Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A049458: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 2.
  1 · -3, 1 · 12, -7, 1 · -60, 47, -12, 1 … Σ 1, -2, 6, -24, 120, -720, 5040, -40320, …
  [5/2, ∞)rational roots
• A038331 → basis A013609 → D
  340, 200 · 8670, 10200, 3000 … Σ 540, 21870, 787320, 26572050, …
  (−∞, 0]nonnegativerational roots
• A130123 → basis A013611 → x ↦ x+1
  1 · 10, 8 · 100, 160, 64 · 1000, 2400, 1920, 512 … Σ 1, 18, 324, 5832, 104976, 1889568, …
  (−∞, −1]nonnegativerational roots
• A081577 → basis A013610 → x ↦ x+1
  1 · 5, 3 · 33, 36, 9 · 205, 333, 171, 27 … Σ 1, 8, 78, 736, 6980, 66144, 626872, …
  (−∞, −1]nonnegative
• A391533 → Pₙ + Pₙ₊₁ → D+1
  20, 24, 9, 1 · 28, 56, 39, 11, 1 … Σ 54, 135, 324, 756, 1728, 3888, 8640, …
  (−∞, 0]nonnegative
• A114687 → basis A007318 → θ+1
  1 · 3, 4 · 11, 28, 12 · 45, 168, 144, 32 … Σ 1, 7, 51, 389, 3069, 24795, 203775, …
  (−∞, 0]nonnegative
• A038321 → basis A013610 → (θ+1)²
  324, 3024, 3969 · 5832, 81648, 214326, 148176 … Σ 7317, 449982, 25310961, 1342151694, …
  (−∞, 0]nonnegative
• A122542 → basis A013610 → (θ+1)²
  1 · 1, 12 · 3, 48, 81 · 7, 156, 567, 432 … Σ 1, 13, 132, 1162, 9434, 72482, 535146, …
  (−∞, 0]nonnegative
• A060187 → θ+1
  1 · 1, 2 · 1, 12, 3 · 1, 46, 69, 4 … Σ 1, 3, 16, 120, 1152, 13440, 184320, …
  (−∞, 0]nonnegative
• A103633 → Veronese 2 (j=1) → D
  1 · 2 · 1, 2 · 1, 6 … Σ 0, 0, 0, 1, 2, 3, 7, 9, …
  (−∞, 0]nonnegative
• A185410 → θ+1
  1 · 1 · 1, 4 · 1, 20, 12 … Σ 1, 1, 5, 33, 285, 3045, 38745, 571725, …
  (−∞, 0]nonnegative
• A075505 → x ↦ x/(x+1)
  100, 230, 131 · 1000, 3700, 4460, 1761 … Σ 461, 10921, 290201, 8906301, 316616421, …
  [−1, 0]nonnegative
• A085880 → basis A013610 → D+1
  1 · 5, 3 · 32, 60, 18 · 220, 720, 675, 135 … Σ 1, 8, 110, 1750, 29750, 525000, …
  (−∞, 0]nonnegativerational roots
• A159854 → basis A008277 → (θ+1)²
  1, 12, 9 · 2, 40, 63, 16 · 4, 128, 342, 192, 25 … Σ 22, 121, 691, 4118, 25655, 167103, …
  (−∞, 0]nonnegative
• A137320 → basis A013611 → (θ+1)²
  1 · 2, 32 · 6, 160, 576 · 24, 832, 5184, 8192 … Σ 1, 34, 742, 14232, 263032, 4865616, …
  (−∞, 0]nonnegative
• A089231 → x ↦ x+1 → x ↦ x+1
  1 · 5, 2 · 37, 30, 6 · 361, 444, 180, 24 … Σ 1, 7, 73, 1009, 17341, 355951, 8488117, …
  (−∞, −1]nonnegative
• A108838 → D+1
  2 · 5, 2 · 12, 12, 2 · 25, 50, 21, 2 … Σ 2, 7, 26, 98, 372, 1419, 5434, 20878, …
  (−∞, 0]nonnegative
• A143494 → θ+1 → reverse
  1 · 2, 2 · 3, 10, 4 · 4, 27, 38, 8 … Σ 1, 4, 17, 77, 372, 1915, 10481, 60814, …
  (−∞, 0]nonnegative
• A144436 → x ↦ x+1
  1 · 2, 1 · 10, 10, 1 · 48, 72, 26, 1 … Σ 1, 3, 21, 147, 1237, 11955, 131253, …
  (−∞, −1]nonnegative
• A147716 → basis A013611 → D+1
  1 · 19, 4 · 345, 152, 16 · 6075, 4140, 912, 64 … Σ 1, 23, 513, 11191, 240065, 5082519, …
  (−∞, 0]nonnegativerational roots
• Hadamard-factor parent of A127773 → θ+1
  1 · 0, 6 · 0, 0, 36 · 0, 0, 0, 240 … Σ 1, 6, 36, 240, 1800, 15120, 141120, …
  (−∞, 0]nonnegativerational roots
• A106246 → reverse
  1 · 1, 2 · 1, 4, 1 · 1, 6, 3 … Σ 1, 3, 6, 10, 15, 21, 28, 36, …
  (−∞, 0]nonnegative
• A136572 → basis A013611 → reverse
  1 · 4, 1 · 32, 16, 2 · 384, 288, 72, 6 … Σ 1, 5, 50, 750, 15000, 375000, 11250000, …
  (−∞, 0]nonnegativerational roots
• A026729 → basis A008277 → reverse
  1 · 1, 1 · 1, 4, 2 · 1, 8, 13, 3 … Σ 1, 2, 7, 25, 102, 456, 2219, 11640, …
  (−∞, 0]nonnegative
• A372868 → θ+1
  1 · 2 · 4, 2 · 8, 12 … Σ 1, 2, 6, 20, 67, 222, 728, 2368, …
  (−∞, 0]nonnegative
• A132812 → basis A007318 → (θ+1)²
  1 · 4, 8 · 15, 60, 27 · 56, 336, 324, 64 … Σ 1, 12, 102, 780, 5655, 39648, 271516, …
  (−∞, 0]nonnegative
• A093375 → x ↦ x+1 → x ↦ x+1
  1 · 4, 1 · 15, 8, 1 · 54, 45, 12, 1 … Σ 1, 5, 24, 112, 512, 2304, 10240, 45056, …
  (−∞, −1]nonnegativerational roots
• A330798 → (θ+1)²
  1 · 2, 8 · 6, 60, 81 · 20, 336, 1008, 768 … Σ 1, 10, 147, 2132, 29925, 408030, …
  (−∞, 0]nonnegative
• A132440 → D+1 → D+1
  1 · 4, 2 · 6, 12, 3 · 0, 24, 24, 4 … Σ 0, 1, 6, 21, 52, 105, 186, 301, …
  (−∞, 0]nonnegative
• A159854 → x ↦ x/(x+1)
  0, 0, 1 · 0, 0, 1, 2 · 0, 0, 1, 4, 4 … Σ 1, 3, 9, 27, 81, 243, 729, 2187, …
  [−1, 0]nonnegativerational roots
• A131222 → D
  1 · 3, 2 · 14, 18, 3 · 90, 166, 54, 4 … Σ 0, 1, 5, 35, 314, 3434, 44304, 658776, …
  (−∞, 0]nonnegative
• A257620 → (θ+1)²
  1 · 3, 12 · 9, 144, 81 · 27, 1188, 2673, 432 … Σ 1, 15, 234, 4320, 92340, 2245320, …
  (−∞, 0]nonnegative
• A099093 → x ↦ x−15
  1 · -45, 3 · 1980, -267, 9 · -87075, 17685, -1197, 27 … Σ 1, -42, 1722, -70560, 2891196, …
  (41/3, 15]
• A063007 → basis A013609 → reverse
  1 · 4, 3 · 24, 36, 13 · 160, 360, 264, 63 … Σ 1, 7, 73, 847, 10321, 129367, 1651609, …
  (−∞, 0]nonnegative
• A257607: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 5.
  1 · 5, 5 · 25, 60, 25 · 125, 535, 535, 125 … Σ 1, 10, 110, 1320, 17160, 240240, …
  (−∞, 0)nonnegative
• A132393 → basis A013611 → θ+1
  1 · 1, 8 · 2, 24, 48 · 6, 88, 288, 256 … Σ 1, 9, 74, 638, 5944, 60216, 662640, …
  (−∞, 0]nonnegative
• A062264 → reverse → D+1
  1 · 6, 1 · 27, 14, 1 · 98, 105, 24, 1 … Σ 1, 7, 42, 228, 1155, 5577, 26026, …
  (−∞, 0]nonnegative
• A094638 → D
  1 · 3, 4 · 6, 22, 18 · 10, 70, 150, 96 … Σ 0, 1, 7, 46, 326, 2556, 22212, 212976, …
  (−∞, 0]nonnegative
• A285061 → θ+1
  1 · 1, 8 · 1, 48, 48 · 1, 248, 720, 256 … Σ 1, 9, 97, 1225, 17665, 284873, 5059553, …
  (−∞, 0]nonnegative
• A225117 → reverse
  1 · 1, 2 · 1, 13, 4 · 1, 60, 93, 8 … Σ 1, 3, 18, 162, 1944, 29160, 524880, …
  (−∞, 0]nonnegative
• A039755 → x ↦ x+1 → D+1
  1 · 3, 1 · 12, 8, 1 · 58, 58, 15, 1 … Σ 1, 4, 21, 132, 953, 7716, 68909, …
  (−∞, 0]nonnegative
• A178820 → (θ+1)²
  1 · 4, 16 · 10, 80, 90 · 20, 240, 540, 320 … Σ 1, 20, 180, 1120, 5600, 24192, 94080, …
  (−∞, 0]nonnegative
• A038329 → x ↦ x+1 → D+1
  315, 108, 9 · 5400, 2835, 486, 27 … Σ 432, 8748, 174960, 3464208, 68024448, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A113214, shifted → Veronese 2 (j=1) → θ+1
  -1 · 2 · -3, -2 · 4, 8 … Σ 0, -1, 2, -5, 12, -28, 64, -144, …
  (−∞, 0]
• A120910 → basis A007318 → θ+1
  3 · 9, 6 · 27, 36, 9 · 81, 162, 81, 12 … Σ 3, 15, 72, 336, 1536, 6912, 30720, …
  (−∞, 0]nonnegativerational roots
• A038328 → basis A013611 → x ↦ x/(x+1)
  196, 616, 484 · 2744, 12936, 20328, 10648 … Σ 1296, 46656, 1679616, 60466176, …
  [−1, 0]nonnegativerational roots
• A027465 → Veronese 2 → D+1
  1 · 3 · 10, 1 · 36, 9 … Σ 1, 3, 11, 45, 192, 828, 3568, 15312, …
  (−∞, 0]nonnegative
• A059364 → basis A013610 → (θ+1)²
  1 · 3, 12 · 15, 132, 162 · 105, 1464, 3807, 2592 … Σ 1, 15, 309, 7968, 246096, 8846316, …
  (−∞, 0]nonnegative
• A280580 → D
  1 · 6, 2 · 30, 30, 3 · 140, 280, 84, 4 … Σ 0, 1, 8, 63, 508, 4175, 34788, 292845, …
  (−∞, 0]nonnegative
• A056939 → x ↦ x+1 → θ+1
  1 · 2, 2 · 6, 12, 3 · 22, 66, 39, 4 … Σ 1, 4, 21, 131, 909, 6798, 53737, …
  (−∞, 0]nonnegative