Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A147716 → basis A013610 → θ+1
  1 · 15, 6 · 225, 180, 27 · 3375, 4050, 1215, 108 … Σ 1, 21, 432, 8748, 174960, 3464208, …
  (−∞, 0]nonnegativerational roots
• A207606 → x ↦ x+1 → x ↦ x/(x+1)
  13, 37, 26 · 34, 146, 207, 97 … Σ 76, 484, 3084, 19652, 125228, 797988, …
  [−1, −1/2)nonnegative
• A088996 → basis A013610 → D+1
  1 · 4, 3 · 18, 51, 18 · 117, 552, 711, 162 … Σ 1, 7, 87, 1542, 35346, 991944, …
  (−∞, 0]nonnegative
• A156289 → (θ+1)²
  1 · 1, 12 · 1, 60, 135 · 1, 252, 1890, 1680 … Σ 1, 13, 196, 3823, 94891, 2921128, …
  (−∞, 0]nonnegative
• A249248 → reverse → D+1
  1 · 4, 1 · 10, 9, 1 · 25, 46, 15, 1 … Σ 1, 5, 20, 87, 367, 1615, 7206, 33149, …
  (−∞, 0]nonnegative
• A167884: Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 8.
  1 · 1, 1 · 1, 18, 1 · 1, 179, 179, 1 … Σ 1, 2, 20, 360, 9360, 318240, 13366080, …
  (−∞, 0)nonnegative
• Veronese parent of A098172 → x ↦ x/(x+1)
  0, 1 · 0, 0, 0, 1, 2 · 0, 0, 0, 0, 0, 1, 4, 4 … Σ 1, 3, 9, 27, 81, 243, 729, 2187, …
  [−1, 0]nonnegativerational roots
• A030528 → D+1
  1 · 2, 1 · 2, 4, 1 · 1, 7, 6, 1 … Σ 1, 3, 7, 15, 30, 58, 109, 201, …
  (−∞, 0]nonnegative
• A118394 → x ↦ x+1
  1 · 1 · 1 · 7, 6 … Σ 1, 1, 1, 13, 49, 121, 1681, 10501, …
  (−∞, −1]nonnegative
• A225470 → (θ+1)²
  1 · 2, 4 · 10, 28, 9 · 80, 264, 135, 16 … Σ 1, 6, 47, 495, 6624, 107892, 2073924, …
  (−∞, 0]nonnegative
• A026729 → basis A013609 → reverse
  1 · 2, 1 · 4, 6, 2 · 8, 20, 14, 3 … Σ 1, 3, 12, 45, 171, 648, 2457, 9315, …
  (−∞, 0]nonnegative
• A225471 → basis A013610 → x ↦ x/(x+1)
  1 · 4, 7 · 32, 100, 77 · 384, 1680, 2424, 1155 … Σ 1, 11, 209, 5643, 197505, 8492715, …
  [−1, 0]nonnegativerational roots
• A099097 → basis A007318 → reverse
  1 · 3, 3 · 9, 19, 10 · 27, 87, 93, 33 … Σ 1, 6, 38, 240, 1516, 9576, 60488, …
  (−∞, 0]nonnegative
• A027467 → basis A013609 → (θ+1)²
  1 · 16, 8 · 256, 256, 36 · 4096, 6144, 1728, 128 … Σ 1, 24, 548, 12096, 260496, 5505408, …
  (−∞, 0]nonnegative
• A038321 → basis A007318 → reverse
  49, 252, 324 · 343, 2646, 6804, 5832 … Σ 625, 15625, 390625, 9765625, 244140625, …
  (−∞, 0]nonnegativerational roots
• A391533 → Pₙ + Pₙ₊₁ → basis A007318
  27, 27, 9, 1 · 54, 81, 45, 11, 1 … Σ 64, 192, 576, 1728, 5184, 15552, 46656, …
  (−∞, −1]nonnegativerational roots
• A116412 → basis A013609 → x ↦ x+1
  1 · 6, 2 · 33, 24, 4 · 183, 204, 72, 8 … Σ 1, 8, 61, 467, 3574, 27353, 209341, …
  (−∞, −1]nonnegative
• A144443 → x ↦ x+1
  1 · 2, 1 · 22, 22, 1 · 320, 480, 162, 1 … Σ 1, 3, 45, 963, 28413, 1078515, …
  (−∞, −1]nonnegative
• A257627 → (θ+1)²
  1 · 3, 12 · 9, 240, 81 · 27, 3012, 6777, 432 … Σ 1, 15, 330, 10248, 396084, 18241080, …
  (−∞, 0]nonnegative
• A038327 → basis A013610 → θ+1
  1 · 13, 6 · 169, 156, 27 · 2197, 3042, 1053, 108 … Σ 1, 19, 352, 6400, 114688, 2031616, …
  (−∞, 0]nonnegativerational roots
• A046739 → D
  7, 2 · 21, 42, 3 · 51, 322, 153, 4 … Σ 9, 66, 530, 4635, 44499, 467236, …
  (−∞, 0]nonnegative
• A225470 → basis A013609 → x ↦ x/(x+1)
  1 · 3, 5 · 18, 54, 40 · 162, 684, 954, 440 … Σ 1, 8, 112, 2240, 58240, 1863680, …
  [−1, 0]nonnegativerational roots
• A161198 → basis A013609 → D+1
  1 · 7, 4 · 47, 64, 16 · 389, 764, 432, 64 … Σ 1, 11, 127, 1649, 24209, 399171, …
  (−∞, 0]nonnegative
• A085841 → reverse
  1 · 4, 3 · 16, 40, 5 · 64, 336, 140, 7 … Σ 1, 7, 61, 547, 4921, 44287, 398581, …
  (−∞, 0]nonnegative
• A100862 → x ↦ x+1 → basis A007318
  1 · 3, 1 · 11, 7, 1 · 45, 42, 12, 1 … Σ 1, 4, 19, 100, 571, 3484, 22501, …
  (−∞, −1]nonnegative
• A100861 → basis A013609 → D+1
  1 · 1 · 4, 2 · 10, 6 … Σ 1, 1, 6, 16, 94, 366, 2236, 10816, …
  (−∞, 0]nonnegative
• A091042 → basis A013610 → basis A007318
  1 · 13, 9 · 121, 150, 45 · 1093, 1911, 1071, 189 … Σ 1, 22, 316, 4264, 56848, 756064, …
  (−∞, −1]nonnegative
• A136124: Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1;…
  1 · 2, 1 · 6, 5, 1 · 24, 26, 9, 1 … Σ 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  (−∞, −1]nonnegativerational roots
• A128813 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 8, 46, 65 · 56, 522, 1605, 1625 … Σ 1, 7, 119, 3808, 198016, 15247232, …
  [−1, 0]nonnegativerational roots
• A114687 → θ+1
  1 · 1, 4 · 1, 12, 12 · 1, 24, 72, 32 … Σ 1, 5, 25, 129, 681, 3653, 19825, …
  (−∞, 0]nonnegative
• A056242 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 10 · 1, 12, 37, 34 … Σ 1, 4, 18, 84, 396, 1872, 8856, 41904, …
  [−1, 0]nonnegative
• A071951 → D+1
  1 · 3, 1 · 12, 10, 1 · 60, 92, 23, 1 … Σ 1, 4, 23, 176, 1697, 20100, 286127, …
  (−∞, 0]nonnegative
• A144436: Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 1, and j = 4, read by rows.
  1 · 1, 1 · 1, 8, 1 · 1, 23, 23, 1 … Σ 1, 2, 10, 48, 280, 1872, 14224, 121280, …
  (−∞, 0)nonnegative
• A026729 → basis A013609 → (θ+1)²
  1 · 1, 8 · 2, 24, 36 · 3, 56, 180, 128 … Σ 1, 9, 62, 367, 1997, 10260, 50589, …
  (−∞, 0]nonnegative
• A049403 → D+1
  3, 5, 1 · 3, 15, 9, 1 · 0, 30, 45, 14, 1 … Σ 9, 28, 90, 306, 1078, 3984, 15228, …
  (−∞, 0]nonnegative
• A143491 → basis A013611 → (θ+1)²
  12, 112, 144 · 60, 752, 1728, 1024 … Σ 268, 3564, 47800, 663528, 9638624, …
  (−∞, 0]nonnegative
• A225471 → basis A013611 → x ↦ x/(x+1)
  1 · 4, 8 · 32, 112, 96 · 384, 1856, 2944, 1536 … Σ 1, 12, 240, 6720, 241920, 10644480, …
  [−1, 0]nonnegativerational roots
• A028338 → D+1
  1 · 2, 1 · 7, 6, 1 · 38, 41, 12, 1 … Σ 1, 3, 14, 92, 784, 8224, 102528, …
  (−∞, 0]nonnegative
• A113953 → x ↦ x−12
  1 · -12, 1 · 120, -22, 1 · -1152, 336, -32, 1 … Σ 1, -11, 99, -847, 7139, -59895, 501787, …
  (4, 12]
• A207606 → reverse → x ↦ x/(x+1)
  2, 11, 13 · 2, 17, 44, 34 · 2, 23, 93, 155, 89 … Σ 26, 97, 362, 1351, 5042, 18817, 70226, …
  [−1, 0]nonnegative
• A173018 → θ+1 → x ↦ x+1
  1 · 1 · 3, 2 · 12, 14, 3 … Σ 1, 1, 5, 29, 209, 1809, 18269, 210853, …
  (−∞, −1]nonnegative
• A209599 → D
  1 · 3 · 7, 2 · 15, 8 … Σ 0, 0, 1, 3, 9, 23, 57, 135, …
  (−∞, 0]nonnegative
• A100861 → basis A013609 → θ+1
  1 · 1 · 2, 4 · 4, 12 … Σ 1, 1, 6, 16, 94, 366, 2236, 10816, …
  (−∞, 0]nonnegative
• A038284 → basis A013610
  196, 504, 324 · 2744, 10584, 13608, 5832 … Σ 1024, 32768, 1048576, 33554432, …
  [−1, 0]nonnegativerational roots
• basis-map coefficients of A026729, shifted → (θ+1)² → D+1
  1 · -4, -4 · 0, 18, 9 · 0, 0, -48, -16 … Σ 1, -8, 27, -64, 125, -216, 343, -512, …
  (−∞, 0]rational roots
• A013613 → x ↦ x/(x+1)
  1 · 1, 7 · 1, 14, 49 · 1, 21, 147, 343 … Σ 1, 8, 64, 512, 4096, 32768, 262144, …
  [−1, 0]nonnegativerational roots
• A290319 → x ↦ x+1 → (θ+1)²
  1 · 2, 4 · 12, 32, 9 · 120, 368, 162, 16 … Σ 1, 6, 53, 666, 10945, 222742, 5410949, …
  (−∞, 0]nonnegative
• A078812 → reverse → x ↦ x+1
  1 · 3, 2 · 8, 10, 3 · 21, 38, 22, 4 … Σ 1, 5, 21, 85, 341, 1365, 5461, 21845, …
  (−∞, −1]nonnegative
• A034870 → D → basis A007318
  4, 2 · 32, 48, 24, 4 · 192, 480, 480, 240, 60, 6 … Σ 0, 6, 108, 1458, 17496, 196830, …
  (−∞, −1]nonnegativerational roots
• A008278 → x ↦ x+1 (shift +1) → x ↦ x+1 → θ+1
  1 · 3, 2 · 11, 14, 3 · 49, 92, 39, 4 … Σ 1, 5, 28, 184, 1415, 12403, 120985, …
  (−∞, 0]nonnegative