Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A062145 → D → D+1
  4 · 30, 20 · 108, 210, 60 · 280, 1092, 840, 140 … Σ 0, 4, 50, 378, 2352, 13200, 69498, …
  (−∞, 0]nonnegative
• A085478 → basis A013610 → θ+1
  1 · 1 · 2, 6 · 5, 30, 27 … Σ 1, 1, 8, 62, 451, 3151, 21410, 142532, …
  (−∞, 0]nonnegative
• A391533 → Veronese 2 (j=1) → θ+1
  4 · 8, 2 · 12, 12 · 16, 38, 3 … Σ 4, 10, 24, 57, 132, 300, 672, 1488, …
  (−∞, 0]nonnegative
• A172431 → D → (θ+1)²
  2 · 4, 24 · 6, 80, 108 · 8, 168, 540, 320 … Σ 0, 2, 28, 194, 1036, 4800, 20268, …
  (−∞, 0]nonnegative
• A030528 → basis A013610 → θ+1
  1 · 2, 6 · 3, 24, 27 · 5, 60, 162, 108 … Σ 1, 8, 54, 335, 1976, 11272, 62784, …
  (−∞, 0]nonnegative
• A256890 → (θ+1)²
  4, 48, 36 · 8, 208, 468, 128 … Σ 88, 812, 8080, 87120, 1016400, …
  (−∞, 0]nonnegative
• A078812 → D → (θ+1)²
  1 · 4, 8 · 10, 48, 27 · 20, 168, 216, 64 … Σ 0, 1, 12, 85, 468, 2220, 9540, 38207, …
  (−∞, 0]nonnegative
• A085881 → basis A013611 → D
  4 · 48, 96 · 720, 2880, 2880 … Σ 0, 4, 144, 6480, 362880, 24494400, …
  (−∞, 0]nonnegativerational roots
• A126216 → basis A007318
  1 · 3, 1 · 11, 7, 1 · 45, 42, 12, 1 … Σ 1, 4, 19, 100, 562, 3304, 20071, …
  (−∞, −1]nonnegative
• delannoy → D+1
  1 · 2, 1 · 4, 5, 1 · 6, 15, 8, 1 … Σ 1, 3, 10, 30, 87, 245, 676, 1836, …
  (−∞, 0]nonnegative
• A176230 → x ↦ x+1
  1 · 2, 1 · 10, 8, 1 · 76, 78, 18, 1 … Σ 1, 3, 19, 173, 2025, 28787, 479851, …
  (−∞, −1]nonnegative
• narayana → γ → D
  1 · 3 · 6, 4 · 10, 20 … Σ 0, 0, 1, 3, 10, 30, 90, 266, …
  (−∞, 0]nonnegative
• A038303 → basis A013610 → D+1
  1 · 14, 3 · 187, 84, 9 · 2420, 1683, 378, 27 … Σ 1, 17, 280, 4508, 71344, 1114064, …
  (−∞, 0]nonnegativerational roots
• A034867 → θ+1
  1 · 2 · 3, 2 · 4, 8 … Σ 1, 2, 5, 12, 28, 64, 144, 320, …
  (−∞, 0]nonnegative
• A028338 → x ↦ x+1 → D
  1 · 6, 2 · 44, 24, 3 · 400, 280, 60, 4 … Σ 0, 1, 8, 71, 744, 9129, 129072, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729 → basis A013611 → reverse
  1 · 4, 1 · 16, 8, 1 · 64, 48, 12, 1 … Σ 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A113214, shifted → Veronese 2 → (θ+1)²
  1 · -1 · 1, 4 · -1, -12 … Σ 1, -1, 5, -13, 34, -86, 212, -512, …
  (−∞, 0]
• A049218 → D+1
  -2, 2, 1 · -8, -8, 3, 1 · 24, -40, -20, 4, 1 … Σ 1, -12, -31, 214, 1289, -7704, -85983, …
  (−∞, ∞)
• A142458 → (θ+1)²
  1 · 1, 4 · 1, 32, 9 · 1, 156, 351, 16 … Σ 1, 5, 42, 524, 8260, 156720, 3467200, …
  (−∞, 0]nonnegative
• A122542 → Veronese 2 → x ↦ x+1
  1 · 1, 1 · 4, 4 · 9, 10, 1 … Σ 1, 0, 2, 8, 20, 56, 168, 488, …
  (−∞, −1]nonnegative
• narayana → basis A013609 → (θ+1)²
  1 · 2, 8 · 5, 40, 36 · 14, 168, 324, 128 … Σ 1, 10, 81, 634, 4922, 38084, 294025, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/4) (shift +1) → x ↦ x+1 → D+1
  1 · 2 · 8, 2 · 40, 16 … Σ 1, 2, 10, 56, 408, 3456, 34016, 378496, …
  (−∞, 0]nonnegative
• A094638 → basis A013611 → basis A007318
  1 · 6, 4 · 66, 92, 32 · 1056, 2264, 1616, 384 … Σ 1, 10, 190, 5320, 196840, 9054640, …
  (−∞, −1]nonnegativerational roots
• A028338 → basis A008277 → x ↦ x+1
  1 · 3, 1 · 16, 9, 1 · 121, 90, 18, 1 … Σ 1, 4, 26, 230, 2558, 34162, 531538, …
  (−∞, −1]nonnegative
• A103327 → D → x ↦ x+1
  1 · 12, 2 · 80, 48, 3 · 448, 480, 120, 4 … Σ 0, 1, 14, 131, 1052, 7813, 55338, …
  (−∞, −1]nonnegative
• A079638 → basis A007318
  1 · 4, 1 · 24, 11, 1 · 192, 122, 21, 1 … Σ 1, 5, 36, 336, 3840, 51840, 806400, …
  (−∞, −1]nonnegative
• A038329 → x ↦ x+1
  225, 90, 9 · 3375, 2025, 405, 27 … Σ 324, 5832, 104976, 1889568, 34012224, …
  (−∞, −1]nonnegativerational roots
• A124932 → reverse → (θ+1)²
  1 · 3, 8 · 6, 36, 27 · 10, 96, 162, 64 … Σ 1, 11, 69, 332, 1360, 4992, 16912, …
  (−∞, 0]nonnegative
• A038335 → x ↦ x+1 → reverse
  1 · 9, 21 · 81, 378, 441 · 729, 5103, 11907, 9261 … Σ 1, 30, 900, 27000, 810000, 24300000, …
  (−∞, 0]nonnegativerational roots
• A091044 → basis A013611 → x ↦ x/(x+1)
  1 · 4, 12 · 16, 96, 128 · 64, 576, 1600, 1344 … Σ 1, 16, 240, 3584, 53504, 798720, …
  [−1, 0]nonnegative
• A008955 → basis A013609 → basis A007318
  1 · 4, 2 · 52, 58, 16 · 1456, 2560, 1492, 288 … Σ 1, 6, 126, 5796, 469476, 59153976, …
  (−∞, −1]nonnegativerational roots
• A120986 → reverse
  1 · 1, 2 · 1, 6, 5 · 1, 12, 28, 14 … Σ 1, 3, 12, 55, 273, 1428, 7752, 43263, …
  (−∞, 0]nonnegative
• A088996: Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.
  1 · 0, 1 · 0, 1, 2 · 0, 2, 7, 6 … Σ 1, 1, 3, 15, 105, 945, 10395, 135135, …
  [−1, 0]nonnegativerational roots
• A007318 → Veronese 2 (j=1) → D
  1 · 4 · 10, 2 · 20, 12 … Σ 0, 0, 0, 1, 4, 12, 32, 80, …
  (−∞, 0]nonnegative
• A152198 → Veronese 2 (j=1) → x ↦ x+1
  1 · 1 · 2 · 2 … Σ 0, 0, 1, 1, 2, 2, 5, 5, …
  (−∞, −1]nonnegative
• A122076 → reverse
  2 · 2, 3 · 2, 8, 7 · 2, 15, 30, 18 … Σ 2, 5, 17, 65, 257, 1025, 4097, 16385, …
  (−∞, 0]nonnegative
• A201637 → reverse (shift +1) → x ↦ x/(x+1)
  1 · 0, 1 · 0, 2, 3 · 0, 6, 20, 15 … Σ 1, 1, 5, 41, 469, 6889, 123605, …
  [−1, 0]nonnegative
• A038324 → reverse
  1 · 10, 11 · 100, 220, 121 · 1000, 3300, 3630, 1331 … Σ 1, 21, 441, 9261, 194481, 4084101, …
  (−∞, 0]nonnegativerational roots
• A142459 → D+1
  1 · 2, 1 · 11, 12, 1 · 60, 177, 62, 1 … Σ 1, 3, 24, 300, 5040, 105840, 2661120, …
  (−∞, 0]nonnegative
• A085880 → D → D+1
  1 · 8, 4 · 45, 60, 15 · 224, 504, 336, 56 … Σ 0, 1, 12, 120, 1120, 10080, 88704, …
  (−∞, 0]nonnegativerational roots
• A008292 → D
  1 · 4, 2 · 11, 22, 3 · 26, 132, 78, 4 … Σ 0, 1, 6, 36, 240, 1800, 15120, 141120, …
  (−∞, 0]nonnegative
• A038314 → x ↦ x+1 → D+1
  1012, 816, 144 · 28072, 36432, 14688, 1728 … Σ 1972, 80920, 3222928, 125615584, …
  (−∞, 0]nonnegativerational roots
• A078812 → reverse → (θ+1)²
  1 · 1, 8 · 1, 16, 27 · 1, 24, 90, 64 … Σ 1, 9, 44, 179, 667, 2352, 7973, 26229, …
  (−∞, 0]nonnegative
• A099097 → D+1
  1 · 3, 3 · 1, 19, 9 · 0, 12, 87, 27 … Σ 1, 6, 29, 126, 516, 2034, 7807, 29382, …
  (−∞, 0]nonnegative
• A102547 → reverse → x ↦ x/(x+1)
  0, 1 · 0, 0, 1 · 0, 0, 0, 1 · 0, 0, 0, 1, 2 … Σ 1, 1, 1, 3, 5, 7, 13, 23, …
  [−1, 0]nonnegative
• A013620 → basis A007318 → θ+1
  1 · 5, 6 · 25, 60, 27 · 125, 450, 405, 108 … Σ 1, 11, 112, 1088, 10240, 94208, 851968, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A297191, shifted → D → D+1
  2, 2 · 0, 0, 12, 4 · 0, 0, 0, 0, 30, 6 … Σ 0, 4, 16, 36, 64, 100, 144, 196, …
  (−∞, 0]nonnegativerational roots
• A185411: A triangular decomposition of the double factorial numbers A001147.
  A185410 → reverse
  1 · 0, 1 · 0, 2, 1 · 0, 4, 10, 1 … Σ 1, 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, 0]nonnegative
• A038271 → basis A013610 → θ+1
  1 · 12, 30 · 144, 720, 675 … Σ 1, 42, 1539, 52488, 1712421, 54206982, …
  (−∞, 0]nonnegativerational roots
• A225470 → basis A013611 → (θ+1)²
  1 · 3, 16 · 18, 144, 144 · 162, 1584, 2592, 1024 … Σ 1, 19, 306, 5362, 106024, 2367288, …
  (−∞, 0]nonnegative