Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A204420 → basis A013610 → reverse
  1 · 3, 1 · 27, 36, 9 · 405, 1215, 1035, 225 … Σ 1, 4, 72, 2880, 201600, 21772800, …
  (−∞, 0]nonnegativerational roots
• A013611 → Veronese 2 → θ+1
  1 · 1 · 1, 32 · 1, 96 … Σ 1, 1, 33, 97, 961, 4161, 28385, 142241, …
  (−∞, 0]nonnegative
• A166341 → x ↦ x+1
  12, 12, 1 · 48, 72, 26, 1 · 240, 480, 294, 54, 1 … Σ 25, 147, 1069, 9291, 94045, 1086987, …
  (−∞, −1]nonnegative
• A038324 → basis A013609 → θ+1
  1 · 21, 40 · 441, 1680, 1200 … Σ 1, 61, 3321, 169781, 8339441, …
  (−∞, 0]nonnegativerational roots
• A130123 → basis A013609
  1 · 2, 4 · 4, 16, 16 · 8, 48, 96, 64 … Σ 1, 6, 36, 216, 1296, 7776, 46656, …
  (−∞, 0]nonnegativerational roots
• A147716 → basis A013609 → D+1
  1 · 17, 2 · 285, 68, 4 · 4725, 1710, 204, 8 … Σ 1, 19, 357, 6647, 122825, 2255067, …
  (−∞, 0]nonnegativerational roots
• A272867 → basis A013611 → θ+1
  1 · 6, 48, 48 · 36, 576, 2304, 3072, 1280 … Σ 1, 102, 7268, 452824, 26280720, …
  (−∞, 0]nonnegative
• A038305 → basis A013611 → x ↦ x/(x+1)
  169, 650, 625 · 2197, 12675, 24375, 15625 … Σ 1444, 54872, 2085136, 79235168, …
  [−1, 0]nonnegativerational roots
• A063007: T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
  1 · 1, 2 · 1, 6, 6 · 1, 12, 30, 20 … Σ 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  (−1, 0)nonnegative
• A038220 → Veronese 2 (j=1) → x ↦ x+1
  2 · 12 · 62, 8 · 312, 96 … Σ 0, 2, 12, 70, 408, 2378, 13860, 80782, …
  (−∞, −1]nonnegative
• A075498 → D+1
  1 · 4, 1 · 18, 11, 1 · 90, 99, 21, 1 … Σ 1, 5, 30, 211, 1691, 15126, 148975, …
  (−∞, 0]nonnegative
• A178821 → Veronese 2 (j=1) → x ↦ x+1
  5 · 30 · 140, 35 · 560, 280 … Σ 0, 5, 30, 175, 840, 3654, 14700, 55770, …
  (−∞, −1]nonnegative
• A116412 → basis A007318 → D
  1 · 8, 2 · 42, 24, 3 · 190, 174, 48, 4 … Σ 0, 1, 10, 69, 416, 2336, 12552, 65452, …
  (−∞, 0]nonnegative
• A059473 → γ → D
  2 · 8 · 24, 8 · 64, 48 … Σ 0, 0, 2, 8, 32, 112, 376, 1216, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/4) → x ↦ x+1
  1 · 2 · 6, 2 · 24, 16 … Σ 1, 2, 8, 40, 256, 1952, 17408, 177280, …
  (−∞, −1]nonnegative
• A113953 → basis A013611 → D
  4 · 16, 32 · 44, 224, 192 · 120, 896, 1920, 1024 … Σ 0, 4, 48, 460, 3960, 32100, 250400, …
  (−∞, 0]nonnegative
• A372868 → x ↦ x+1
  1 · 2 · 5, 1 · 14, 6 … Σ 1, 2, 6, 20, 68, 232, 792, 2704, …
  (−∞, −1]nonnegative
• A180013: Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have exactly k cycles; n>=1, 1<=k<=n.
  A028338 → x ↦ 2x−1
  1 · 0, 2 · 0, 3, 3 · 0, 8, 12, 4 … Σ 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, 1/2]nonnegativerational roots
• A102662 → reverse → D
  1 · 5, 2 · 11, 14, 3 · 17, 46, 27, 4 … Σ 0, 1, 7, 28, 94, 291, 857, 2442, …
  (−∞, 0]nonnegative
• A152842 → θ+1
  1 · 1, 6 · 1, 8, 9 · 1, 14, 45, 36 … Σ 1, 7, 18, 96, 224, 1088, 2432, 11264, …
  (−∞, 0]nonnegative
• A137597 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 15, 67, 98, 47 … Σ 1, 5, 31, 227, 1897, 17693, 181351, …
  [−1, 0]nonnegative
• A172431 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 6, 8 · 1, 9, 25, 21 … Σ 1, 4, 15, 56, 209, 780, 2911, 10864, …
  [−1, 0]nonnegative
• A008292 → γ (shift +1) → x ↦ x+1 → x ↦ x+1
  1 · 1 · 5, 2 · 17, 8 … Σ 1, 1, 7, 25, 211, 1381, 14167, 135505, …
  (−∞, −1]nonnegative
• A269939 → D+1
  1 · 1, 1 · 1, 7, 3 · 1, 21, 55, 15 … Σ 1, 2, 11, 92, 1022, 14100, 231996, …
  (−∞, 0]nonnegative
• A008278 → x ↦ x+1 (shift +1) → reverse → D+1
  1 · 3, 2 · 6, 15, 5 · 11, 56, 68, 15 … Σ 1, 5, 26, 150, 963, 6807, 52448, …
  (−∞, 0]nonnegative
• A089231 → x ↦ x+1 → D
  2 · 18, 12 · 156, 216, 72 · 1460, 3120, 2160, 480 … Σ 0, 2, 30, 444, 7220, 130830, 2633442, …
  (−∞, 0]nonnegative
• A272867 → x ↦ x+1 → reverse
  1 · 1, 6, 6 · 1, 12, 48, 72, 36 … Σ 1, 13, 169, 2197, 28561, 371293, …
  (−∞, 0]nonnegative
• basis-map coefficients of A371080, shifted → basis A013611 → D
  4 · 20, 32 · 156, 384, 192 · 1672, 5088, 4224, 1024 … Σ 0, 4, 52, 732, 12008, 228776, 4994400, …
  (−∞, 0]nonnegative
• A272866 → D+1
  1 · 4, 5, 1 · 7, 28, 29, 10, 1 … Σ 1, 10, 75, 500, 3125, 18750, 109375, …
  (−∞, 0]nonnegative
• A038327 → x ↦ x+1 → D
  1 · 26, 2 · 507, 78, 3 · 8788, 2028, 156, 4 … Σ 0, 1, 28, 588, 10976, 192080, 3226944, …
  (−∞, 0]nonnegativerational roots
• A161198 → basis A013609 → (θ+1)²
  1 · 3, 16 · 15, 128, 144 · 105, 1136, 2160, 1024 … Σ 1, 19, 287, 4425, 73489, 1332043, …
  (−∞, 0]nonnegative
• A191935 → reverse
  1 · 2, 1 · 4, 8, 1 · 8, 52, 20, 1 … Σ 1, 3, 13, 81, 669, 6955, 88505, …
  (−∞, 0]nonnegative
• A156992 → basis A013611
  1 · 4, 8 · 24, 96, 96 · 192, 1152, 2304, 1536 … Σ 1, 12, 216, 5184, 155520, 5598720, …
  [−1, 0]nonnegativerational roots
• A156992 → γ
  1 · 2 · 6 · 24 … Σ 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, −1][−1, 0][−3−2√2, −3+2√2]+2
• A079510 → basis A007318
  1 · 2, 2 · 9, 15, 6 · 44, 112, 92, 24 … Σ 1, 4, 30, 272, 3020, 39504, 595336, …
  (−∞, −1]nonnegative
• A121574 → x ↦ x+1 → basis A007318
  1 · 4, 1 · 18, 9, 1 · 80, 60, 14, 1 … Σ 1, 5, 28, 155, 859, 4760, 26377, …
  (−∞, −1]nonnegative
• basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
  1 · 0, -1 · 0, 0, 1 · 0, 0, 0, -1 … Σ 1, -1, 1, -1, 1, -1, 1, -1, …
  [−1, 0]rational roots
• A034870 → D → θ+1
  2, 4 · 4, 24, 36, 16 · 6, 60, 180, 240, 150, 36 … Σ 0, 6, 80, 672, 4608, 28160, 159744, …
  (−∞, 0]nonnegativerational roots
• A159854 → Veronese 2 (j=1) → D
  1 · 2 · 3, 2 · 4, 8 … Σ 0, 1, 2, 5, 12, 28, 64, 144, …
  (−∞, 0]nonnegative
• A178822 → basis A013611 → (θ+1)²
  1 · 12, 96 · 84, 1344, 3024 … Σ 1, 108, 4452, 116928, 2340576, …
  (−∞, 0]nonnegative
• A038279 → basis A013609 → D+1
  1 · 11, 2 · 117, 44, 4 · 1215, 702, 132, 8 … Σ 1, 13, 165, 2057, 25289, 307461, …
  (−∞, 0]nonnegativerational roots
• A038330 → basis A007318 → (θ+1)²
  256, 512, 144 · 4096, 12288, 6912, 1024 … Σ 912, 24320, 620800, 15360000, …
  (−∞, 0]nonnegative
• A094305 → D
  3 · 12, 12 · 30, 60, 30 · 60, 180, 180, 60 … Σ 0, 3, 24, 120, 480, 1680, 5376, 16128, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A297191, shifted → basis A013611 → x ↦ x+1
  1 · 25, 40, 16 · 625, 2000, 2400, 1280, 256 … Σ 1, 81, 6561, 531441, 43046721, …
  (−∞, −1]nonnegativerational roots
• A257621 → x ↦ x+1
  1 · 6, 3 · 60, 60, 9 · 840, 1260, 474, 27 … Σ 1, 9, 129, 2601, 67521, 2142729, …
  (−∞, −1]nonnegative
• A145596 → D → (θ+1)²
  2 · 8, 24 · 20, 160, 108 · 40, 600, 1080, 320 … Σ 0, 2, 32, 288, 2040, 12650, 72072, …
  (−∞, 0]nonnegative
• A079638 → D
  1 · 9, 2 · 83, 36, 3 · 870, 550, 90, 4 … Σ 0, 1, 11, 122, 1514, 21264, 336216, …
  (−∞, 0]nonnegative
• A123519 → basis A007318
  11, 14, 4 · 41, 76, 44, 8 · 153, 372, 324, 120, 16 … Σ 29, 169, 985, 5741, 33461, 195025, …
  (−3, −1)nonnegative
• A038208 → Veronese 2 → θ+1
  1 · 2 · 4, 8 · 8, 48 … Σ 1, 2, 12, 56, 256, 1152, 5120, 22528, …
  (−∞, 0]nonnegative
• basis-map coefficients of A030528, shifted → basis A013610 → x ↦ x+1
  1 · -4, -3 · 16, 24, 9 · -64, -144, -108, -27 … Σ 1, -7, 49, -343, 2401, -16807, 117649, …
  (−∞, −1]rational roots