Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A099093 → basis A013609 → D
  6 · 42, 72 · 234, 792, 648 · 1170, 5904, 9720, 5184 … Σ 0, 6, 114, 1674, 21978, 271188, …
  (−∞, 0]nonnegative
• A113953 → basis A013610 → reverse
  1 · 3, 1 · 9, 12, 3 · 27, 63, 33, 5 … Σ 1, 4, 24, 128, 704, 3840, 20992, …
  (−∞, 0]nonnegative
• A103633 → θ+1
  1 · 0, 2 · 0, 2, 3 · 0, 0, 3, 4 … Σ 1, 2, 5, 7, 16, 20, 44, 52, …
  (−∞, 0]nonnegativerational roots
• A114656 → basis A013609 → (θ+1)²
  1 · 3, 8 · 11, 64, 36 · 45, 408, 540, 128 … Σ 1, 11, 111, 1121, 11357, 115255, …
  (−∞, 0]nonnegative
• A062264: Coefficient triangle of certain polynomials N(4; m,x).
  1 · 1, 5 · 1, 12, 15 · 1, 21, 63, 35 … Σ 1, 6, 28, 120, 495, 2002, 8008, 31824, …
  (−∞, 0)nonnegative
• A059344 → D+1
  1 · 1 · 3, 2 · 7, 6 … Σ 1, 1, 5, 13, 61, 221, 1081, 4705, …
  (−∞, 0]nonnegative
• A166961 → D
  1 · 9, 2 · 59, 84, 3 · 361, 1850, 462, 4 … Σ 0, 1, 11, 146, 2677, 63823, 1917362, …
  (−∞, 0]nonnegative
• A038221 → x ↦ x+1 → D+1
  1 · 9, 3 · 72, 54, 9 · 540, 648, 243, 27 … Σ 1, 12, 135, 1458, 15309, 157464, …
  (−∞, 0]nonnegativerational roots
• A038321 → basis A013611 → x ↦ x/(x+1)
  324, 1656, 2116 · 5832, 44712, 114264, 97336 … Σ 4096, 262144, 16777216, 1073741824, …
  [−1, 0]nonnegativerational roots
• A143491 → basis A013610 → D
  21, 18 · 141, 216, 81 · 1026, 2142, 1458, 324 … Σ 39, 438, 4950, 58572, 735012, 9818064, …
  (−∞, 0]nonnegative
• A120909 → basis A013610 → x ↦ x/(x+1)
  3 · 9, 27 · 27, 162, 243 · 81, 729, 2187, 2187 … Σ 3, 36, 432, 5184, 62208, 746496, …
  [−1, 0]nonnegativerational roots
• A038279 → basis A013609 → x ↦ x/(x+1)
  1 · 9, 11 · 81, 198, 121 · 729, 2673, 3267, 1331 … Σ 1, 20, 400, 8000, 160000, 3200000, …
  [−1, 0]nonnegativerational roots
• A166973 → D+1
  1 · 2, 1 · 8, 9, 1 · 44, 79, 21, 1 … Σ 1, 3, 18, 145, 1383, 15262, 191431, …
  (−∞, 0]nonnegative
• A075504 → θ+1
  81, 54, 3 · 729, 1134, 162, 4 … Σ 138, 2029, 34871, 684402, 14974639, …
  (−∞, 0]nonnegative
• A208509 → θ+1
  1 · 3 · 5, 2 · 7, 10 … Σ 1, 3, 7, 17, 40, 92, 208, 464, …
  (−∞, 0]nonnegative
• derivative of Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.
  2, 2 · 6, 20, 18, 4 · 14, 94, 204, 188, 70, 6 … Σ 0, 4, 48, 576, 7680, 115200, 1935360, …
  (−∞, 0]nonnegative
• A038220 → Veronese 2 → D+1
  1 · 3 · 13, 4 · 63, 36 … Σ 1, 3, 17, 99, 561, 3123, 17185, 93747, …
  (−∞, 0]nonnegative
• A027466 → basis A013609 → D+1
  1 · 10, 2 · 96, 40, 4 · 896, 576, 120, 8 … Σ 1, 12, 140, 1600, 18000, 200000, …
  (−∞, 0]nonnegativerational roots
• A127773 → θ+1 → x ↦ x+1
  1 · 6, 6 · 18, 36, 18 · 40, 120, 120, 40 … Σ 1, 12, 72, 320, 1200, 4032, 12544, …
  (−∞, −1]nonnegativerational roots
• A008955 → basis A007318 → D
  1 · 13, 8 · 220, 314, 108 · 5340, 12378, 9372, 2304 … Σ 0, 1, 21, 642, 29394, 1922319, …
  (−∞, 0]nonnegative
• A008288 → basis A007318 → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 8, 18, 12 … Σ 1, 3, 11, 39, 139, 495, 1763, 6279, …
  (−∞, 0]nonnegative
• A039756 → x ↦ x+1 → D
  1 · 6, 2 · 38, 32, 3 · 256, 368, 132, 4 … Σ 0, 1, 8, 73, 760, 8617, 105840, …
  (−∞, 0]nonnegative
• A038319 → basis A013610 → basis A007318
  961, 930, 225 · 29791, 43245, 20925, 3375 … Σ 2116, 97336, 4477456, 205962976, …
  (−∞, −1]nonnegativerational roots
• A008459 → basis A013609 → D+1
  1 · 4, 2 · 18, 20, 4 · 80, 156, 72, 8 … Σ 1, 6, 42, 316, 2430, 18804, 145684, …
  (−∞, 0]nonnegative
• A112486 → reverse
  1 · 1, 1 · 3, 5, 2 · 15, 35, 26, 6 … Σ 1, 2, 10, 82, 938, 13778, 247210, …
  (−∞, −1]nonnegative
• A051338: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 5.
  1 · -6, 1 · 42, -13, 1 · -336, 146, -21, 1 … Σ 1, -5, 30, -210, 1680, -15120, 151200, …
  [11/2, ∞)rational roots
• A166972 → x ↦ x+1
  1 · 2, 1 · 12, 12, 1 · 126, 168, 44, 1 … Σ 1, 3, 25, 339, 6769, 185283, 6572041, …
  (−∞, −1]nonnegative
• A048993 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 1, 5, 5 … Σ 1, 1, 3, 11, 49, 257, 1539, 10299, …
  [−1, 0]nonnegative
• A091044 → x ↦ x/(x+1)
  1 · 2, 4 · 3, 16, 16 · 4, 40, 96, 64 … Σ 1, 6, 35, 204, 1189, 6930, 40391, …
  [−1, 0]nonnegative
• A111125 → basis A013611 → reverse
  1 · 4, 4 · 16, 28, 11 · 64, 160, 124, 29 … Σ 1, 8, 55, 377, 2584, 17711, 121393, …
  (−∞, 0]nonnegative
• A063967 → x ↦ x−3
  1 · -2, 1 · 2, -3, 1 · 0, 4, -4, 1 … Σ 1, -1, 0, 1, -1, 0, 1, -1, …
  (−2, 2]
• Veronese parent of A098172 → D
  0, 2 · 0, 0, 0, 4, 5 · 0, 0, 0, 0, 0, 6, 14, 8 … Σ 2, 9, 28, 76, 192, 464, 1088, 2496, …
  (−∞, 0]nonnegativerational roots
• A103406 → Veronese 2 → x ↦ x+1
  1 · 1 · 4, 3 · 10, 9 … Σ 1, 1, 7, 19, 57, 161, 447, 1219, …
  (−∞, −1]nonnegative
• A034870 → D → D
  2 · 12, 24, 12 · 30, 120, 180, 120, 30 … Σ 0, 2, 48, 480, 3584, 23040, 135168, …
  (−∞, 0]nonnegativerational roots
• A178820 → basis A013610
  1 · 8, 12 · 40, 120, 90 · 160, 720, 1080, 540 … Σ 1, 20, 250, 2500, 21875, 175000, …
  [−1, 0]nonnegativerational roots
• A038329 → x ↦ x+1 → x ↦ x+1
  324, 108, 9 · 5832, 2916, 486, 27 … Σ 441, 9261, 194481, 4084101, 85766121, …
  (−∞, −1]nonnegativerational roots
• A128813 → basis A013609 → reverse
  1 · 2, 2 · 12, 20, 8 · 144, 324, 236, 56 … Σ 1, 4, 40, 760, 23560, 1083760, …
  (−∞, 0]nonnegativerational roots
• A048993 → D+1 → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 1, 28, 54, 16 … Σ 1, 5, 22, 99, 471, 2386, 12867, 73681, …
  (−∞, 0]nonnegative
• A062145 → reverse → θ+1
  1 · 4, 2 · 10, 20, 3 · 20, 90, 54, 4 … Σ 1, 6, 33, 168, 810, 3762, 17017, 75504, …
  (−∞, 0]nonnegative
• A038271 → basis A007318 → D+1
  1 · 17, 5 · 264, 170, 25 · 3888, 3960, 1275, 125 … Σ 1, 22, 459, 9248, 181781, 3507882, …
  (−∞, 0]nonnegativerational roots
• A035607 → basis A013611 → θ+1
  1 · 1 · 3, 16 · 7, 64, 96 … Σ 1, 1, 19, 167, 1425, 11593, 91571, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A013611 → D+1
  1 · 9, 40, 16 · 17, 208, 864, 1280, 256 … Σ 1, 65, 2625, 90625, 2890625, 87890625, …
  (−∞, 0]nonnegativerational roots
• A066325 → D
  1 · 0, 2 · -3, 0, 3 · 0, -12, 0, 4 … Σ 0, 1, 2, 0, -8, -10, 36, 112, …
  (−∞, ∞)
• row-factor cofactor of A101842 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 14, 36, 24 … Σ 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  [−1, 0]nonnegative
• A133087 → reverse
  1 · 1, 2 · 1, 2, 1 · 1, 4, 5, 2 … Σ 1, 3, 4, 12, 16, 48, 64, 192, …
  (−∞, 0]nonnegativerational roots
• A094638 → θ+1 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 8, 13 · 1, 15, 60, 70 … Σ 1, 4, 22, 146, 1116, 9612, 92016, …
  [−1, 0]nonnegative
• A130534 → D²∘x (shift +1) → x ↦ x+1
  1 · 6, 3 · 35, 30, 6 · 225, 255, 90, 10 … Σ 1, 9, 71, 580, 5104, 48860, 509004, …
  (−∞, −1]nonnegative
• A145596 → D → D
  6 · 40, 24 · 150, 240, 60 · 420, 1260, 840, 120 … Σ 0, 0, 6, 64, 450, 2640, 14014, 69888, …
  (−∞, 0]nonnegative
• A008288 → basis A013610 → θ+1
  1 · 2, 6 · 5, 30, 27 · 12, 108, 216, 108 … Σ 1, 8, 62, 444, 3050, 20344, 132838, …
  (−∞, 0]nonnegative
• A059364 → basis A013610 → D
  3 · 33, 36 · 366, 846, 486 · 4554, 16398, 19602, 7776 … Σ 0, 3, 69, 1698, 48330, 1589220, …
  (−∞, 0]nonnegative