Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A225472 → (θ+1)²
  1 · 2, 12 · 4, 84, 162 · 8, 468, 2430, 2592 … Σ 1, 14, 250, 5498, 144202, 4398554, …
  (−∞, 0]nonnegative
• A059473 → γ
  1 · 2 · 4, 2 · 8, 8 … Σ 1, 2, 6, 16, 44, 120, 328, 896, …
  (−∞, 0]nonnegative
• A128813 → reverse
  1 · 1, 1 · 3, 4, 1 · 18, 27, 10, 1 … Σ 1, 2, 8, 56, 616, 9856, 216832, …
  (−∞, 0]nonnegativerational roots
• A299989 → x ↦ x/(x+1)
  0, 1 · 0, 3, 10, 8 · 0, 9, 60, 148, 160, 64 … Σ 1, 21, 441, 9261, 194481, 4084101, …
  [−1, 0]nonnegativerational roots
• A356546 → basis A013609
  1 · 4, 4 · 24, 48, 24 · 160, 480, 480, 160 … Σ 1, 8, 96, 1280, 17920, 258048, 3784704, …
  (−∞, 0]nonnegativerational roots
• A304236 → basis A007318 → D+1
  1 · 1 · 7, 3 · 13, 6 … Σ 1, 1, 10, 19, 100, 253, 1018, 2935, …
  (−∞, 0]nonnegative
• A125662 → basis A013610 → (θ+1)²
  1 · 4, 12 · 15, 96, 81 · 56, 552, 972, 432 … Σ 1, 16, 192, 2012, 19463, 178464, …
  (−∞, 0]nonnegative
• A204420 → basis A013610 → D+1
  1 · 4, 3 · 45, 90, 27 · 1260, 3465, 2430, 405 … Σ 1, 7, 162, 7560, 589680, 69128640, …
  (−∞, 0]nonnegative
• A116412 → basis A013609 → θ+1
  1 · 4, 4 · 13, 32, 12 · 43, 168, 144, 32 … Σ 1, 8, 57, 387, 2538, 16245, 102105, …
  (−∞, 0]nonnegative
• A375853 → D
  8 · 56, 40 · 216, 432, 120 · 616, 2376, 1848, 280 … Σ 0, 8, 96, 768, 5120, 30720, 172032, …
  (−∞, 0]nonnegative
• A124932 → reverse → x ↦ x/(x+1)
  1 · 3, 5 · 6, 21, 18 · 10, 54, 96, 56 … Σ 1, 8, 45, 216, 945, 3888, 15309, 58320, …
  [−1, 0]nonnegativerational roots
• delannoy → x ↦ x+1 → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 8, 18, 12 … Σ 1, 3, 11, 39, 139, 495, 1763, 6279, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729, shifted → (θ+1)² → reverse
  1 · -4 · 9 · -16 … Σ 1, -4, 9, -16, 25, -36, 49, -64, …
  (−∞, 0]rational roots
• A317497: Triangle T(n,k) = 3*T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.
  A102547 → x ↦ (1/27)x
  1 · 3 · 9 · 27, 1 … Σ 1, 3, 9, 28, 87, 270, 838, 2601, …
  (−∞, 0]nonnegative
• A386789 → reverse
  1 · 2 · 6, 3 · 20, 20, 4 … Σ 1, 2, 9, 44, 225, 1182, 6321, 34232, …
  (−∞, 0]nonnegative
• A286718 → basis A013610 → (θ+1)²
  1 · 2, 12 · 10, 84, 81 · 80, 792, 1215, 432 … Σ 1, 14, 175, 2519, 42520, 832316, …
  (−∞, 0]nonnegative
• A108838 → θ+1
  2 · 3, 4 · 4, 16, 6 · 5, 40, 45, 8 … Σ 2, 7, 26, 98, 372, 1419, 5434, 20878, …
  (−∞, 0]nonnegative
• A152828 → basis A007318
  1 · 1 · 1 · 2, 1 … Σ 1, 1, 1, 3, 3, 3, 9, 9, …
  (−∞, −1]nonnegativerational roots
• A144697 → D+1
  1 · 4, 3 · 11, 28, 9 · 26, 159, 148, 27 … Σ 1, 7, 48, 360, 3000, 27720, 282240, …
  (−∞, 0]nonnegative
• A038208 → Veronese 2 (j=1) → θ+1
  2 · 8 · 24, 16 · 64, 128 … Σ 0, 2, 8, 40, 192, 896, 4096, 18432, …
  (−∞, 0]nonnegative
• A163936 → reverse → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 3 · 0, 1, 10, 15 … Σ 1, 1, 4, 26, 236, 2752, 39208, 660032, …
  [−1, 0]nonnegative
• A081580 → γ → x ↦ x+1
  1 · 1 · 6, 5 · 11, 10 … Σ 1, 1, 11, 21, 131, 341, 1651, 5061, …
  (−∞, −1]nonnegative
• A038221 → (θ+1)²
  1 · 3, 12 · 9, 72, 81 · 27, 324, 729, 432 … Σ 1, 15, 162, 1512, 12960, 104976, …
  (−∞, 0]nonnegative
• A139548 → x ↦ x/(x+1)
  1 · 2, 8, 8 · 4, 32, 96, 128, 64 … Σ 1, 18, 324, 5832, 104976, 1889568, …
  [−1, 0]nonnegativerational roots
• A081579 → γ → D
  4 · 8 · 12, 32 · 16, 96 … Σ 0, 0, 4, 8, 44, 112, 404, 1112, …
  (−∞, 0]nonnegative
• A081577 → basis A013611 → θ+1
  1 · 2, 8 · 6, 48, 48 · 16, 192, 480, 256 … Σ 1, 10, 102, 944, 8396, 72408, 611080, …
  (−∞, 0]nonnegative
• A001263 → θ+1 → x ↦ x+1
  1 · 3, 2 · 10, 12, 3 · 35, 60, 30, 4 … Σ 1, 5, 25, 129, 681, 3653, 19825, …
  (−∞, −1]nonnegative
• A225466 → D
  3 · 21, 18 · 117, 270, 81 · 609, 2862, 2106, 324 … Σ 0, 3, 39, 468, 5901, 79863, 1161738, …
  (−∞, 0]nonnegative
• A156366 → D
  12, 18 · 33, 198, 81 · 78, 1188, 2106, 324 … Σ 30, 312, 3696, 49752, 753552, 12715392, …
  (−∞, 0]nonnegative
• A395972 → D+1
  20, 32, 8 · 80, 360, 288, 48 … Σ 60, 776, 12048, 216672, 4414912, …
  (−∞, 0]nonnegative
• A035342 → D+1
  1 · 4, 1 · 24, 11, 1 · 192, 123, 21, 1 … Σ 1, 5, 36, 337, 3875, 52776, 830095, …
  (−∞, 0]nonnegative
• A172431 → reverse
  1 · 2, 1 · 3, 4, 1 · 4, 10, 6, 1 … Σ 1, 3, 8, 21, 55, 144, 377, 987, …
  (−∞, 0]nonnegative
• A007318 → −D²+(−1+n)D+1+n → D+1
  1 · 4, 2 · 11, 14, 3 · 22, 54, 30, 4 … Σ 1, 6, 28, 110, 384, 1232, 3712, 10656, …
  (−∞, 0]nonnegative
• A119808 → x ↦ x+1
  1 · 3, 1 · 9, 5 · 27, 21, 2 … Σ 1, 4, 14, 50, 178, 634, 2258, 8042, …
  (−∞, −1]nonnegative
• A063967 → basis A007318 → (θ+1)²
  1 · 2, 4 · 6, 20, 9 · 16, 80, 72, 16 … Σ 1, 6, 35, 184, 920, 4428, 20727, 94932, …
  (−∞, 0]nonnegative
• A135838: Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
  1 · 2, 2 · 2, 4, 2 · 4, 12, 12, 4 … Σ 1, 4, 8, 32, 64, 256, 512, 2048, …
  (−∞, 0)nonnegativerational roots
• A256890 → x ↦ x+1
  20, 20, 4 · 120, 180, 76, 8 … Σ 44, 384, 3896, 45096, 586664, 8475144, …
  (−∞, −1]nonnegative
• A094638 → basis A013609 → D
  2 · 14, 16 · 92, 232, 144 · 652, 2632, 3504, 1536 … Σ 0, 2, 30, 468, 8324, 169584, 3920976, …
  (−∞, 0]nonnegative
• A028338 → basis A013611 → reverse
  1 · 4, 2 · 16, 24, 8 · 64, 192, 176, 48 … Σ 1, 6, 48, 480, 5760, 80640, 1290240, …
  (−∞, 0]nonnegativerational roots
• A039756 → reverse → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 24, 34, 12, 1 … Σ 1, 3, 13, 71, 457, 3355, 27509, 248127, …
  (−∞, −1]nonnegative
• A060920 → Veronese 2 → D
  1 · 9 · 51, 2 · 233, 40 … Σ 0, 0, 1, 9, 53, 273, 1325, 6217, …
  (−∞, 0]nonnegative
• A124848 → x ↦ x/(x+1)
  1 · 1, 5 · 1, 10, 19 · 1, 15, 57, 63 … Σ 1, 6, 30, 136, 579, 2358, 9288, 35640, …
  [−1, 0]nonnegative
• A103327 → D → (θ+1)²
  1 · 10, 8 · 35, 168, 27 · 84, 1008, 972, 64 … Σ 0, 1, 18, 230, 2128, 16416, 112640, …
  (−∞, 0]nonnegative
• A147716: Triangle of coefficients in expansion of (14 + x)^n.
  1 · 14, 1 · 196, 28, 1 · 2744, 588, 42, 1 … Σ 1, 15, 225, 3375, 50625, 759375, …
  (−∞, −1]nonnegativerational roots
• A028326 → basis A013611
  2 · 4, 8 · 8, 32, 32 · 16, 96, 192, 128 … Σ 2, 12, 72, 432, 2592, 15552, 93312, …
  [−1, 0]nonnegativerational roots
• A013610 → D → D+1
  3 · 24, 18 · 63, 216, 81 · 120, 756, 1296, 324 … Σ 0, 3, 42, 360, 2496, 15360, 87552, …
  (−∞, 0]nonnegativerational roots
• A086645 → basis A007318 → reverse
  1 · 1, 2 · 1, 8, 8 · 1, 18, 48, 32 … Σ 1, 3, 17, 99, 577, 3363, 19601, 114243, …
  (−∞, 0]nonnegative
• A179900 → D+1
  1 · 4, -1 · 14, -8, 1 · 42, -43, 12, -1 … Σ 1, 3, 7, 10, -23, -311, -2001, -10604, …
  (−∞, ∞)
• A081579 → x ↦ x+1 → D+1
  1 · 3, 1 · 16, 10, 1 · 60, 64, 17, 1 … Σ 1, 4, 27, 142, 785, 4152, 21863, …
  (−∞, 0]nonnegative
• A008517 → (θ+1)²
  1 · 1, 8 · 1, 32, 54 · 1, 88, 522, 384 … Σ 1, 9, 87, 995, 13265, 202545, 3489255, …
  (−∞, 0]nonnegative