Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A280580 → x ↦ x+1
  1 · 2, 1 · 9, 8, 1 · 51, 63, 18, 1 … Σ 1, 3, 18, 133, 1094, 9614, 88372, …
  (−∞, −1]nonnegative
• A019538 → D
  2 · 6, 12 · 14, 72, 72 · 30, 300, 720, 480 … Σ 0, 2, 18, 158, 1530, 16622, 201978, …
  (−∞, 0]nonnegative
• A274496 → θ+1
  1 · 2 · 2, 4 · 4, 8 … Σ 1, 2, 6, 12, 32, 64, 160, 320, …
  (−∞, 0]nonnegativerational roots
• A013609 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 9 · 9, 54, 81 · 27, 243, 729, 729 … Σ 1, 12, 144, 1728, 20736, 248832, …
  [−1, 0]nonnegativerational roots
• A027467 → basis A013611 → x ↦ x+1
  1 · 20, 4 · 400, 160, 16 · 8000, 4800, 960, 64 … Σ 1, 24, 576, 13824, 331776, 7962624, …
  (−∞, −1]nonnegativerational roots
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
  1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 1, 1, 1, 1, …
  (−∞, −1][−1, 0][−3−2√2, −3+2√2]+2
• A008556 → D+1
  1 · 2 · 8, 2 · 32, 12 … Σ 1, 2, 10, 44, 208, 992, 4784, 23200, …
  (−∞, 0]nonnegative
• A159854 → Veronese 2 (j=1) → (θ+1)²
  0, 4 · 0, 8 · 0, 12, 9 · 0, 16, 36 … Σ 0, 4, 8, 21, 52, 126, 300, 704, …
  (−∞, 0]nonnegative
• A026729: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
  1 · 0, 1 · 0, 1, 1 · 0, 0, 2, 1 … Σ 1, 1, 2, 3, 5, 8, 13, 21, …
  (−4, 0]nonnegative
• A103327 → basis A013610 → x ↦ x+1
  1 · 7, 3 · 61, 54, 9 · 547, 753, 297, 27 … Σ 1, 10, 124, 1624, 21520, 285856, …
  (−∞, −1]nonnegative
• A132393 → basis A013609 → θ+1
  1 · 1, 4 · 2, 12, 12 · 6, 44, 72, 32 … Σ 1, 5, 26, 154, 1044, 8028, 69264, …
  (−∞, 0]nonnegative
• A269945 → D+1
  1 · 1, 1 · 1, 3, 1 · 1, 11, 8, 1 … Σ 1, 2, 5, 21, 126, 1001, 10129, 126006, …
  (−∞, 0]nonnegative
• A391533 → x ↦ x+1
  9, 6, 1 · 18, 21, 8, 1 · 36, 60, 37, 10, 1 … Σ 16, 48, 144, 432, 1296, 3888, 11664, …
  (−∞, −1]nonnegativerational roots
• A027467 → D+1
  1 · 16, 1 · 255, 32, 1 · 4050, 765, 48, 1 … Σ 1, 17, 288, 4864, 81920, 1376256, …
  (−∞, 0]nonnegativerational roots
• A085880 → basis A013609 → θ+1
  1 · 2, 4 · 8, 32, 24 · 40, 240, 360, 160 … Σ 1, 6, 64, 800, 10752, 150528, 2162688, …
  (−∞, 0]nonnegativerational roots
• A207824 → basis A013609
  1 · 6, 2 · 35, 24, 4 · 204, 212, 72, 8 … Σ 1, 8, 63, 496, 3905, 30744, 242047, …
  (−4, −2)nonnegative
• A178822 → x ↦ x+1 → reverse
  1 · 6, 12 · 21, 84, 84 · 56, 336, 672, 448 … Σ 1, 18, 189, 1512, 10206, 61236, 336798, …
  (−∞, 0]nonnegativerational roots
• A108838 → x ↦ x+1 → θ+1
  2 · 5, 4 · 14, 24, 6 · 42, 112, 63, 8 … Σ 2, 9, 44, 225, 1182, 6321, 34232, …
  (−∞, 0]nonnegative
• A091044 → ⊙ C(n+2,k+1) → D+1
  1 · 2, 1 · 6, 7, 1 · 15, 42, 17, 1 … Σ 1, 3, 14, 75, 438, 2702, 17324, 114291, …
  (−∞, 0]nonnegative
• A154233 → θ+1
  1 · 1, 2 · 1, 14, 3 · 1, 38, 57, 4 … Σ 1, 3, 18, 100, 753, 5817, 54460, …
  (−∞, 0]nonnegative
• A269944 → x ↦ x+1 → reverse
  1 · 1, 1 · 1, 3, 2 · 1, 8, 17, 10 … Σ 1, 2, 6, 36, 396, 7128, 192456, …
  (−∞, 0]nonnegativerational roots
• A116414 → x ↦ x/(x+1)
  1 · 4, 5 · 13, 34, 22 · 40, 162, 216, 95 … Σ 1, 9, 69, 513, 3789, 27945, 206037, …
  [−1, 0)nonnegative
• A081578 → θ+1
  1 · 1, 2 · 1, 10, 3 · 1, 18, 27, 4 … Σ 1, 3, 14, 50, 183, 637, 2188, 7380, …
  (−∞, 0]nonnegative
• A137597 → D+1
  1 · 3, 1 · 10, 7, 1 · 37, 40, 12, 1 … Σ 1, 4, 18, 90, 494, 2946, 18926, 130066, …
  (−∞, −1]nonnegative
• A125092 → basis A007318
  1 · 5, 4 · 18, 26, 9 · 56, 114, 75, 16 … Σ 1, 9, 53, 261, 1161, 4833, 19197, …
  (−∞, −1]nonnegative
• A085478 → reverse → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 1, 20, 54, 16 … Σ 1, 5, 22, 91, 349, 1260, 4349, 14509, …
  (−∞, 0]nonnegative
• A225476 → (θ+1)²
  1 · 1, 4 · 1, 16, 18 · 1, 52, 162, 96 … Σ 1, 5, 35, 311, 3341, 42065, 607055, …
  (−∞, 0]nonnegative
• A038326 → basis A013609 → basis A007318
  1 · 47, 24 · 2209, 2256, 576 … Σ 1, 71, 5041, 357911, 25411681, …
  (−∞, −1]nonnegativerational roots
• A091044 → basis A013610 → reverse
  1 · 6, 4 · 27, 48, 16 · 108, 360, 288, 64 … Σ 1, 10, 91, 820, 7381, 66430, 597871, …
  (−∞, 0]nonnegative
• A152828 → D+1
  1 · 1 · 1 · 2, 1 … Σ 1, 1, 1, 3, 3, 3, 8, 8, …
  (−∞, 0]nonnegativerational roots
• A063007 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 3, 5 · 13, 44, 37 · 63, 321, 543, 305 … Σ 1, 8, 94, 1232, 16966, 240368, 3468844, …
  [−1, −1/2)nonnegative
• A121574 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 6 · 10, 41, 40 · 33, 204, 408, 264 … Σ 1, 9, 91, 909, 9091, 90909, 909091, …
  [−1, 0]nonnegative
• A013609 → basis A008288 → θ+1
  1 · 3, 4 · 9, 32, 12 · 27, 164, 156, 32 … Σ 1, 7, 53, 379, 2617, 17631, 116621, …
  (−∞, 0]nonnegative
• A082985 → θ+1
  1 · 1, 6 · 1, 10, 15 · 1, 14, 42, 28 … Σ 1, 7, 26, 85, 265, 804, 2393, 7019, …
  (−∞, 0]nonnegative
• A391533 → reverse
  1, 4, 4 · 1, 5, 8, 4 · 1, 6, 13, 12, 4 … Σ 9, 18, 36, 72, 144, 288, 576, 1152, …
  (−∞, 0]nonnegativerational roots
• A081577 → basis A013610 → reverse
  1 · 3, 2 · 9, 18, 6 · 27, 90, 72, 16 … Σ 1, 5, 33, 205, 1289, 8085, 50737, …
  (−∞, 0]nonnegative
• A130123 → θ+1 → basis A007318
  1 · 4, 4 · 12, 24, 12 · 32, 96, 96, 32 … Σ 1, 8, 48, 256, 1280, 6144, 28672, …
  (−∞, −1]nonnegativerational roots
• A228955 → θ+1
  1 · 1, 2 · 2, 12 · 6, 72, 18 … Σ 1, 3, 14, 96, 864, 9600, 126720, …
  (−∞, 0]nonnegative
• A122542 → reverse → x ↦ x+1
  1 · 1 · 3, 2 · 7, 8, 2 … Σ 1, 1, 5, 17, 61, 217, 773, 2753, …
  (−∞, −1]nonnegative
• A367270 → D → θ+1
  2 · 6, 12 · 12, 72, 36 · 20, 240, 360, 80 … Σ 0, 0, 2, 18, 120, 700, 3780, 19404, …
  (−∞, 0]nonnegative
• A099040 → basis A007318 → (θ+1)²
  1 · 2, 8 · 6, 40, 36 · 16, 160, 288, 128 … Σ 1, 10, 82, 592, 3976, 25408, 156608, …
  (−∞, 0]nonnegative
• A112493 → reverse
  1 · 1, 1 · 3, 4, 1 · 15, 25, 11, 1 … Σ 1, 2, 8, 52, 472, 5504, 78416, 1320064, …
  (−∞, 0]nonnegative
• A154986 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 14, 36, 24 … Σ 1, 3, 13, 75, 537, 4611, 46053, 525483, …
  [−1, 0]nonnegative
• A136124 → basis A008277
  1 · 3, 1 · 12, 8, 1 · 60, 60, 15, 1 … Σ 1, 4, 21, 136, 1045, 9276, 93289, …
  (−∞, 0]nonnegative
• A038301 → basis A013609 → (θ+1)²
  400, 3520, 4356 · 8000, 105600, 261360, 170368 … Σ 8276, 545328, 32916240, 1874722752, …
  (−∞, 0]nonnegative
• A243662 → x ↦ x+1
  1 · 4, 1 · 21, 10, 1 · 126, 88, 18, 1 … Σ 1, 5, 32, 233, 1833, 15180, 130392, …
  (−∞, −1]nonnegative
• A038328 → basis A013609 → x ↦ x+1
  324, 144, 16 · 5832, 3888, 864, 64 … Σ 484, 10648, 234256, 5153632, 113379904, …
  (−∞, −1]nonnegativerational roots
• reversed rows → D
  2 · 3 · 4, 4 · 5, 10 … Σ 0, 2, 3, 8, 15, 30, 56, 104, …
  (−∞, 0]nonnegative
• A133080 → basis A013609 → x ↦ x+1
  1 · 4, 2 · 9, 12, 4 · 36, 66, 40, 8 … Σ 1, 6, 25, 150, 625, 3750, 15625, 93750, …
  (−∞, −1]nonnegativerational roots
• A038328 → x ↦ x+1 → θ+1
  196, 112, 12 · 2744, 2352, 504, 32 … Σ 320, 5632, 98304, 1703936, 29360128, …
  (−∞, 0]nonnegativerational roots