Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A178822 → basis A013610 → x ↦ x/(x+1)
  1 · 12, 30 · 84, 420, 525 · 448, 3360, 8400, 7000 … Σ 1, 42, 1029, 19208, 302526, 4235364, …
  [−1, 0]nonnegativerational roots
• A132393 → Veronese 2 → x ↦ x+1
  1 · 1, 1 · 3, 3 · 12, 13, 1 … Σ 1, 0, 2, 6, 26, 140, 896, 6636, …
  (−∞, −1]nonnegative
• A376827 → D
  1 · 6, 6 · 21, 42, 21 · 76, 228, 228, 76 … Σ 0, 1, 12, 84, 608, 4080, 27072, 176064, …
  (−∞, 0]nonnegativerational roots
• A371081 → reverse
  1 · 1, 16 · 1, 52, 400 · 1, 116, 2948, 14400 … Σ 1, 17, 453, 17465, 921233, 63789145, …
  (−∞, 0]nonnegative
• A207823 → basis A013609 → D+1
  1 · 7, 2 · 44, 28, 4 · 261, 266, 84, 8 … Σ 1, 9, 76, 619, 4915, 38296, 294069, …
  (−∞, 0]nonnegative
• A085880 → x ↦ x+1 → reverse
  1 · 1, 2 · 2, 8, 8 · 5, 30, 60, 40 … Σ 1, 3, 18, 135, 1134, 10206, 96228, …
  (−∞, 0]nonnegativerational roots
• A109692 → basis A007318 → x ↦ x/(x+1)
  1 · 2, 3 · 8, 26, 21 · 48, 244, 412, 231 … Σ 1, 5, 55, 935, 21505, 623645, 21827575, …
  [−1, 0]nonnegativerational roots
• A172431 → D → D
  6 · 20, 24 · 42, 120, 60 · 72, 336, 420, 120 … Σ 0, 0, 6, 44, 222, 948, 3672, 13332, …
  (−∞, 0]nonnegative
• A169803 → basis A013610 → basis A007318
  1 · 5, 3 · 9, 6 · 29, 33, 9 … Σ 1, 8, 15, 71, 176, 673, 1905, 6616, …
  (−∞, −1]nonnegative
• A128813 → basis A013609 → D
  2 · 20, 24 · 236, 648, 432 … Σ 0, 2, 44, 1316, 55996, 3282616, …
  (−∞, 0]nonnegative
• A038321 → basis A013610 → θ+1
  324, 1512, 1323 · 5832, 40824, 71442, 37044 … Σ 3159, 155142, 7296237, 333135504, …
  (−∞, 0]nonnegativerational roots
• A159854 → x ↦ x+1
  1, 2, 1 · 2, 5, 4, 1 · 4, 12, 13, 6, 1 … Σ 4, 12, 36, 108, 324, 972, 2916, 8748, …
  (−∞, −1]nonnegativerational roots
• A261711 → (θ+1)²
  1 · 4 · 15, 4 · 56, 32 … Σ 1, 4, 19, 88, 402, 1816, 8128, 36096, …
  (−∞, 0]nonnegative
• A026729 → basis A013610 → θ+1
  1 · 1, 6 · 2, 18, 27 · 3, 42, 135, 108 … Σ 1, 7, 47, 288, 1688, 9584, 53200, …
  (−∞, 0]nonnegative
• A207824 → basis A013610 → D
  3 · 36, 18 · 318, 324, 81 · 2484, 3834, 1944, 324 … Σ 0, 3, 54, 723, 8586, 95508, 1019466, …
  (−∞, 0]nonnegative
• Veronese parent of A098172 → D+1
  1, 1 · 0, 0, 3, 5, 1 · 0, 0, 0, 0, 5, 13, 9, 1 … Σ 2, 9, 28, 76, 192, 464, 1088, 2496, …
  (−∞, 0]nonnegative
• Veronese parent of A098172 → D
  0, 2, 3 · 0, 0, 0, 4, 10, 6 … Σ 0, 5, 20, 60, 160, 400, 960, 2240, …
  (−∞, 0]nonnegativerational roots
• A249139 → basis A013610 → D
  3 · 6 · 27, 18 · 66, 54 … Σ 0, 3, 6, 45, 120, 555, 1674, 6393, …
  (−∞, 0]nonnegative
• A062190 → reverse → D+1
  1 · 7, 1 · 35, 16, 1 · 140, 132, 27, 1 … Σ 1, 8, 52, 300, 1595, 8008, 38584, …
  (−∞, 0]nonnegative
• A013610 → Veronese 2 → D
  9 · 27 · 54, 162 · 90, 810 … Σ 0, 0, 9, 27, 216, 900, 4752, 21168, …
  (−∞, 0]nonnegative
• A199881 → x ↦ x/(x+1)
  1, 2 · 0, 1, 2 · 0, 1, 4, 4 · 0, 0, 2, 7, 6 … Σ 3, 3, 9, 15, 33, 63, 129, 255, …
  [−1, 0]nonnegative
• A091044: One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
  1 · 2, 2 · 3, 10, 3 · 4, 28, 28, 4 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  (−∞, 0)nonnegative
• A033282 → x ↦ x+1 → D+1
  1 · 5, 2 · 26, 25, 5 · 138, 219, 105, 14 … Σ 1, 7, 56, 476, 4187, 37645, 343630, …
  (−∞, 0]nonnegative
• Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(…
  1, 8, 1 · 1, 19, 19, 1 · 1, 42, 114, 42, 1 … Σ 10, 40, 200, 1200, 8400, 67200, 604800, …
  (−∞, 0)nonnegative
• A059473 → x ↦ x+1 → D+1
  1 · 6, 2 · 36, 26, 4 · 200, 232, 80, 8 … Σ 1, 8, 66, 520, 3992, 30048, 222784, …
  (−∞, 0]nonnegative
• A131222 → D+1
  1 · 1, 1 · 3, 5, 1 · 14, 32, 12, 1 … Σ 1, 2, 9, 59, 506, 5354, 67344, 981336, …
  (−∞, 0]nonnegative
• A038255: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).
  1 · 6, 1 · 36, 12, 1 · 216, 108, 18, 1 … Σ 1, 7, 49, 343, 2401, 16807, 117649, …
  (−∞, −1]nonnegativerational roots
• A064861 → Veronese 2 → D
  2 · 8 · 13, 8 · 25, 56 … Σ 2, 8, 21, 81, 189, 675, 1512, 5184, …
  (−∞, 0]nonnegative
• A038330 → basis A013609 → reverse
  64, 256, 256 · 512, 3072, 6144, 4096 … Σ 576, 13824, 331776, 7962624, 191102976, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A026729, shifted → D
  -1 · 0, 2 · 0, 0, -3 · 0, 0, 0, 4 … Σ 0, -1, 2, -3, 4, -5, 6, -7, …
  (−∞, 0]rational roots
• A086915 → x ↦ x+1
  2 · 8, 4 · 44, 40, 8 · 304, 384, 144, 16 … Σ 2, 12, 92, 848, 9072, 110048, 1488608, …
  (−∞, −1]nonnegative
• A113214 → x ↦ x−6
  1 · -4, 1 · 18, -9, 1 · -84, 62, -14, 1 … Σ 1, -3, 10, -35, 125, -450, 1625, -5875, …
  (2, 6]
• A059473 → x ↦ x/(x+1)
  1 · 2, 4 · 4, 18, 18 · 8, 56, 120, 80 … Σ 1, 6, 40, 264, 1744, 11520, 76096, …
  [−1, 0]nonnegative
• A100862 → x ↦ x+1 → D+1
  1 · 3, 1 · 10, 7, 1 · 35, 39, 12, 1 … Σ 1, 4, 18, 87, 447, 2418, 13692, 80769, …
  (−∞, 0]nonnegative
• A286718 → D+1
  1 · 2, 1 · 9, 7, 1 · 67, 63, 15, 1 … Σ 1, 3, 17, 146, 1686, 24484, 428548, …
  (−∞, 0]nonnegative
• A064861 → reverse → x ↦ x/(x+1)
  2, 7, 6 · 4, 20, 33, 18 · 4, 28, 73, 84, 36 … Σ 15, 75, 225, 1125, 3375, 16875, 50625, …
  [−1, 0]nonnegativerational roots
• A099040 → x ↦ x−15
  1 · -30, 2 · 870, -118, 4 · -25200, 5160, -352, 8 … Σ 1, -28, 756, -20384, 549584, -14817600, …
  (13, 15]
• basis-map coefficients of A026729, shifted → θ+1 → D
  -2 · 0, 6 · 0, 0, -12 · 0, 0, 0, 20 … Σ 0, -2, 6, -12, 20, -30, 42, -56, …
  (−∞, 0]rational roots
• A272867 → basis A013610 → D
  18, 18 · 216, 864, 972, 324 … Σ 0, 36, 2376, 117612, 5174928, …
  (−∞, 0]nonnegative
• A156992 → basis A013609 → D+1
  1 · 8, 4 · 72, 96, 24 · 768, 1728, 1152, 192 … Σ 1, 12, 192, 3840, 92160, 2580480, …
  (−∞, 0]nonnegativerational roots
• A200536 → reverse
  1 · 2, 3, 1 · 4, 12, 13, 6, 1 … Σ 1, 6, 36, 216, 1296, 7776, 46656, …
  (−∞, 0]nonnegativerational roots
• A027465 → basis A008277 → x ↦ x/(x+1)
  1 · 4, 5 · 16, 41, 26 · 64, 253, 329, 141 … Σ 1, 9, 83, 787, 7697, 77865, 816627, …
  [−1, 0]nonnegative
• A038337 → D+1
  408, 506, 121 · 6480, 13464, 8349, 1331 … Σ 1035, 29624, 815189, 21827598, …
  (−∞, 0]nonnegativerational roots
• A013613 → x ↦ x+1 → θ+1
  1 · 7, 12 · 49, 168, 108 · 343, 1764, 2268, 864 … Σ 1, 19, 325, 5239, 81289, 1228123, …
  (−∞, 0]nonnegativerational roots
• A272866 → D → D
  2 · 22, 36, 12 · 60, 270, 360, 180, 30 … Σ 0, 2, 70, 900, 8500, 68750, 506250, …
  (−∞, 0]nonnegative
• A166960 → D+1
  1 · 2, 1 · 7, 8, 1 · 28, 69, 24, 1 … Σ 1, 3, 16, 122, 1272, 17204, 291028, …
  (−∞, 0]nonnegative
• A088996 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 3 · 0, 2, 11, 15 … Σ 1, 1, 4, 28, 280, 3640, 58240, 1106560, …
  [−1, 0]nonnegativerational roots
• A052553 → basis A008277 → D+1
  1 · 1 · 3, 1 · 5, 2 … Σ 1, 1, 4, 7, 20, 43, 113, 267, …
  (−∞, 0]nonnegative
• narayana → basis A013610 → D
  3 · 15, 18 · 63, 162, 81 · 252, 1008, 1134, 324 … Σ 0, 3, 33, 306, 2718, 23805, 207495, …
  (−∞, 0]nonnegative
• A207823 → basis A013609 → D
  2 · 20, 8 · 146, 120, 24 · 940, 1176, 480, 64 … Σ 0, 2, 28, 290, 2660, 22840, 188132, …
  (−∞, 0]nonnegative