Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A013622 → basis A013610 → basis A007318
  1 · 23, 15 · 529, 690, 225 … Σ 1, 38, 1444, 54872, 2085136, 79235168, …
  (−∞, −1]nonnegativerational roots
• A081578 → basis A013609 → D+1
  1 · 4, 2 · 21, 22, 4 · 80, 156, 72, 8 … Σ 1, 6, 47, 316, 2109, 13634, 86731, …
  (−∞, 0]nonnegative
• A093375 → Veronese 2 → θ+1
  1 · 2 · 3, 2 · 4, 12 … Σ 1, 2, 5, 16, 44, 116, 296, 736, …
  (−∞, 0]nonnegative
• A290319 → basis A013611 → reverse
  1 · 4, 2 · 16, 32, 12 · 64, 288, 368, 120 … Σ 1, 6, 60, 840, 15120, 332640, 8648640, …
  (−∞, 0]nonnegativerational roots
• A098158 → reverse
  1 · 1 · 1, 1 · 1, 3 … Σ 1, 1, 2, 4, 8, 16, 32, 64, …
  (−∞, 0]nonnegative
• A013613 → (θ+1)²
  1 · 1, 24 · 1, 48, 324 · 1, 72, 972, 3456 … Σ 1, 25, 373, 4501, 48265, 479857, …
  (−∞, 0]nonnegative
• A249247 → (θ+1)²
  1 · 1, 4 · 2, 12 · 6, 40, 9 … Σ 1, 5, 14, 55, 255, 1405, 9016, 66281, …
  (−∞, 0]nonnegative
• A089503 → (θ+1)²
  1 · 1, 16 · 1, 48, 270 · 1, 96, 1512, 5376 … Σ 1, 17, 319, 6985, 177101, 5129761, …
  (−∞, 0]nonnegative
• A027466 → basis A013611 → D+1
  1 · 12, 4 · 128, 96, 16 · 1280, 1536, 576, 64 … Σ 1, 16, 240, 3456, 48384, 663552, …
  (−∞, 0]nonnegativerational roots
• A013620 → θ+1
  1 · 2, 6 · 4, 24, 27 · 8, 72, 162, 108 … Σ 1, 8, 55, 350, 2125, 12500, 71875, …
  (−∞, 0]nonnegativerational roots
• A272867 → basis A013610 → D+1
  1 · 24, 36, 9 · 252, 1080, 1404, 648, 81 … Σ 1, 69, 3465, 153549, 6360849, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A013610
  1 · 1, 6, 9 · 1, 12, 54, 108, 81 … Σ 1, 16, 256, 4096, 65536, 1048576, …
  [−1, 0]nonnegativerational roots
• A113953 → x ↦ x−15
  1 · -15, 1 · 195, -28, 1 · -2475, 555, -41, 1 … Σ 1, -14, 168, -1960, 22736, -263424, …
  (7, 15]
• basis-map coefficients of A026729, shifted → D → (θ+1)²
  -1 · 0, 8 · 0, 0, -27 · 0, 0, 0, 64 … Σ 0, -1, 8, -27, 64, -125, 216, -343, …
  (−∞, 0]rational roots
• A106246 → D → θ+1
  1 · 4, 4 · 3, 24, 9 · 0, 24, 72, 16 … Σ 0, 1, 8, 36, 112, 275, 576, 1078, …
  (−∞, 0]nonnegative
• A059374: Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.
  A089231 → x ↦ x+1
  1 · 3, 2 · 13, 18, 6 · 73, 156, 108, 24 … Σ 1, 5, 37, 361, 4361, 62701, 1044205, …
  (−∞, −1]nonnegative
• A089503 → D
  4 · 12, 60 · 24, 336, 1008 · 40, 1080, 8640, 20160 … Σ 0, 4, 72, 1368, 29920, 755100, …
  (−∞, 0]nonnegative
• A178821 → basis A013610 → reverse
  1 · 15, 10 · 135, 180, 60 · 945, 1890, 1260, 280 … Σ 1, 25, 375, 4375, 43750, 393750, …
  (−∞, 0]nonnegativerational roots
• A038301 → x ↦ x+1 → D+1
  840, 682, 121 · 21200, 27720, 11253, 1331 … Σ 1643, 61504, 2234325, 79422806, …
  (−∞, 0]nonnegativerational roots
• image under θ+1 → (θ+1)²
  1 · 0, 8 · 0, 0, 27 · 0, 0, 0, 64 … Σ 1, 8, 27, 64, 125, 216, 343, 512, …
  [−1, 0]nonnegativerational roots
• A079621 → θ+1
  1 · 4, 2 · 24, 24, 3 · 192, 288, 72, 4 … Σ 1, 6, 51, 556, 7365, 114546, 2042551, …
  (−∞, 0]nonnegative
• A023531 → Veronese 2 → D
  1 · 0, 2 · 0, 0, 3 · 0, 0, 0, 4 … Σ 0, 0, 1, 0, 2, 0, 3, 0, …
  (−∞, 0]nonnegativerational roots
• A081579 → θ+1
  1 · 1, 2 · 1, 12, 3 · 1, 22, 33, 4 … Σ 1, 3, 16, 60, 240, 896, 3328, 12096, …
  (−∞, 0]nonnegative
• A376827 → basis A007318
  1 · 2, 1 · 12, 12, 3 · 56, 84, 42, 7 … Σ 1, 3, 27, 189, 1539, 12393, 102789, …
  (−∞, −1]nonnegativerational roots
• A367270 → basis A013609 → (θ+1)²
  1 · 1 · 3, 16 · 10, 96, 108 … Σ 1, 1, 19, 214, 2107, 19606, 176494, …
  (−∞, 0]nonnegative
• A027466 → basis A013610 → basis A013609
  1 · 11, 6 · 121, 132, 36 · 1331, 2178, 1188, 216 … Σ 1, 17, 289, 4913, 83521, 1419857, …
  (−∞, −1]nonnegativerational roots
• A225470 → D
  1 · 7, 2 · 66, 30, 3 · 806, 462, 78, 4 … Σ 0, 1, 9, 99, 1350, 22194, 428652, …
  (−∞, 0]nonnegative
• A008277 → θ+2 (shift +1) → x ↦ x+1
  2 · 5, 3 · 15, 17, 4 · 52, 84, 39, 5 … Σ 2, 8, 36, 180, 988, 5892, 37852, …
  (−∞, −1]nonnegative
• A099097 → x ↦ x−9
  1 · -27, 3 · 720, -161, 9 · -19197, 6453, -723, 27 … Σ 1, -24, 568, -13440, 318016, -7524864, …
  (77/9, 9]
• A102547 → x ↦ (1/27)x → basis A013609 → x ↦ x+1
  1 · 3 · 9 · 30, 2 … Σ 1, 3, 9, 32, 111, 378, 1294, 4437, …
  (−∞, −1]nonnegative
• A105793: Expansion of e.g.f. (1 + y)^(1 + x).
  1 · 1, 1 · 0, 1, 1 · 0, -1, 0, 1 … Σ 1, 2, 2, 0, 0, 0, 0, 0, …
  [−3/2, ∞)rational roots
• A207606 → x ↦ x+1 → D+1
  24, 15, 2 · 78, 78, 23, 2 · 244, 341, 162, 31, 2 … Σ 41, 181, 780, 3301, 13775, 56841, …
  (−∞, 0]nonnegative
• A038333 → D
  168, 98 · 3024, 3528, 1029 … Σ 266, 7581, 192052, 4561235, 103996158, …
  (−∞, 0]nonnegativerational roots
• A122542 → x ↦ x+1
  1 · 1, 1 · 3, 4, 1 · 7, 13, 7, 1 … Σ 1, 2, 8, 28, 100, 356, 1268, 4516, …
  (−∞, −1]nonnegative
• A038284 → basis A013609 → D+1
  532, 624, 144 · 9800, 19152, 11232, 1728 … Σ 1300, 41912, 1300624, 39299936, …
  (−∞, 0]nonnegativerational roots
• A272867 → basis A013609 → reverse
  1 · 4, 12, 6 · 16, 96, 192, 144, 36 … Σ 1, 22, 484, 10648, 234256, 5153632, …
  (−∞, 0]nonnegative
• A106246 → D → x ↦ x+1
  1 · 6, 2 · 18, 18, 3 · 40, 72, 36, 4 … Σ 0, 1, 8, 39, 152, 520, 1632, 4816, …
  (−∞, −1]nonnegative
• A039756 → (θ+1)²
  1 · 1, 4 · 1, 16, 9 · 1, 36, 117, 16 … Σ 1, 5, 26, 170, 1252, 9972, 85448, …
  (−∞, 0]nonnegative
• A085881 → basis A007318 → reverse
  1 · 1, 2 · 3, 12, 12 · 15, 90, 180, 120 … Σ 1, 3, 27, 405, 8505, 229635, 7577955, …
  (−∞, 0]nonnegativerational roots
• A113413: A Riordan array of coordination sequences.
  1 · 2, 1 · 2, 4, 1 · 2, 8, 6, 1 … Σ 1, 3, 7, 17, 41, 99, 239, 577, …
  (−∞, 0)nonnegative
• A094816 → basis A013610 → θ+1
  1 · 2, 6 · 5, 30, 27 · 16, 138, 243, 108 … Σ 1, 8, 62, 505, 4433, 42248, 437338, …
  (−∞, 0]nonnegative
• A137320 → basis A013609 → D+1
  1 · 6, 4 · 26, 52, 16 · 128, 392, 336, 64 … Σ 1, 10, 94, 920, 9624, 108336, 1312656, …
  (−∞, 0]nonnegative
• A049020 → x ↦ x+1
  1 · 2, 1 · 6, 5, 1 · 22, 25, 9, 1 … Σ 1, 3, 12, 57, 309, 1866, 12351, 88563, …
  (−∞, −1]nonnegative
• A133081 → reverse (shift +1) → D
  1, 2 · 0, 0, 0, 4 · 0, 0, 0, 0, 5, 6 … Σ 0, 3, 4, 11, 8, 19, 12, 27, …
  (−∞, 0]nonnegativerational roots
• A013609 → D
  2 · 4, 8 · 6, 24, 24 · 8, 48, 96, 64 … Σ 0, 2, 12, 54, 216, 810, 2916, 10206, …
  (−∞, 0]nonnegativerational roots
• A249139 → θ+1
  1 · 3, 2 · 5, 4 · 11, 14, 3 … Σ 1, 5, 9, 28, 62, 165, 391, 983, …
  (−∞, 0]nonnegative
• A112493 → D
  1 · 4, 6 · 11, 50, 45 · 26, 260, 630, 420 … Σ 0, 1, 10, 106, 1336, 19944, 346368, …
  (−∞, 0]nonnegative
• A059473 → basis A013610 → x ↦ x/(x+1)
  1 · 4, 10 · 18, 90, 108 · 80, 600, 1464, 1160 … Σ 1, 14, 216, 3304, 50576, 774144, …
  [−1, 0]nonnegative
• A090582 → x ↦ x+1
  1 · 3, 1 · 13, 8, 1 · 75, 67, 17, 1 … Σ 1, 4, 22, 160, 1456, 15904, 202672, …
  (−∞, −1]nonnegative
• A191935 → D
  2 · 8, 8 · 20, 104, 24 · 40, 584, 960, 64 … Σ 0, 2, 16, 148, 1648, 21630, 333408, …
  (−∞, 0]nonnegative