Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A143495 → D+1 → D
  1 · 9, 2 · 61, 30, 3 · 369, 302, 66, 4 … Σ 0, 1, 11, 94, 741, 5691, 43642, 338535, …
  (−∞, 0]nonnegative
• A085478 → basis A013610 → D+1
  1 · 1 · 5, 3 · 20, 33, 9 … Σ 1, 1, 8, 62, 451, 3151, 21410, 142532, …
  (−∞, 0]nonnegative
• A094638 → basis A013611 → D
  4 · 28, 64 · 184, 928, 1152 … Σ 0, 4, 92, 2264, 64440, 2118960, …
  (−∞, 0]nonnegative
• A178822 → Veronese 2 (j=1) → D
  56 · 504 · 2520, 504 · 9240, 5544 … Σ 0, 0, 0, 56, 504, 3024, 14784, 63360, …
  (−∞, 0]nonnegative
• A038329 → x ↦ x+1 → D
  90, 18 · 2025, 810, 81 · 40500, 24300, 4860, 324 … Σ 108, 2916, 69984, 1574640, 34012224, …
  (−∞, 0]nonnegativerational roots
• A178821 → basis A013609 → D
  10 · 120, 120 · 840, 1680, 840 … Σ 0, 10, 240, 3360, 35840, 322560, …
  (−∞, 0]nonnegativerational roots
• A154929 → D
  1 · 4, 2 · 10, 12, 3 · 22, 42, 24, 4 … Σ 0, 1, 6, 25, 92, 316, 1040, 3324, …
  (−∞, 0]nonnegative
• image under θ+1 → θ+1 → D+1 → D+1
  1 · 8, 4 · 18, 36, 9 · 0, 96, 96, 16 … Σ 1, 12, 63, 208, 525, 1116, 2107, 3648, …
  (−∞, 0]nonnegative
• A059473 → basis A013609 → D
  4 · 36, 32 · 240, 448, 192 · 1424, 4064, 3648, 1024 … Σ 0, 4, 68, 880, 10160, 110128, 1146800, …
  (−∞, 0]nonnegative
• A099097 → basis A013609 → x ↦ x/(x+1)
  1 · 3, 9 · 10, 58, 84 · 33, 285, 819, 783 … Σ 1, 12, 152, 1920, 24256, 306432, …
  [−1, 0]nonnegative
• A136532 → D+1
  -24, -8, 4 · -125, 35, 35, -5 … Σ -28, -60, 264, 5096, 50496, 396000, …
  (−∞, ∞)
• A257617 → (θ+1)²
  1 · 2, 8 · 4, 144, 36 · 8, 1552, 3492, 128 … Σ 1, 10, 184, 5180, 185680, 8054640, …
  (−∞, 0]nonnegative
• A106800 → x ↦ x+1
  1 · 1 · 2, 1 · 5, 5, 1 … Σ 1, 1, 3, 11, 49, 257, 1539, 10299, …
  (−∞, −1]nonnegative
• basis-map coefficients of A026729, shifted → reverse
  1 · -1 · 1 · -1 … Σ 1, -1, 1, -1, 1, -1, 1, -1, …
  (−∞, 0]rational roots
• A272867 → basis A013609 → x ↦ x+1
  1 · 22, 20, 4 · 484, 880, 576, 160, 16 … Σ 1, 46, 2116, 97336, 4477456, 205962976, …
  (−∞, −1]nonnegative
• A085478 → D → x ↦ x+1
  1 · 5, 2 · 19, 16, 3 · 65, 84, 33, 4 … Σ 0, 0, 1, 7, 38, 186, 859, 3821, …
  (−∞, −1]nonnegative
• A208509 → x ↦ x+1
  1 · 3 · 6, 1 · 12, 5 … Σ 1, 3, 7, 17, 41, 99, 239, 577, …
  (−∞, −1]nonnegative
• basis-map coefficients of A026729 → D → D
  2 · 0, 6 · 0, 0, 12 · 0, 0, 0, 20 … Σ 0, 0, 2, 6, 12, 20, 30, 42, …
  (−∞, 0]nonnegativerational roots
• A132774 → (θ+1)²
  1 · 2, 12 · 0, 16, 45 · 0, 0, 54, 112 … Σ 1, 14, 61, 166, 353, 646, 1069, 1646, …
  (−∞, 0]nonnegativerational roots
• A013611 → basis A007318 → θ+1
  1 · 5, 8 · 25, 80, 48 · 125, 600, 720, 256 … Σ 1, 13, 153, 1701, 18225, 190269, …
  (−∞, 0]nonnegativerational roots
• A204420 → basis A013609 → θ+1
  1 · 1, 4 · 9, 48, 36 · 225, 1380, 1620, 480 … Σ 1, 5, 93, 3705, 255465, 27077085, …
  (−∞, 0]nonnegative
• A108838 → reverse
  2 · 2, 3 · 2, 8, 4 · 2, 15, 20, 5 … Σ 2, 5, 14, 42, 132, 429, 1430, 4862, …
  (−∞, 0]nonnegative
• A249139 → basis A007318 → basis A007318
  1 · 5, 1 · 9, 2 · 29, 11, 1 … Σ 1, 6, 11, 41, 96, 301, 781, 2286, …
  (−∞, −1]nonnegative
• A038322 → basis A013609 → D
  16 · 608, 512 · 17328, 29184, 12288 … Σ 0, 16, 1120, 58800, 2744000, 120050000, …
  (−∞, 0]nonnegativerational roots
• A257611 → (θ+1)²
  1 · 3, 12 · 9, 120, 81 · 27, 852, 1917, 432 … Σ 1, 15, 210, 3228, 55104, 1040640, …
  (−∞, 0]nonnegative
• A245962 → θ+1
  1 · 1 · 3, 4 · 4, 6 … Σ 1, 1, 7, 10, 29, 56, 131, 274, …
  (−∞, 0]nonnegative
• A143491 → θ+1
  6, 10, 3 · 24, 52, 27, 4 · 120, 308, 213, 56, 5 … Σ 19, 107, 702, 5274, 44712, 422568, …
  (−∞, 0]nonnegative
• A130123 → basis A013611 → θ+1
  1 · 2, 16 · 4, 64, 192 · 8, 192, 1152, 2048 … Σ 1, 18, 260, 3400, 42000, 500000, …
  (−∞, 0]nonnegativerational roots
• A105868 → (θ+1)²
  1 · 0, 4 · 0, 8, 9 · 0, 0, 54, 16 … Σ 1, 4, 17, 70, 271, 1016, 3699, 13182, …
  (−∞, 0]nonnegative
• A063967: Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
  1 · 1, 1 · 2, 3, 1 · 3, 7, 5, 1 … Σ 1, 2, 6, 16, 44, 120, 328, 896, …
  (−5, −1]nonnegative
• A069466 → basis A007318 → x ↦ x+1
  1 · 6, 2 · 78, 48, 6 · 1260, 1140, 300, 20 … Σ 1, 8, 132, 2720, 62020, 1497888, …
  (−∞, −1]nonnegative
• A038338 → θ+1
  144, 576, 432 · 1728, 10368, 15552, 6912 … Σ 1152, 34560, 995328, 27869184, …
  (−∞, 0]nonnegativerational roots
• A060920 → D
  1 · 5, 2 · 20, 18, 3 · 71, 102, 42, 4 … Σ 0, 1, 7, 41, 219, 1101, 5311, 24881, …
  (−∞, −1]nonnegative
• A008292 → γ (scale 1/4) → x ↦ x+1 → x ↦ x+1
  1 · 2 · 8, 2 · 40, 16 … Σ 1, 2, 10, 56, 424, 3728, 38800, 458624, …
  (−∞, −1]nonnegative
• A013610 → Veronese 2 → (θ+1)²
  1 · 1 · 1, 36 · 1, 108 … Σ 1, 1, 37, 109, 946, 4006, 23140, …
  (−∞, 0]nonnegative
• A225470 → basis A013611 → reverse
  1 · 4, 3 · 16, 36, 18 · 64, 288, 396, 162 … Σ 1, 7, 70, 910, 14560, 276640, 6086080, …
  (−∞, 0]nonnegativerational roots
• A116414 → basis A007318 → basis A007318
  1 · 6, 1 · 33, 12, 1 · 180, 102, 18, 1 … Σ 1, 7, 46, 301, 1969, 12880, 84253, …
  (−10, −2)nonnegative
• A257610: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 2.
  1 · 2, 2 · 4, 20, 4 · 8, 132, 132, 8 … Σ 1, 4, 28, 280, 3640, 58240, 1106560, …
  (−∞, 0)nonnegative
• A085881: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).
  1 · 1, 1 · 3, 6, 3 · 15, 45, 45, 15 … Σ 1, 2, 12, 120, 1680, 30240, 665280, …
  (−∞, 0)nonnegativerational roots
• A013610 → basis A013610 → x ↦ x+1
  1 · 13, 9 · 169, 234, 81 · 2197, 4563, 3159, 729 … Σ 1, 22, 484, 10648, 234256, 5153632, …
  (−∞, −1]nonnegativerational roots
• A144697 → x ↦ x+1
  1 · 4, 3 · 20, 28, 9 · 120, 240, 148, 27 … Σ 1, 7, 57, 535, 5721, 68887, 924057, …
  (−∞, −1]nonnegative
• A122848 → (θ+1)²
  1 · 0, 4 · 0, 4, 9 · 0, 0, 27, 16 … Σ 1, 4, 13, 43, 148, 526, 1954, 7498, …
  (−∞, 0]nonnegative
• A166972 → reverse
  1 · 1, 1 · 1, 10, 1 · 1, 41, 83, 1 … Σ 1, 2, 12, 126, 1902, 39852, 1092924, …
  (−∞, 0]nonnegative
• A114655 → x ↦ x+1 → x ↦ x/(x+1)
  2 · 6, 8 · 22, 60, 40 · 90, 372, 504, 224 … Σ 2, 14, 122, 1190, 12434, 136094, …
  [−1, 0]nonnegative
• A169803 → D+1
  1 · 2, 1 · 3, 2 · 4, 5, 1 … Σ 1, 3, 5, 10, 18, 33, 59, 105, …
  (−∞, 0]nonnegative
• A367270 → x ↦ x+1 → x ↦ x+1
  1 · 1 · 5, 2 · 25, 18, 3 … Σ 1, 1, 7, 46, 307, 2086, 14374, 100108, …
  (−∞, −1]nonnegative
• A257622 → x ↦ x/(x+1)
  1 · 4, 8 · 16, 88, 88 · 64, 744, 1848, 1232 … Σ 1, 12, 192, 3888, 95472, 2756592, …
  [−1, 0]nonnegative
• A116412 → D+1
  1 · 4, 1 · 12, 8, 1 · 33, 39, 12, 1 … Σ 1, 5, 21, 85, 332, 1266, 4741, 17507, …
  (−∞, 0]nonnegative
• A225473 → reverse
  1 · 4, 3 · 32, 40, 9 · 384, 672, 316, 27 … Σ 1, 7, 81, 1399, 32289, 931687, …
  (−∞, 0]nonnegative
• A132393 → Veronese 2 → D
  1 · 3 · 11, 2 · 50, 20 … Σ 0, 0, 1, 3, 13, 70, 447, 3297, …
  (−∞, 0]nonnegative