Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A038281 → x ↦ x/(x+1)
  64, 176, 121 · 512, 2112, 2904, 1331 … Σ 361, 6859, 130321, 2476099, 47045881, …
  [−1, 0]nonnegativerational roots
• A125662 → basis A013609 → D
  2 · 16, 8 · 92, 96, 24 · 464, 744, 384, 64 … Σ 0, 2, 24, 212, 1656, 12102, 84816, …
  (−∞, 0]nonnegative
• A036969 → θ+1
  1 · 1, 2 · 1, 10, 3 · 1, 42, 42, 4 … Σ 1, 3, 14, 89, 737, 7696, 98093, …
  (−∞, 0]nonnegative
• A137597 → D
  1 · 5, 2 · 22, 18, 3 · 99, 122, 42, 4 … Σ 0, 1, 7, 43, 267, 1731, 11819, 85107, …
  (−∞, −1]nonnegative
• basis-map coefficients of A026729, shifted → basis A013610 → D
  -3 · 6, 18 · -9, -54, -81 · 12, 108, 324, 324 … Σ 0, -3, 24, -144, 768, -3840, 18432, …
  (−∞, 0]rational roots
• A122542 → basis A013611 → θ+1
  1 · 1, 8 · 3, 32, 48 · 7, 104, 336, 256 … Σ 1, 9, 83, 703, 5713, 45073, 348083, …
  (−∞, 0]nonnegative
• A075497 → θ+1 → D+1
  1 · 4, 2 · 16, 18, 3 · 64, 128, 48, 4 … Σ 1, 6, 37, 244, 1741, 13442, 111833, …
  (−∞, 0]nonnegative
• A038320 → x ↦ x+1 → x ↦ x/(x+1)
  289, 782, 529 · 4913, 19941, 26979, 12167 … Σ 1600, 64000, 2560000, 102400000, …
  [−1, 0]nonnegativerational roots
• A038322 → θ+1
  1 · 11, 16 · 121, 352, 192 · 1331, 5808, 6336, 2048 … Σ 1, 27, 665, 15523, 349809, 7688939, …
  (−∞, 0]nonnegativerational roots
• A094638 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 6, 33, 45 · 24, 210, 609, 585 … Σ 1, 7, 84, 1428, 31416, 848232, …
  [−1, 0]nonnegativerational roots
• A131689 → θ+1 → (θ+1)²
  1 · 0, 8 · 0, 8, 54 · 0, 8, 162, 384 … Σ 1, 8, 62, 554, 5690, 66338, 867002, …
  (−∞, 0]nonnegative
• A110441 → basis A007318 → D
  1 · 8, 2 · 44, 24, 3 · 208, 180, 48, 4 … Σ 0, 1, 10, 71, 440, 2537, 13990, 74843, …
  (−∞, 0]nonnegative
• A281000 → D+1
  1 · 4, 1 · 20, 12, 1 · 105, 112, 24, 1 … Σ 1, 5, 33, 242, 1871, 14933, 121739, …
  (−∞, 0]nonnegative
• A167884 → x ↦ x+1
  1 · 2, 1 · 20, 20, 1 · 360, 540, 182, 1 … Σ 1, 3, 41, 1083, 40721, 1998003, …
  (−∞, −1]nonnegative
• A114656 → θ+1 → x ↦ x/(x+1)
  1 · 2, 4 · 4, 20, 19 · 8, 72, 156, 96 … Σ 1, 6, 43, 332, 2661, 21810, 181455, …
  [−1, 0]nonnegative
• type_b_eulerian → γ (scale 1/4) → x ↦ x+1 → (θ+1)²
  1 · 1 · 2, 4 · 6, 20 … Σ 1, 1, 6, 26, 181, 1389, 12934, 136274, …
  (−∞, 0]nonnegative
• A099093 → x ↦ x−7
  1 · -21, 3 · 420, -123, 9 · -8379, 3717, -549, 27 … Σ 1, -18, 306, -5184, 87804, -1487160, …
  (17/3, 7]
• A121574 → reverse → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 10 · 1, 11, 35, 33 … Σ 1, 4, 18, 80, 356, 1584, 7048, 31360, …
  [−1, 0]nonnegative
• A099040 → θ+1 → (θ+1)²
  1 · 0, 16 · 0, 16, 108 · 0, 0, 216, 512 … Σ 1, 16, 124, 728, 3644, 16448, 69024, …
  (−∞, 0]nonnegative
• A144440 → D+1
  1 · 2, 1 · 12, 13, 1 · 55, 162, 57, 1 … Σ 1, 3, 26, 275, 3747, 62356, 1226476, …
  (−∞, 0]nonnegative
• A100861 → basis A007318 → θ+1
  1 · 1 · 2, 2 · 4, 6 … Σ 1, 1, 4, 10, 43, 151, 706, 3004, …
  (−∞, 0]nonnegative
• A078812 → basis A013611 → D+1
  1 · 7, 4 · 32, 56, 16 · 121, 388, 336, 64 … Σ 1, 11, 104, 909, 7575, 61136, 482201, …
  (−∞, 0]nonnegative
• A103633 → basis A007318
  1 · 1, 1 · 2, 3, 1 · 2, 5, 4, 1 … Σ 1, 2, 6, 12, 36, 72, 216, 432, …
  (−∞, −1]nonnegativerational roots
• A103633 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 0, 1, 2 … Σ 1, 1, 3, 3, 9, 9, 27, 27, …
  [−1, 0]nonnegativerational roots
• A370258 → reverse
  15, 10, 1 · 84, 84, 21, 1 · 495, 660, 270, 36, 1 … Σ 26, 190, 1462, 11584, 93536, 765314, …
  (−∞, −1]nonnegative
• A113953 → θ+1 → D+1
  1 · 2, 2 · 4, 10, 3 · 0, 24, 24, 4 … Σ 1, 4, 17, 52, 157, 428, 1137, 2892, …
  (−∞, 0]nonnegative
• A085881 → basis A013611
  1 · 2, 4 · 12, 48, 48 · 120, 720, 1440, 960 … Σ 1, 6, 108, 3240, 136080, 7348320, …
  [−1, 0]nonnegativerational roots
• A132393 → basis A013610 → D
  3 · 9, 18 · 33, 108, 81 · 150, 630, 810, 324 … Σ 0, 3, 27, 222, 1914, 17832, 180648, …
  (−∞, 0]nonnegative
• A119462 → D+1
  1 · 2 · 4, 2 · 8, 6 … Σ 1, 2, 6, 14, 32, 72, 160, 352, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → basis A013611 → θ+1
  1, 16, 48 · 1, 24, 144, 256 · 1, 32, 288, 1024, 1280 … Σ 65, 425, 2625, 15625, 90625, 515625, …
  (−∞, 0]nonnegativerational roots
• A034870 → Veronese 2 (j=1) → θ+1
  2 · 4, 8 · 6, 40, 18 · 8, 112, 168, 32 … Σ 0, 2, 12, 64, 320, 1536, 7168, 32768, …
  (−∞, 0]nonnegative
• A038277 → x ↦ x/(x+1)
  1 · 7, 18 · 49, 252, 324 · 343, 2646, 6804, 5832 … Σ 1, 25, 625, 15625, 390625, 9765625, …
  [−1, 0]nonnegativerational roots
• A143685 → γ → θ+1
  1 · 1 · 1, 18 · 1, 36 … Σ 1, 1, 19, 37, 298, 802, 4465, 14203, …
  (−∞, 0]nonnegative
• A125092 → D+1 → reverse
  1 · 4, 5 · 9, 26, 9 · 16, 75, 66, 13 … Σ 1, 9, 44, 170, 576, 1792, 5248, 14688, …
  (−∞, 0]nonnegative
• A286718 → basis A013611 → x ↦ x+1
  1 · 6, 4 · 54, 60, 16 · 648, 936, 432, 64 … Σ 1, 10, 130, 2080, 39520, 869440, …
  (−∞, −1]nonnegativerational roots
• Veronese parent of A109446 → D
  1 · 1 · 1, 4, 3 · 3, 6, 3 … Σ 1, 1, 8, 12, 48, 80, 256, 448, …
  (−∞, 0]nonnegativerational roots
• A114656 → basis A007318 → x ↦ x+1
  1 · 4, 1 · 20, 10, 1 · 112, 84, 18, 1 … Σ 1, 5, 31, 215, 1597, 12425, 99955, …
  (−∞, −1]nonnegative
• A038243 → basis A013611 → D+1
  1 · 10, 4 · 84, 80, 16 · 648, 1008, 480, 64 … Σ 1, 14, 180, 2200, 26000, 300000, …
  (−∞, 0]nonnegativerational roots
• A113413 → reverse → x ↦ x/(x+1)
  1 · 1, 3 · 1, 6, 7 · 1, 9, 23, 17 … Σ 1, 4, 14, 50, 178, 634, 2258, 8042, …
  [−1, 0]nonnegative
• A131222 → x ↦ x+1
  1 · 1, 1 · 4, 5, 1 · 24, 35, 12, 1 … Σ 1, 2, 10, 72, 672, 7680, 103680, …
  (−∞, −1]nonnegative
• A139548 → x ↦ x+1 → (θ+1)²
  1 · 8, 32, 18 · 64, 512, 864, 512, 100 … Σ 1, 58, 2052, 60264, 1597968, 39680928, …
  (−∞, 0]nonnegative
• A038314 → x ↦ x+1
  484, 528, 144 · 10648, 17424, 9504, 1728 … Σ 1156, 39304, 1336336, 45435424, …
  (−∞, −1]nonnegativerational roots
• A272866 → basis A013611 → θ+1
  1 · 5, 40, 48 · 25, 400, 1680, 2560, 1280 … Σ 1, 93, 5945, 331157, 17161329, …
  (−∞, 0]nonnegative
• A249139 → basis A013610 → θ+1
  1 · 4, 6 · 7, 12 · 19, 54, 27 … Σ 1, 10, 19, 100, 253, 1018, 2935, 10432, …
  (−∞, 0]nonnegative
• A119808 → D+1
  1 · 3, 1 · 9, 5 · 25, 21, 2 … Σ 1, 4, 14, 48, 162, 540, 1782, 5832, …
  (−∞, 0]nonnegative
• A274496 → basis A013610 → D
  6 · 12 · 48, 72 · 96, 144 … Σ 0, 0, 6, 12, 120, 240, 1800, 3600, …
  (−∞, 0]nonnegativerational roots
• A159854 → Veronese 2 (j=1) → basis A007318
  1, 1 · 2, 2 · 4, 5, 1 · 8, 12, 4 … Σ 0, 2, 4, 10, 24, 58, 140, 338, …
  (−∞, −1]nonnegative
• Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.
  9, 24, 9 · 27, 141, 141, 27 · 81, 726, 1410, 726, 81 … Σ 42, 336, 3024, 30240, 332640, 3991680, …
  (−∞, 0)nonnegative
• A272866 → basis A013611 → x ↦ x+1
  1 · 41, 52, 16 · 1681, 4264, 4016, 1664, 256 … Σ 1, 109, 11881, 1295029, 141158161, …
  (−∞, −1]nonnegative
• A121574 → basis A013609 → D+1
  1 · 5, 2 · 24, 22, 4 · 103, 158, 68, 8 … Σ 1, 7, 50, 337, 2201, 14044, 88061, …
  (−∞, 0]nonnegative