Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A292219 → x ↦ x/(x+1)
  1 · 6, 7 · 60, 140, 81 · 840, 2940, 3402, 1303 … Σ 1, 13, 281, 8485, 328753, 15539261, …
  [−1, 0]nonnegative
• A345013 → (θ+1)²
  1 · 4, 12 · 15, 80, 54 · 56, 420, 540, 160 … Σ 1, 16, 149, 1176, 8621, 60628, 415033, …
  (−∞, 0]nonnegative
• A269947 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 8, 25, 18 … Σ 1, 1, 3, 51, 2805, 361845, 90823095, …
  [−1, 0]nonnegativerational roots
• A145324 → basis A013610 → θ+1
  1 · 3, 12 · 12, 102, 162 · 60, 798, 2646, 2592 … Σ 1, 15, 276, 6096, 157851, 4698117, …
  (−∞, 0]nonnegative
• A085880: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
  1 · 1, 1 · 2, 4, 2 · 5, 15, 15, 5 … Σ 1, 2, 8, 40, 224, 1344, 8448, 54912, …
  (−∞, 0)nonnegativerational roots
• A085478 → reverse → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 8, 19, 13 … Σ 1, 3, 11, 41, 153, 571, 2131, 7953, …
  [−1, 0]nonnegative
• A299989 → D
  1 · 3, 8, 3 · 9, 48, 66, 32, 5 … Σ 1, 14, 160, 1664, 16384, 155648, …
  (−∞, 0]nonnegative
• A272867 → basis A013609 → (θ+1)²
  1 · 6, 48, 36 · 36, 576, 1728, 1536, 400 … Σ 1, 90, 4276, 162184, 5453712, …
  (−∞, 0]nonnegative
• A038279 → x ↦ x/(x+1)
  1 · 8, 9 · 64, 144, 81 · 512, 1728, 1944, 729 … Σ 1, 17, 289, 4913, 83521, 1419857, …
  [−1, 0]nonnegativerational roots
• A046716 → basis A013610 → θ+1
  1 · 2, 6 · 5, 30, 27 · 16, 150, 297, 108 … Σ 1, 8, 62, 571, 7247, 129266, 2994532, …
  (−∞, 0]nonnegative
• A038319 → basis A013611 → θ+1
  256, 1280, 1200 · 4096, 30720, 57600, 32000 … Σ 2736, 124416, 5412096, 228427776, …
  (−∞, 0]nonnegativerational roots
• narayana → basis A007318 → D
  1 · 5, 2 · 21, 18, 3 · 84, 112, 42, 4 … Σ 0, 1, 7, 42, 242, 1375, 7773, 43876, …
  (−∞, −1]nonnegative
• A154227 → x ↦ x+1
  1 · 2, 1 · 10, 10, 1 · 40, 60, 22, 1 … Σ 1, 3, 21, 123, 999, 8163, 80433, …
  (−∞, −1]nonnegative
• A172431 → Veronese 2 → x ↦ x+1
  1 · 1 · 4, 3 · 11, 10 … Σ 1, 1, 7, 21, 63, 213, 671, 2149, …
  (−∞, −1]nonnegative
• A038277 → x ↦ x+1 → x ↦ x+1
  1 · 29, 11 · 841, 638, 121 … Σ 1, 40, 1600, 64000, 2560000, 102400000, …
  (−∞, −1]nonnegativerational roots
• A035607 → D+1
  1 · 1 · 3, 2 · 5, 8, 2 … Σ 1, 1, 5, 15, 45, 129, 361, 991, …
  (−∞, 0]nonnegative
• A135065 → θ+1
  1 · 4, 8 · 9, 36, 27 · 16, 96, 144, 64 … Σ 1, 12, 72, 320, 1200, 4032, 12544, …
  (−∞, 0]nonnegativerational roots
• A102662 → reverse → θ+1
  1 · 3, 2 · 3, 10, 3 · 3, 22, 21, 4 … Σ 1, 5, 16, 50, 147, 419, 1166, 3188, …
  (−∞, 0]nonnegative
• A038325 → D
  11 · 242, 242 · 3993, 7986, 3993 … Σ 0, 11, 484, 15972, 468512, 12884080, …
  (−∞, 0]nonnegativerational roots
• A272867 → basis A013609 → θ+1
  1 · 6, 24, 12 · 36, 288, 576, 384, 80 … Σ 1, 42, 1364, 39688, 1086096, 28579232, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A013611 → D
  8, 32 · 16, 192, 768, 1024 … Σ 0, 40, 2000, 75000, 2500000, 78125000, …
  (−∞, 0]nonnegativerational roots
• A038305 → x ↦ x+1 → D+1
  247, 96, 9 · 3718, 2223, 432, 27 … Σ 352, 6400, 114688, 2031616, 35651584, …
  (−∞, 0]nonnegativerational roots
• A352880 → x ↦ x/(x+1)
  8 · 12, 30 · 16, 96, 112 · 20, 210, 560, 420 … Σ 8, 42, 224, 1210, 6600, 36274, 200576, …
  [−1, 0]nonnegative
• A013613 → basis A013611 → θ+1
  1 · 7, 48 · 49, 672, 1728 · 343, 7056, 36288, 55296 … Σ 1, 55, 2449, 98983, 3783457, 139451671, …
  (−∞, 0]nonnegativerational roots
• row-factor cofactor of A101842 → D+1
  1 · 2, 1 · 5, 6, 1 · 12, 33, 14, 1 … Σ 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  (−∞, 0]nonnegative
• A135065 → x ↦ x/(x+1)
  1 · 4, 8 · 9, 36, 36 · 16, 96, 192, 128 … Σ 1, 12, 81, 432, 2025, 8748, 35721, …
  [−1, 0]nonnegativerational roots
• A038291 → basis A013609 → x ↦ x/(x+1)
  1 · 10, 12 · 100, 240, 144 · 1000, 3600, 4320, 1728 … Σ 1, 22, 484, 10648, 234256, 5153632, …
  [−1, 0]nonnegativerational roots
• A178820 → basis A013610 → x ↦ x/(x+1)
  1 · 8, 20 · 40, 200, 250 · 160, 1200, 3000, 2500 … Σ 1, 28, 490, 6860, 84035, 941192, …
  [−1, 0]nonnegativerational roots
• A122076 → D
  2 · 8, 4 · 30, 30, 6 · 104, 160, 72, 8 … Σ 0, 2, 12, 66, 344, 1710, 8196, 38234, …
  (−∞, 0]nonnegative
• A038271 → x ↦ x/(x+1) → D+1
  1 · 19, 12 · 217, 456, 144 · 2107, 7812, 8208, 1728 … Σ 1, 31, 817, 19855, 459553, 10295359, …
  (−∞, 0]nonnegativerational roots
• A118933 → x ↦ x+1
  1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 13, 61, 181, 421, …
  (−∞, −1]nonnegative
• A131431 → x ↦ x+1 → D
  4 · 14, 14 · 30, 60, 30 · 52, 156, 156, 52 … Σ 0, 4, 28, 120, 416, 1280, 3648, 9856, …
  (−∞, −1]nonnegativerational roots
• A331333 → reverse
  1 · 2, 1 · 4, 8, 2 · 8, 36, 36, 6 … Σ 1, 3, 14, 86, 648, 5752, 58576, 671568, …
  (−∞, 0]nonnegative
• A286718 → basis A013611 → x ↦ x/(x+1)
  1 · 2, 6 · 10, 48, 54 · 80, 504, 1008, 648 … Σ 1, 8, 112, 2240, 58240, 1863680, …
  [−1, 0]nonnegativerational roots
• A049020 → reverse → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 6 · 1, 9, 25, 22 … Σ 1, 3, 12, 57, 309, 1866, 12351, 88563, …
  [−1, 0]nonnegative
• A126351 → x ↦ x+1 → D
  2 · 13, 8 · 71, 86, 24 · 383, 692, 387, 64 … Σ 0, 2, 21, 181, 1526, 13252, 120647, …
  (−∞, 0]nonnegative
• A120910 → Veronese 2 → D
  3 · 18 · 72, 6 · 240, 60 … Σ 0, 0, 3, 18, 78, 300, 1089, 3822, …
  (−∞, 0]nonnegative
• A008288 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 5, 25, 29 · 12, 90, 216, 165 … Σ 1, 7, 59, 483, 3971, 32627, 268099, …
  [−1, 0]nonnegative
• A108838 → D
  2 · 8, 4 · 20, 30, 6 · 40, 120, 72, 8 … Σ 0, 2, 12, 56, 240, 990, 4004, 16016, …
  (−∞, 0]nonnegative
• A114655 → reverse → (θ+1)²
  2 · 2, 16 · 2, 48, 72 · 2, 96, 432, 256 … Σ 2, 18, 122, 786, 4962, 30946, 191354, …
  (−∞, 0]nonnegative
• A137320 → basis A013611 → D+1
  1 · 10, 8 · 46, 168, 64 · 232, 1360, 2112, 512 … Σ 1, 18, 278, 4216, 65368, 1052432, …
  (−∞, 0]nonnegative
• A038324 → basis A007318 → θ+1
  1 · 21, 20 · 441, 840, 300 · 9261, 26460, 18900, 4000 … Σ 1, 41, 1581, 58621, 2115161, 74805201, …
  (−∞, 0]nonnegativerational roots
• A144442: Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 17, 1 · 1, 118, 118, 1 … Σ 1, 2, 19, 238, 4141, 92292, 2512589, …
  (−∞, 0)nonnegative
• A027466 → basis A013610 → θ+1
  1 · 8, 6 · 64, 96, 27 · 512, 1152, 648, 108 … Σ 1, 14, 187, 2420, 30613, 380666, …
  (−∞, 0]nonnegativerational roots
• A038259 → basis A013611 → x ↦ x/(x+1)
  1 · 11, 31 · 121, 682, 961 … Σ 1, 42, 1764, 74088, 3111696, 130691232, …
  [−1, 0]nonnegativerational roots
• A099097 → (θ+1)²
  1 · 0, 12 · 0, 4, 81 · 0, 0, 54, 432 … Σ 1, 12, 85, 486, 2466, 11592, 51667, …
  (−∞, 0]nonnegative
• A166962 → (θ+1)²
  1 · 1, 4 · 1, 48, 9 · 1, 412, 621, 16 … Σ 1, 5, 58, 1050, 27948, 990472, …
  (−∞, 0]nonnegative
• A114656 → basis A013609 → D
  2 · 16, 8 · 102, 120, 24 · 608, 1168, 576, 64 … Σ 0, 2, 24, 246, 2416, 23370, 224760, …
  (−∞, 0]nonnegative
• A201701 → θ+1
  1 · 1 · 2, 2 · 4, 6 … Σ 1, 1, 4, 10, 27, 71, 186, 484, …
  (−∞, 0]nonnegative
• A286718 → x ↦ x/(x+1)
  1 · 1, 2 · 4, 13, 10 · 28, 123, 174, 80 … Σ 1, 3, 27, 405, 8505, 229635, 7577955, …
  [−1, 0]nonnegativerational roots