Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A249248 → x ↦ x+1
  1 · 4, 3 · 11, 13, 3 · 38, 65, 31, 3 … Σ 1, 7, 27, 137, 681, 3687, 20595, …
  (−∞, −1]nonnegative
• A078812 → reverse → D+1
  1 · 3, 2 · 5, 10, 3 · 7, 26, 22, 4 … Σ 1, 5, 18, 59, 185, 564, 1685, 4957, …
  (−∞, 0]nonnegative
• A038284 → basis A013609 → D
  336, 288 · 7056, 12096, 5184 … Σ 624, 24336, 843648, 27418560, …
  (−∞, 0]nonnegativerational roots
• A079510 → θ+1
  1 · 0, 4 · 0, 6, 18 · 0, 0, 60, 96 … Σ 1, 4, 24, 156, 1165, 9780, 91448, …
  (−∞, 0]nonnegative
• A357367 → D+1
  1 · 2, 2 · 6, 30, 12 · 24, 264, 480, 120 … Σ 1, 4, 48, 888, 21960, 676800, 24927840, …
  (−∞, 0]nonnegative
• A196347 → basis A013611 → (θ+1)²
  1 · 2, 16 · 8, 128, 288 · 48, 1152, 5184, 6144 … Σ 1, 18, 424, 12528, 445824, 18558720, …
  (−∞, 0]nonnegative
• A391045: Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.
  1 · 1, 1 · 1, 4, 1 · 1, 9, 10, 1 … Σ 1, 2, 6, 21, 80, 322, 1347, 5798, …
  (−∞, 0)nonnegative
• A056242 → Veronese 2 → x ↦ x+1
  1 · 1 · 5, 4 · 17, 16 … Σ 1, 1, 9, 33, 147, 619, 2647, 11271, …
  (−∞, −1]nonnegative
• A013609 → basis A013610 → x ↦ x+1
  1 · 9, 6 · 81, 108, 36 · 729, 1458, 972, 216 … Σ 1, 15, 225, 3375, 50625, 759375, …
  (−∞, −1]nonnegativerational roots
• A008459 → basis A013611 → D
  4 · 24, 32 · 120, 384, 192 · 560, 2880, 3840, 1024 … Σ 0, 4, 56, 696, 8304, 96600, 1104144, …
  (−∞, 0]nonnegative
• A053117 → x ↦ x+1 → x ↦ x+1
  1 · 4, 2 · 15, 16, 4 · 56, 92, 48, 8 … Σ 1, 6, 35, 204, 1189, 6930, 40391, …
  (−3, −1)nonnegative
• A091042 → basis A013609 → (θ+1)²
  1 · 4, 24 · 16, 160, 180 · 64, 896, 2016, 896 … Σ 1, 28, 356, 3872, 39376, 383296, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A013609
  1 · 1, 4, 4 · 1, 8, 24, 32, 16 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, 0]nonnegativerational roots
• A084534 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 6, 7 · 1, 9, 24, 18 … Σ 1, 4, 14, 52, 194, 724, 2702, 10084, …
  [−1, 0]nonnegative
• A391533 → basis A013609 → basis A007318
  25, 20, 4 · 100, 130, 56, 8 · 400, 720, 484, 144, 16 … Σ 49, 294, 1764, 10584, 63504, 381024, …
  (−∞, −1]nonnegativerational roots
• A143491 → basis A013611 → reverse
  16, 28, 12 · 64, 192, 188, 60 … Σ 56, 504, 5040, 55440, 665280, 8648640, …
  (−∞, 0]nonnegativerational roots
• A290319 → basis A013611 → D+1
  1 · 6, 4 · 44, 64, 16 · 488, 944, 480, 64 … Σ 1, 10, 124, 1976, 38928, 916896, …
  (−∞, 0]nonnegative
• A013622 → (θ+1)²
  1 · 3, 20 · 9, 120, 225 · 27, 540, 2025, 2000 … Σ 1, 23, 354, 4592, 54016, 595968, …
  (−∞, 0]nonnegative
• A125092 → reverse → D+1
  1 · 5, 1 · 17, 10, 1 · 43, 51, 15, 1 … Σ 1, 6, 28, 110, 384, 1232, 3712, 10656, …
  (−∞, 0]nonnegative
• A391533 → reverse → x ↦ x/(x+1)
  1, 6, 9 · 1, 8, 21, 18 · 1, 10, 37, 60, 36 … Σ 16, 48, 144, 432, 1296, 3888, 11664, …
  [−1, 0]nonnegativerational roots
• A372868 → D
  1 · 6 · 24, 2 · 80, 20 … Σ 0, 0, 1, 6, 26, 100, 363, 1274, …
  (−∞, 0]nonnegative
• A091042 → Veronese 2 → x ↦ x+1
  1 · 1 · 6, 5 · 36, 35 … Σ 1, 1, 11, 71, 289, 1321, 6683, 32111, …
  (−∞, −1]nonnegative
• A144439 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 10, 10 · 1, 34, 96, 64 … Σ 1, 3, 21, 195, 2341, 34611, 609269, …
  [−1, 0]nonnegative
• A038291 → basis A013610 → basis A013609
  1 · 13, 6 · 169, 156, 36 · 2197, 3042, 1404, 216 … Σ 1, 19, 361, 6859, 130321, 2476099, …
  (−∞, −1]nonnegativerational roots
• A038320 → basis A013610 → D+1
  901, 1260, 324 · 20519, 48654, 34020, 5832 … Σ 2485, 109025, 4587625, 187578125, …
  (−∞, 0]nonnegativerational roots
• A038291 → basis A013610 → (θ+1)²
  1 · 10, 12 · 100, 240, 81 · 1000, 3600, 2430, 432 … Σ 1, 22, 421, 7462, 125905, 2051998, …
  (−∞, 0]nonnegative
• A159854 → basis A013609 → D
  4, 8 · 10, 32, 24 · 24, 104, 144, 64 … Σ 12, 66, 336, 1632, 7680, 35328, 159744, …
  (−∞, 0]nonnegativerational roots
• A130850 → (θ+1)²
  1 · 1, 4 · 2, 12, 9 · 6, 48, 63, 16 … Σ 1, 5, 23, 133, 979, 8761, 92007, …
  (−∞, 0]nonnegative
• A008517 → θ+1
  1 · 1, 4 · 1, 16, 18 · 1, 44, 174, 96 … Σ 1, 5, 35, 315, 3465, 45045, 675675, …
  (−∞, 0]nonnegative
• A111125 → D → D
  2 · 14, 6 · 54, 54, 12 · 154, 264, 132, 20 … Σ 0, 0, 2, 20, 120, 570, 2370, 9048, …
  (−∞, 0]nonnegative
• A038220 → basis A007318 → D+1
  1 · 7, 2 · 45, 28, 4 · 275, 270, 84, 8 … Σ 1, 9, 77, 637, 5145, 40817, 319333, …
  (−∞, 0]nonnegativerational roots
• A137320 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 2, 6 · 0, 4, 20, 24 … Σ 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  [−1, 0]nonnegativerational roots
• A178820 → basis A013610 → D
  12 · 120, 180 · 720, 2160, 1620 … Σ 0, 12, 300, 4500, 52500, 525000, …
  (−∞, 0]nonnegativerational roots
• A322944 → θ+1
  1 · 1, 2 · 2, 12, 3 · 6, 76, 45, 4 … Σ 1, 3, 17, 131, 1249, 14091, 182913, …
  (−∞, 0]nonnegative
• A178820 → basis A013611 → basis A007318
  1 · 24, 16 · 360, 480, 160 · 4320, 8640, 5760, 1280 … Σ 1, 40, 1000, 20000, 350000, 5600000, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A371080, shifted → basis A013609 → D+1
  1 · 3, 2 · 14, 18, 4 · 106, 174, 72, 8 … Σ 1, 5, 36, 360, 4644, 73548, 1382184, …
  (−∞, 0]nonnegative
• A038336 → x ↦ x+1 → D
  440, 200 · 14520, 13200, 3000 … Σ 640, 30720, 1310720, 52428800, …
  (−∞, 0]nonnegativerational roots
• A091044 → ⊙ C(n+2,k+1) → D
  1 · 5, 2 · 14, 28, 3 · 30, 168, 90, 4 … Σ 0, 1, 7, 45, 292, 1930, 12993, 88893, …
  (−∞, 0]nonnegative
• A131431 → D+1 → θ+1
  1 · 4, 8 · 0, 28, 21 · 0, 0, 90, 40 … Σ 1, 12, 49, 130, 273, 496, 817, 1254, …
  (−∞, 0]nonnegativerational roots
• A062145 → reverse
  1 · 4, 1 · 10, 10, 1 · 20, 45, 18, 1 … Σ 1, 5, 21, 84, 330, 1287, 5005, 19448, …
  (−∞, 0]nonnegative
• A075500 → (θ+1)²
  25, 60, 9 · 125, 700, 270, 16 … Σ 94, 1111, 14575, 209786, 3287579, …
  (−∞, 0]nonnegative
• A125662 → basis A013609 → D+1
  1 · 6, 2 · 31, 24, 4 · 148, 188, 72, 8 … Σ 1, 8, 59, 416, 2845, 19032, 125207, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729, shifted → basis A013609 → (θ+1)²
  1 · -1, -8 · 1, 16, 36 · -1, -24, -108, -128 … Σ 1, -9, 53, -261, 1161, -4833, 19197, …
  (−∞, 0]
• A159854 → basis A008277 → D+1
  4, 5, 1 · 12, 24, 10, 1 · 36, 108, 74, 16, 1 … Σ 10, 47, 235, 1248, 7023, 41767, 261792, …
  (−∞, 0]nonnegative
• A136124 → basis A013611 → reverse
  1 · 4, 3 · 16, 28, 12 · 64, 192, 188, 60 … Σ 1, 7, 56, 504, 5040, 55440, 665280, …
  (−∞, 0]nonnegativerational roots
• Veronese parent of A098172 → (θ+1)²
  0, 4 · 0, 0, 0, 16, 25 · 0, 0, 0, 0, 0, 36, 98, 64 … Σ 4, 41, 198, 728, 2320, 6768, 18592, …
  (−∞, 0]nonnegative
• A112007 → D
  1 · 8, 2 · 58, 44, 3 · 444, 656, 156, 4 … Σ 0, 1, 10, 105, 1260, 17325, 270270, …
  (−∞, 0]nonnegative
• A013610 → θ+1
  1 · 1, 6 · 1, 12, 27 · 1, 18, 81, 108 … Σ 1, 7, 40, 208, 1024, 4864, 22528, …
  (−∞, 0]nonnegativerational roots
• A081578 → D
  1 · 5, 2 · 9, 18, 3 · 13, 66, 39, 4 … Σ 0, 1, 7, 30, 122, 455, 1641, 5740, …
  (−∞, 0]nonnegative
• type_b_eulerian → γ (scale 1/4) → x ↦ x+1
  1 · 1 · 2, 1 · 6, 5 … Σ 1, 1, 3, 11, 57, 361, 2763, 24611, …
  (−∞, −1]nonnegative