, 256 … Σ 1, 20, 336, 5440, 87296, 1397760, …
  (−∞, −1]nonnegative
• Veronese parent of A098172 → x ↦ x/(x+1)
  1 · 0, 0, 1, 2 · 0, 0, 0, 0, 1, 4, 4 … Σ 1, 3, 9, 27, 81, 243, 729, 2187, …
  [−1, 0]nonnegativerational roots
• A038336 → x ↦ x+1
  484, 440, 100 · 10648, 14520, 6600, 1000 … Σ 1024, 32768, 1048576, 33554432, …
  (−∞, −1]nonnegativerational roots
• A008277 → (θ+1)² → x ↦ x/(x+1)
  1 · 1, 5 · 1, 14, 22 · 1, 31, 113, 99 … Σ 1, 6, 37, 244, 1741, 13442, 111833, …
  [−1, 0]nonnegative
• A152198 → x ↦ x+1 → θ+1
  1 · 1 · 2, 2 · 2, 2 … Σ 1, 1, 4, 4, 15, 15, 54, 54, …
  (−∞, 0]nonnegativerational roots
• A237765: Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
  A007318 → ×(−(1/2)n+(1/2)n²)
  1, 2, 1 · 3, 9, 9, 3 · 6, 24, 36, 24, 6 … Σ 0, 0, 4, 24, 96, 320, 960, 2688, …
  (−∞, 0]nonnegativerational roots
• A120910 → Veronese 2 (j=1) → θ+1
  3 · 12 · 36, 6 · 96, 48 … Σ 0, 3, 12, 42, 144, 489, 1644, 5472, …
  (−∞, 0]nonnegative
• A078812 → basis A013610 → reverse
  1 · 3, 3 · 9, 18, 8 · 27, 81, 75, 21 … Σ 1, 6, 35, 204, 1189, 6930, 40391, …
  (−∞, 0]nonnegative
• A081577 → x ↦ x+1 → D
  1 · 6, 2 · 24, 20, 3 · 88, 116, 42, 4 … Σ 0, 1, 8, 47, 250, 1245, 5964, 27787, …
  (−∞, 0]nonnegative
• A225467 → D+1
  1 · 7, 4 · 49, 72, 16 · 343, 988, 528, 64 … Σ 1, 11, 137, 1923, 30097, 519035, …
  (−∞, 0]nonnegative
• A358735 → basis A007318 → (θ+1)²
  1 · 2, 4 · 7, 32, 18 · 33, 240, 306, 96 … Σ 1, 6, 57, 675, 9457, 152304, 2766161, …
  (−∞, 0]nonnegative
• A013620 → basis A013610
  1 · 5, 9 · 25, 90, 81 · 125, 675, 1215, 729 … Σ 1, 14, 196, 2744, 38416, 537824, …
  [−1, 0]nonnegativerational roots
• A128813 → basis A013609 → (θ+1)²
  1 · 2, 8 · 8, 80, 108 · 56, 944, 2916, 2304 … Σ 1, 10, 196, 6220, 291060, 18868920, …
  (−∞, 0]nonnegative
• A200536 → basis A007318 → D+1
  1 · 13, 11, 2 · 120, 230, 157, 44, 4 … Σ 1, 26, 555, 10800, 199125, 3543750, …
  (−∞, 0]nonnegative
• A358735 → (θ+1)²
  1 · 1, 4 · 1, 16, 18 · 1, 40, 144, 96 … Σ 1, 5, 35, 281, 2527, 25113, 273245, …
  (−∞, 0]nonnegative
• A104684 → basis A013609 → D
  2 · 16, 8 · 114, 120, 24 · 768, 1248, 576, 64 … Σ 0, 2, 24, 258, 2656, 26610, 261720, …
  (−∞, 0]nonnegative
• A120986 → θ+1
  1 · 2, 2 · 5, 12, 3 · 14, 56, 36, 4 … Σ 1, 4, 20, 110, 637, 3808, 23256, …
  (−∞, 0]nonnegative
• A136668 → D
  0, 4 · -11, 0, 24 · 0, -148, 0, 192 … Σ 4, 13, 44, 143, -124, -16673, -418364, …
  (−∞, ∞)
• A225471 → basis A008277 → reverse
  1 · 1, 4 · 1, 13, 32 · 1, 27, 201, 384 … Σ 1, 5, 46, 613, 10679, 229576, 5868715, …
  (−∞, 0]nonnegative
• A172431 → basis A007318 → x ↦ x/(x+1)
  1 · 3, 5 · 8, 26, 21 · 21, 101, 161, 85 … Σ 1, 8, 55, 368, 2449, 16280, 108199, …
  [−1, 0]nonnegative
• A035607 → reverse → x ↦ x+1
  1 · 3, 1 · 7, 6, 1 · 17, 23, 9, 1 … Σ 1, 4, 14, 50, 178, 634, 2258, 8042, …
  (−∞, −1]nonnegative
• A094816 → basis A013609 → D+1
  1 · 4, 2 · 15, 18, 4 · 62, 118, 60, 8 … Σ 1, 6, 37, 248, 1837, 15074, 136593, …
  (−∞, 0]nonnegative
• A200536 → basis A007318
  1 · 6, 7, 2 · 36, 84, 73, 28, 4 … Σ 1, 15, 225, 3375, 50625, 759375, …
  (−∞, −1]nonnegativerational roots
• A139548 → D
  4, 4 · 16, 48, 48, 16 · 48, 240, 480, 480, 240, 48 … Σ 0, 8, 128, 1536, 16384, 163840, …
  (−∞, 0]nonnegativerational roots
• A038314 → θ+1
  100, 480, 432 · 1000, 7200, 12960, 6912 … Σ 1012, 28072, 745360, 19208992, …
  (−∞, 0]nonnegativerational roots
• A204420 → basis A013609 → (θ+1)²
  1 · 1, 8 · 9, 96, 108 · 225, 2760, 4860, 1920 … Σ 1, 9, 213, 9765, 740985, 84340305, …
  (−∞, 0]nonnegative
• A114655 → θ+1 → x ↦ x+1
  2 · 8, 4 · 38, 36, 6 · 192, 264, 96, 8 … Σ 2, 12, 80, 560, 4032, 29568, 219648, …
  (−∞, −1]nonnegative
• A113214 → θ+1 → x ↦ x/(x+1)
  1 · 2, 4 · 0, 6, 9 · 0, 4, 20, 20 … Σ 1, 6, 15, 44, 105, 258, 595, 1368, …
  [−1, 0]nonnegative
• A099040 → basis A013611 → D+1
  1 · 10, 8 · 46, 168, 64 · 176, 1184, 2048, 512 … Σ 1, 18, 278, 3920, 52460, 678200, …
  (−∞, 0]nonnegative
• A038314 → basis A013609 → x ↦ x+1
  2116, 2208, 576 · 97336, 152352, 79488, 13824 … Σ 4900, 343000, 24010000, 1680700000, …
  (−∞, −1]nonnegativerational roots
• A046716 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 12, 24, 4 … Σ 1, 3, 10, 41, 209, 1287, 9270, 76225, …
  (−∞, 0]nonnegative
• A075504 → reverse
  1, 27, 81 · 1, 54, 567, 729 … Σ 109, 1351, 19612, 333451, 6493069, …
  (−∞, 0]nonnegative
• A013610 → basis A013610 → (θ+1)²
  1 · 4, 36 · 16, 288, 729 · 64, 1728, 8748, 11664 … Σ 1, 40, 1033, 22204, 430105, 7786168, …
  (−∞, 0]nonnegative
• A038319 → x ↦ x+1 → D
  160, 50 · 3840, 2400, 375 · 81920, 76800, 24000, 2500 … Σ 210, 6615, 185220, 4862025, 122523030, …
  (−∞, 0]nonnegativerational roots
• A027466 → basis A013610
  1 · 8, 3 · 64, 48, 9 · 512, 576, 216, 27 … Σ 1, 11, 121, 1331, 14641, 161051, …
  (−∞, −1]nonnegativerational roots
• A034870 → basis A013611 → θ+1
  1 · 4, 32, 48 · 16, 256, 1152, 2048, 1280 … Σ 1, 84, 4752, 233280, 10637568, …
  (−∞, 0]nonnegativerational roots