Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A156992 → basis A007318 → (θ+1)²
  1 · 4, 8 · 24, 96, 54 · 192, 1152, 1296, 384 … Σ 1, 12, 174, 3024, 61560, 1438560, …
  (−∞, 0]nonnegative
• A321434 → x ↦ x+1
  1 · 1, 1 · 1, 1 · 3, 5, 2 … Σ 1, 2, 2, 10, 10, 74, 74, 730, …
  (−∞, −1]nonnegative
• A123160 → (θ+1)²
  1 · 1, 4 · 1, 16, 27 · 1, 36, 162, 160 … Σ 1, 5, 44, 359, 2760, 20337, 145220, …
  (−∞, 0]nonnegative
• A331333: Interpolating the factorial and the powers of 2. Triangle read by rows, T(n, k) for 0 <= k <= n.
  1 · 1, 2 · 2, 8, 4 · 6, 36, 36, 8 … Σ 1, 3, 14, 86, 648, 5752, 58576, 671568, …
  (−∞, 0)nonnegative
• A038324 → θ+1
  1 · 11, 20 · 121, 440, 300 · 1331, 7260, 9900, 4000 … Σ 1, 31, 861, 22491, 564921, 13808151, …
  (−∞, 0]nonnegativerational roots
• A322944 → D+1
  1 · 2, 1 · 8, 8, 1 · 44, 68, 18, 1 … Σ 1, 3, 17, 131, 1249, 14091, 182913, …
  (−∞, 0]nonnegative
• A225470 → basis A008277 → D
  1 · 10, 2 · 118, 42, 3 · 1696, 824, 108, 4 … Σ 0, 1, 12, 163, 2632, 49885, 1090196, …
  (−∞, 0]nonnegative
• A172431 → basis A007318 → reverse
  1 · 2, 3 · 3, 10, 8 · 4, 22, 38, 21 … Σ 1, 5, 21, 85, 341, 1365, 5461, 21845, …
  (−∞, 0]nonnegative
• A085881 → basis A013609 → θ+1
  1 · 2, 4 · 12, 48, 36 · 120, 720, 1080, 480 … Σ 1, 6, 96, 2400, 80640, 3386880, …
  (−∞, 0]nonnegativerational roots
• A115951 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 2, 8 · 0, 0, 12, 32 … Σ 1, 2, 10, 44, 214, 1052, 5284, 26840, …
  [−1, 0]nonnegative
• A367270 → basis A013609 → D+1
  1 · 1 · 7, 4 · 34, 48, 12 … Σ 1, 1, 11, 94, 763, 6086, 48094, 377644, …
  (−∞, 0]nonnegative
• A134082 → reverse → x ↦ x+1
  1 · 3, 2 · 5, 4 · 7, 6 … Σ 1, 5, 9, 13, 17, 21, 25, 29, …
  (−∞, −1]nonnegativerational roots
• A116414 → (θ+1)²
  1 · 4, 4 · 13, 32, 9 · 40, 168, 108, 16 … Σ 1, 8, 54, 332, 1921, 10656, 57301, …
  (−∞, 0]nonnegative
• A008957 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 7, 7 · 1, 17, 52, 37 … Σ 1, 3, 15, 107, 1015, 12243, 181311, …
  [−1, 0]nonnegative
• A142459 → x ↦ x+1
  1 · 2, 1 · 12, 12, 1 · 120, 180, 62, 1 … Σ 1, 3, 25, 363, 7345, 190803, 6056905, …
  (−∞, −1]nonnegative
• A026729 → basis A013610 → (θ+1)²
  1 · 1, 12 · 2, 36, 81 · 3, 84, 405, 432 … Σ 1, 13, 119, 924, 6530, 43352, 275272, …
  (−∞, 0]nonnegative
• A094816 → x ↦ x+1 → D+1
  1 · 3, 1 · 10, 7, 1 · 39, 41, 12, 1 … Σ 1, 4, 18, 93, 553, 3760, 28946, 249697, …
  (−∞, 0]nonnegative
• A028338 → basis A008277 → (θ+1)²
  1 · 2, 4 · 8, 28, 9 · 48, 228, 135, 16 … Σ 1, 6, 45, 427, 4932, 67221, 1056211, …
  (−∞, 0]nonnegative
• A143495 → D
  1 · 7, 2 · 37, 24, 3 · 175, 194, 54, 4 … Σ 0, 1, 9, 64, 427, 2821, 18890, 129683, …
  (−∞, 0]nonnegative
• A078812 → reverse → D
  2 · 4, 6 · 6, 20, 12 · 8, 42, 60, 20 … Σ 0, 2, 10, 38, 130, 420, 1308, 3970, …
  (−∞, 0]nonnegative
• A080248 → D
  1 · 4, 2 · 13, 20, 3 · 40, 146, 60, 4 … Σ 0, 1, 6, 36, 250, 2041, 19444, 213989, …
  (−∞, 0]nonnegative
• A008277 → θ+1 → D+1
  1 · 3, 2 · 7, 12, 3 · 15, 50, 30, 4 … Σ 1, 5, 22, 99, 471, 2386, 12867, 73681, …
  (−∞, 0]nonnegative
• A103406 → Veronese 2 (j=1) → x ↦ x+1
  2 · 4 · 10, 4 · 24, 16 … Σ 0, 2, 4, 14, 40, 114, 316, 862, …
  (−∞, −1]nonnegative
• A113214 → x ↦ x−15
  1 · -13, 1 · 180, -27, 1 · -2505, 557, -41, 1 … Σ 1, -12, 154, -1988, 25676, -331632, …
  (11, 15]
• A113413 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 3, 4 · 7, 20, 14 · 17, 74, 106, 50 … Σ 1, 7, 41, 247, 1481, 8887, 53321, …
  [−1, 0]nonnegative
• A119900 → D+1
  1 · 2, 2 · 1, 7, 3 · 0, 8, 16, 4 … Σ 1, 4, 11, 28, 68, 160, 368, 832, …
  (−∞, 0]nonnegative
• A038333 → x ↦ x+1 → D+1
  627, 364, 49 · 14440, 13167, 3822, 343 … Σ 1040, 31772, 949104, 27875536, …
  (−∞, 0]nonnegativerational roots
• A028326 → Veronese 2 (j=1) → D
  2 · 8 · 20, 4 · 40, 24 … Σ 0, 0, 0, 2, 8, 24, 64, 160, …
  (−∞, 0]nonnegative
• A038322 → reverse
  1 · 8, 11 · 64, 176, 121 · 512, 2112, 2904, 1331 … Σ 1, 19, 361, 6859, 130321, 2476099, …
  (−∞, 0]nonnegativerational roots
• A038319 → x ↦ x+1 → reverse
  25, 160, 256 · 125, 1200, 3840, 4096 … Σ 441, 9261, 194481, 4084101, 85766121, …
  (−∞, 0]nonnegativerational roots
• A391403: Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).
  1 · 1 · 1, 2 · 1, 6 … Σ 1, 1, 3, 7, 19, 51, 141, 393, …
  (−∞, 0)nonnegative
• A123519 → basis A013611 → D
  56, 128 · 304, 1408, 1536 · 1488, 10368, 23040, 16384 … Σ 184, 3248, 51280, 760920, 10852968, …
  (−∞, 0]nonnegative
• A038321 → basis A013611 → reverse
  784, 1008, 324 · 21952, 42336, 27216, 5832 … Σ 2116, 97336, 4477456, 205962976, …
  (−∞, 0]nonnegativerational roots
• A034870 → x ↦ x+1
  1 · 4, 4, 1 · 16, 32, 24, 8, 1 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, −1]nonnegativerational roots
• A272867 → basis A013611 → x ↦ x/(x+1)
  1 · 6, 36, 46 · 36, 432, 1848, 3312, 2116 … Σ 1, 88, 7744, 681472, 59969536, …
  [−1, 0]nonnegative
• A154227: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.
  1 · 1, 1 · 1, 8, 1 · 1, 19, 19, 1 … Σ 1, 2, 10, 40, 230, 1300, 9040, 64880, …
  (−∞, 0)nonnegative
• A082985 → x ↦ x+1 → basis A007318
  1 · 7, 3 · 31, 25, 5 · 127, 147, 56, 7 … Σ 1, 10, 61, 337, 1810, 9637, 51169, …
  (−∞, −1]nonnegative
• A013622 → basis A013611 → x ↦ x/(x+1)
  1 · 8, 28 · 64, 448, 784 · 512, 5376, 18816, 21952 … Σ 1, 36, 1296, 46656, 1679616, 60466176, …
  [−1, 0]nonnegativerational roots
• A125092 → x ↦ x/(x+1)
  1 · 1, 5 · 1, 10, 18 · 1, 15, 54, 56 … Σ 1, 6, 29, 126, 513, 1998, 7533, 27702, …
  [−1, 0]nonnegative
• A135838 → D+1
  1 · 4, 2 · 6, 8, 2 · 16, 36, 24, 4 … Σ 1, 6, 16, 80, 192, 896, 2048, 9216, …
  (−∞, 0]nonnegativerational roots
• A120910 → basis A013611 → D
  12 · 72, 96 · 324, 864, 576 · 1296, 5184, 6912, 3072 … Σ 0, 12, 168, 1764, 16464, 144060, …
  (−∞, 0]nonnegativerational roots
• A207823 → basis A013611 → D+1
  1 · 9, 4 · 64, 72, 16 · 407, 772, 432, 64 … Σ 1, 13, 152, 1675, 17767, 183504, …
  (−∞, 0]nonnegative
• A143494 → reverse → (θ+1)²
  1 · 1, 8 · 1, 20, 36 · 1, 36, 171, 128 … Σ 1, 9, 57, 336, 1992, 12193, 77725, …
  (−∞, 0]nonnegative
• A039755 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 16, 38, 24 … Σ 1, 3, 13, 79, 601, 5339, 53861, 607527, …
  [−1, 0]nonnegative
• A038330 → basis A013610 → D+1
  640, 672, 144 · 13312, 23040, 12096, 1728 … Σ 1456, 50176, 1668352, 54089728, …
  (−∞, 0]nonnegativerational roots
• A038334 → x ↦ x+1 → D+1
  720, 448, 64 · 17600, 17280, 5376, 512 … Σ 1232, 40768, 1317120, 41796608, …
  (−∞, 0]nonnegativerational roots
• A133214 → basis A013609 → θ+1
  13, 64, 48 · 63, 456, 720, 256 … Σ 125, 1495, 18305, 225947, 2793997, …
  (−∞, 0]nonnegative
• A225466 → x ↦ x+1
  1 · 5, 3 · 34, 39, 9 · 287, 468, 216, 27 … Σ 1, 8, 82, 998, 13870, 215054, 3661390, …
  (−∞, −1]nonnegative
• A038336 → x ↦ x/(x+1) → D
  528, 968 · 9504, 34848, 31944 … Σ 1496, 76296, 3458752, 146996960, …
  (−∞, 0]nonnegativerational roots
• A290595 → D
  2 · 19, 8 · 222, 294, 24 · 3194, 8256, 2661, 64 … Σ 0, 2, 27, 540, 14175, 459270, 17681895, …
  (−∞, 0]nonnegative