Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A093375 → x ↦ x+1
  1 · 3, 1 · 8, 6, 1 · 20, 24, 9, 1 … Σ 1, 4, 15, 54, 189, 648, 2187, 7290, …
  (−∞, −1]nonnegativerational roots
• A085478 → basis A013609 → D
  2 · 10, 8 · 38, 64, 24 · 130, 336, 264, 64 … Σ 0, 0, 2, 18, 126, 794, 4716, 26964, …
  (−∞, 0]nonnegative
• A257621 → D
  3 · 42, 18 · 393, 786, 81 · 3156, 17292, 9468, 324 … Σ 0, 3, 60, 1260, 30240, 831600, …
  (−∞, 0]nonnegative
• A098158 → basis A007318
  1 · 1, 1 · 2, 3, 1 · 4, 9, 6, 1 … Σ 1, 2, 6, 20, 68, 232, 792, 2704, …
  (−∞, −1]nonnegative
• A089627: T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.
  1 · 1 · 1, 2 · 1, 6 … Σ 1, 1, 3, 7, 19, 51, 141, 393, …
  (−∞, 0)nonnegative
• A054142 → D → (θ+1)²
  1 · 3, 8 · 5, 48, 27 · 7, 120, 270, 64 … Σ 0, 1, 11, 80, 461, 2263, 9960, 40577, …
  (−∞, 0]nonnegative
• A249252 → reverse → x ↦ x+1
  1 · 5, 1 · 17, 10, 1 · 61, 59, 15, 1 … Σ 1, 6, 28, 136, 656, 3168, 15296, 73856, …
  (−∞, −1]nonnegative
• A356546 → (θ+1)²
  1 · 2, 8 · 6, 48, 54 · 20, 240, 540, 320 … Σ 1, 10, 108, 1120, 11200, 108864, …
  (−∞, 0]nonnegative
• A367270 → basis A013610 → x ↦ x+1
  1 · 1 · 9, 6 · 73, 90, 27 … Σ 1, 1, 15, 190, 2339, 28806, 357078, …
  (−∞, −1]nonnegative
• A161198 → x ↦ x/(x+1)
  1 · 1, 3 · 3, 14, 15 · 15, 91, 173, 105 … Σ 1, 4, 32, 384, 6144, 122880, 2949120, …
  [−1, 0]nonnegativerational roots
• A027465 → D → (θ+1)²
  1 · 6, 8 · 27, 72, 27 · 108, 432, 324, 64 … Σ 0, 1, 14, 126, 928, 6080, 36864, …
  (−∞, 0]nonnegative
• A026729 → D+1
  1 · 1, 1 · 1, 3, 1 · 0, 4, 5, 1 … Σ 1, 2, 5, 10, 20, 38, 71, 130, …
  (−∞, 0]nonnegative
• A132812 → basis A007318 → D
  2 · 15, 6 · 84, 72, 12 · 420, 560, 210, 20 … Σ 0, 2, 21, 168, 1210, 8250, 54411, …
  (−∞, −1]nonnegative
• A008278 → x ↦ x+1 (shift +1) → x ↦ x+1 → x ↦ x+1
  1 · 4, 1 · 19, 9, 1 · 109, 75, 16, 1 … Σ 1, 5, 29, 201, 1657, 15821, 170389, …
  (−∞, −1]nonnegative
• A028338 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 3 · 8, 22, 15 · 48, 188, 244, 105 … Σ 1, 5, 45, 585, 9945, 208845, 5221125, …
  [−1, 0]nonnegativerational roots
• A038305 → basis A013609 → D+1
  325, 228, 36 · 5239, 5850, 2052, 216 … Σ 589, 13357, 294937, 6385729, 136185445, …
  (−∞, 0]nonnegativerational roots
• A034870 → D → D+1
  4, 2 · 16, 36, 24, 4 · 36, 150, 240, 180, 60, 6 … Σ 0, 6, 80, 672, 4608, 28160, 159744, …
  (−∞, 0]nonnegativerational roots
• A094262 → (θ+1)²
  1 · 1, 8, 9 · 1, 24, 108, 160, 75 … Σ 1, 18, 368, 9120, 269632, 9305856, …
  (−∞, 0]nonnegative
• A272867 → (θ+1)²
  1 · 1, 16, 9 · 1, 32, 162, 128, 25 … Σ 1, 26, 348, 3672, 34128, 292896, …
  (−∞, 0]nonnegative
• A038319 → θ+1
  121, 220, 75 · 1331, 3630, 2475, 500 … Σ 416, 7936, 147456, 2686976, 48234496, …
  (−∞, 0]nonnegativerational roots
• A121574 → (θ+1)²
  1 · 2, 4 · 4, 20, 9 · 8, 64, 72, 16 … Σ 1, 6, 33, 160, 726, 3144, 13159, 53652, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729 → basis A013610 → θ+1
  1 · 1, 6 · 1, 12, 27 · 1, 18, 81, 108 … Σ 1, 7, 40, 208, 1024, 4864, 22528, …
  (−∞, 0]nonnegativerational roots
• A013612 → reverse
  25, 10, 1 · 125, 75, 15, 1 · 625, 500, 150, 20, 1 … Σ 36, 216, 1296, 7776, 46656, 279936, …
  (−∞, 0]nonnegativerational roots
• A249252 → basis A007318 → x ↦ x/(x+1)
  1 · 5, 9 · 17, 58, 49 · 61, 307, 511, 281 … Σ 1, 14, 124, 1160, 10768, 100064, …
  [−1, 0]nonnegative
• A038305 → x ↦ x+1
  169, 78, 9 · 2197, 1521, 351, 27 … Σ 256, 4096, 65536, 1048576, 16777216, …
  (−∞, −1]nonnegativerational roots
• A038220 → basis A013611 → x ↦ x/(x+1)
  1 · 5, 13 · 25, 130, 169 · 125, 975, 2535, 2197 … Σ 1, 18, 324, 5832, 104976, 1889568, …
  [−1, 0]nonnegativerational roots
• A131436 → D
  3 · 0, 14 · 0, 0, 45 · 0, 0, 0, 124 … Σ 0, 3, 14, 45, 124, 315, 762, 1785, …
  (−∞, 0]nonnegativerational roots
• A249139 → basis A007318 → θ+1
  1 · 4, 2 · 7, 4 · 19, 18, 3 … Σ 1, 6, 11, 40, 93, 282, 719, 2028, …
  (−∞, 0]nonnegative
• A100861 → D → D
  6 · 30 · 90, 90 · 210, 630 … Σ 0, 0, 0, 0, 6, 30, 180, 840, …
  (−∞, 0]nonnegative
• A013610 → basis A007318 → D+1
  1 · 7, 3 · 40, 42, 9 · 208, 360, 189, 27 … Σ 1, 10, 91, 784, 6517, 52822, 420175, …
  (−∞, 0]nonnegativerational roots
• A321434 → D
  1 · 1 · 1, 4 · 1, 4 … Σ 0, 1, 1, 5, 5, 31, 31, 233, …
  (−∞, 0]nonnegative
• A225472 → D+1
  1 · 5, 3 · 25, 57, 18 · 125, 657, 756, 162 … Σ 1, 8, 100, 1700, 36388, 937988, …
  (−∞, 0]nonnegative
• A113413 → basis A013609 → x ↦ x/(x+1)
  1 · 3, 5 · 7, 26, 23 · 17, 97, 179, 107 … Σ 1, 8, 56, 400, 2848, 20288, 144512, …
  [−1, 0]nonnegative
• narayana → basis A013609 → D+1
  1 · 4, 2 · 15, 18, 4 · 56, 114, 60, 8 … Σ 1, 6, 37, 238, 1578, 10684, 73429, …
  (−∞, 0]nonnegative
• A145324 → basis A013611 → x ↦ x+1
  1 · 11, 8 · 176, 260, 96 · 3696, 8276, 6176, 1536 … Σ 1, 19, 532, 19684, 905464, 49800520, …
  (−∞, −1]nonnegativerational roots
• A225470 → θ+1
  1 · 2, 2 · 10, 14, 3 · 80, 132, 45, 4 … Σ 1, 4, 27, 261, 3294, 51354, 953532, …
  (−∞, 0]nonnegative
• A054142 → reverse
  1 · 1, 1 · 1, 3, 1 · 1, 6, 5, 1 … Σ 1, 2, 5, 13, 34, 89, 233, 610, …
  (−∞, 0]nonnegative
• A290598 → x ↦ x+1
  1 · 5, 1 · 43, 16, 1 · 521, 273, 33, 1 … Σ 1, 6, 60, 828, 14472, 304992, 7506432, …
  (−∞, −1]nonnegative
• A128099 → x ↦ x+1 → D+1
  1 · 1 · 5, 2 · 9, 4 … Σ 1, 1, 7, 13, 51, 121, 383, 997, …
  (−∞, 0]nonnegative
• A116414 → basis A007318 → reverse
  1 · 1, 5 · 1, 10, 22 · 1, 15, 69, 95 … Σ 1, 6, 33, 180, 981, 5346, 29133, …
  (−1, 0]nonnegative
• A038319 → D
  110, 50 · 1815, 1650, 375 · 26620, 36300, 16500, 2500 … Σ 160, 3840, 81920, 1638400, 31457280, …
  (−∞, 0]nonnegativerational roots
• A099093 → x ↦ x−6
  1 · -18, 3 · 306, -105, 9 · -5184, 2700, -468, 27 … Σ 1, -15, 210, -2925, 40725, -567000, …
  (14/3, 6]
• A126351 → reverse → x ↦ x/(x+1)
  1 · 2, 3 · 4, 13, 10 · 8, 43, 71, 37 … Σ 1, 5, 27, 159, 1025, 7221, 55307, …
  [−1, 0]nonnegative
• A008955 → basis A013610 → θ+1
  1 · 2, 6 · 10, 78, 108 · 100, 1320, 4239, 3888 … Σ 1, 8, 196, 9547, 771515, 93254890, …
  (−∞, 0]nonnegative
• A257622 → x ↦ x+1
  1 · 8, 4 · 88, 88, 16 · 1232, 1848, 744, 64 … Σ 1, 12, 192, 3888, 95472, 2756592, …
  (−∞, −1]nonnegative
• A091044 → (θ+1)²
  1 · 2, 8 · 3, 40, 27 · 4, 112, 252, 64 … Σ 1, 10, 70, 432, 2464, 13312, 69120, …
  (−∞, 0]nonnegative
• A306364 → θ+1
  1 · 3 · 12, 6 · 60, 90 … Σ 1, 3, 18, 150, 1575, 19845, 291060, …
  (−∞, 0]nonnegative
• A100861 → D
  1 · 3 · 6, 6 · 10, 30 … Σ 0, 0, 1, 3, 12, 40, 150, 546, …
  (−∞, 0]nonnegative
• A113413 → basis A013609 → θ+1
  1 · 3, 4 · 7, 24, 12 · 17, 92, 108, 32 … Σ 1, 7, 43, 249, 1385, 7495, 39747, …
  (−∞, 0]nonnegative
• A156992 → x ↦ x/(x+1)
  1 · 2, 4 · 6, 24, 24 · 24, 144, 288, 192 … Σ 1, 6, 54, 648, 9720, 174960, 3674160, …
  [−1, 0]nonnegativerational roots