Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A161198 → reverse
  1 · 2, 1 · 4, 8, 3 · 8, 36, 46, 15 … Σ 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, 0]nonnegativerational roots
• A038322 → basis A013610 → D+1
  1 · 43, 24 · 1273, 2064, 576 … Σ 1, 67, 3913, 212635, 11051473, …
  (−∞, 0]nonnegativerational roots
• A038291 → reverse
  1 · 1, 9 · 1, 18, 81 · 1, 27, 243, 729 … Σ 1, 10, 100, 1000, 10000, 100000, …
  (−∞, 0]nonnegativerational roots
• A225472 → reverse
  1 · 3, 2 · 18, 21, 4 · 162, 270, 117, 8 … Σ 1, 5, 43, 557, 9643, 208685, 5419243, …
  (−∞, 0]nonnegative
• A143497 → x ↦ x/(x+1)
  20, 50, 31 · 120, 450, 558, 229 … Σ 101, 1357, 21225, 378641, 7584589, …
  [−1, 0]nonnegative
• A056939 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 22, 33, 13, 1 … Σ 1, 3, 13, 69, 417, 2763, 19609, 146793, …
  (−∞, −1]nonnegative
• A027466 → basis A013611 → x ↦ x/(x+1)
  1 · 8, 12 · 64, 192, 144 · 512, 2304, 3456, 1728 … Σ 1, 20, 400, 8000, 160000, 3200000, …
  [−1, 0]nonnegativerational roots
• A241171 → x ↦ x/(x+1)
  1 · 1, 7 · 1, 32, 121 · 1, 129, 1515, 3907 … Σ 1, 8, 154, 5552, 321616, 27325088, …
  [−1, 0]nonnegative
• A136572 → basis A013609 → (θ+1)²
  1 · 1, 8 · 2, 32, 72 · 6, 144, 648, 768 … Σ 1, 9, 106, 1566, 27864, 579960, …
  (−∞, 0]nonnegative
• A086645 → γ → D+1
  1 · 1 · 5, 4 · 13, 12 … Σ 1, 1, 9, 25, 97, 321, 1097, 3641, …
  (−∞, 0]nonnegative
• A144697 → reverse
  1 · 3, 1 · 9, 10, 1 · 27, 67, 25, 1 … Σ 1, 4, 20, 120, 840, 6720, 60480, …
  (−∞, 0]nonnegative
• A081577 → x ↦ x+1 → reverse
  1 · 1, 2 · 1, 6, 6 · 1, 10, 24, 16 … Σ 1, 3, 13, 51, 205, 819, 3277, 13107, …
  (−∞, 0]nonnegative
• A114687 → basis A007318 → reverse
  1 · 2, 3 · 4, 14, 11 · 8, 48, 84, 45 … Σ 1, 5, 29, 185, 1257, 8925, 65445, …
  (−∞, 0]nonnegative
• A207606 → basis A013609 → reverse
  8, 22, 13 · 16, 68, 88, 34 · 32, 184, 372, 310, 89 … Σ 43, 206, 987, 4729, 22658, 108561, …
  (−∞, 0]nonnegative
• A001263 → θ+1 → reverse
  1 · 2, 1 · 3, 6, 1 · 4, 18, 12, 1 … Σ 1, 3, 10, 35, 126, 462, 1716, 6435, …
  (−∞, 0]nonnegative
• A120986 → D+1
  1 · 3, 1 · 11, 8, 1 · 42, 52, 15, 1 … Σ 1, 4, 20, 110, 637, 3808, 23256, …
  (−∞, 0]nonnegative
• A103327 → x ↦ x+1 → (θ+1)²
  1 · 4, 4 · 16, 48, 9 · 64, 320, 216, 16 … Σ 1, 8, 73, 616, 4873, 36864, 269921, …
  (−∞, 0]nonnegative
• A173018 → x ↦ x+1 → D
  1 · 6, 2 · 36, 28, 3 · 240, 300, 90, 4 … Σ 0, 0, 1, 8, 67, 634, 6793, 81780, …
  (−∞, −1]nonnegative
• A126351 → θ+1
  1 · 1, 4 · 1, 10, 12 · 1, 18, 57, 32 … Σ 1, 5, 23, 108, 534, 2803, 15623, 92249, …
  (−∞, 0]nonnegative
• A038336 → x ↦ x+1 → x ↦ x/(x+1)
  484, 1408, 1024 · 10648, 46464, 67584, 32768 … Σ 2916, 157464, 8503056, 459165024, …
  [−1, 0]nonnegativerational roots
• A219512 → (θ+1)²
  1 · 1, 12 · 1, 48, 135 · 1, 132, 1269, 1680 … Σ 1, 13, 184, 3082, 60292, 1353688, …
  (−∞, 0]nonnegative
• A207824 → basis A013611 → D+1
  1 · 10, 4 · 83, 80, 16 · 628, 1000, 480, 64 … Σ 1, 14, 179, 2172, 25461, 291242, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729, shifted → basis A007318 → θ+1
  1 · -1, -2 · 1, 4, 3 · -1, -6, -9, -4 … Σ 1, -3, 8, -20, 48, -112, 256, -576, …
  (−∞, 0]rational roots
• A106246 → reverse → θ+1
  1 · 1, 4 · 1, 8, 3 · 1, 12, 9 … Σ 1, 5, 12, 22, 35, 51, 70, 92, …
  (−∞, 0]nonnegative
• A371259 → D+1
  1 · 37, 1 · 1864, 102, 1 · 122760, 10264, 203, 1 … Σ 1, 38, 1967, 133228, 11446741, …
  (−∞, 0]nonnegative
• A169803: Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).
  1 · 1, 1 · 1, 2 · 1, 3, 1 … Σ 1, 2, 3, 5, 8, 13, 21, 34, …
  (−∞, 0)nonnegative
• A038321 → basis A007318 → θ+1
  324, 504, 147 · 5832, 13608, 7938, 1372 … Σ 975, 28750, 828125, 23437500, …
  (−∞, 0]nonnegativerational roots
• A167884 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 20, 20 · 1, 182, 540, 360 … Σ 1, 3, 41, 1083, 40721, 1998003, …
  [−1, 0]nonnegative
• A094816 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 5, 25, 29 · 16, 117, 267, 193 … Σ 1, 7, 59, 593, 7001, 95527, 1483939, …
  [−1, 0]nonnegative
• A034870 → Veronese 2 (j=1) → D
  4 · 20, 12 · 56, 112, 24 · 120, 504, 360, 40 … Σ 0, 0, 4, 32, 192, 1024, 5120, 24576, …
  (−∞, 0]nonnegative
• A081579 → D+1
  1 · 2, 1 · 7, 8, 1 · 12, 33, 14, 1 … Σ 1, 3, 16, 60, 240, 896, 3328, 12096, …
  (−∞, 0]nonnegative
• A199881 → D
  1 · 1, 2 · 1, 4, 3 · 0, 4, 9, 4 … Σ 1, 3, 8, 17, 35, 68, 129, 239, …
  (−∞, 0]nonnegative
• A056242 → (θ+1)²
  1 · 1, 8 · 1, 20, 36 · 1, 36, 144, 128 … Σ 1, 9, 57, 309, 1530, 7134, 31864, …
  (−∞, 0]nonnegative
• delannoy → basis A013610 → D+1
  1 · 5, 3 · 20, 33, 9 · 66, 198, 153, 27 … Σ 1, 8, 62, 444, 3050, 20344, 132838, …
  (−∞, 0]nonnegative
• A128540 → θ+1
  1 · 1, 2 · 0, 4, 6 · 0, 0, 9, 12 … Σ 1, 3, 10, 21, 45, 88, 169, 315, …
  (−∞, 0]nonnegativerational roots
• A094262 → x ↦ x/(x+1)
  1 · 1, 4, 4 · 1, 10, 36, 56, 32 … Σ 1, 9, 135, 3159, 102789, 4293081, …
  [−1, 0]nonnegative
• A272866 → basis A013609 → basis A007318
  1 · 19, 18, 4 · 361, 684, 476, 144, 16 … Σ 1, 41, 1681, 68921, 2825761, 115856201, …
  (−∞, −1]nonnegative
• A038333 → x ↦ x+1 → reverse
  49, 266, 361 · 343, 2793, 7581, 6859 … Σ 676, 17576, 456976, 11881376, …
  (−∞, 0]nonnegativerational roots
• A013618 → D
  22, 242 · 33, 726, 3993 · 44, 1452, 15972, 58564 … Σ 264, 4752, 76032, 1140480, 16422912, …
  (−∞, 0]nonnegativerational roots
• A114656 → θ+1 → (θ+1)²
  1 · 2, 8 · 4, 48, 27 · 8, 192, 324, 64 … Σ 1, 10, 79, 588, 4221, 29518, 202315, …
  (−∞, 0]nonnegative
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
  1 · 2, 3 · 4, 12, 9 · 8, 36, 54, 27 … Σ 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  (−∞, 0)[−1, 0]nonnegative+1
• A079621 → D+1
  1 · 5, 1 · 36, 14, 1 · 336, 192, 27, 1 … Σ 1, 6, 51, 556, 7365, 114546, 2042551, …
  (−∞, 0]nonnegative
• A113953 → θ+1 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 4, 7 · 0, 0, 12, 16 … Σ 1, 2, 11, 28, 101, 278, 863, 2416, …
  [−1, 0]nonnegative
• A094816 → basis A013609 → D
  2 · 10, 8 · 46, 72, 24 · 232, 520, 336, 64 … Σ 0, 2, 18, 142, 1152, 10018, 94514, …
  (−∞, 0]nonnegative
• A099093 → x ↦ x+1 → reverse
  1 · 3, 3 · 9, 21, 12 · 27, 99, 117, 45 … Σ 1, 6, 42, 288, 1980, 13608, 93528, …
  (−∞, 0]nonnegative
• A013618 → θ+1
  1, 44, 363 · 1, 66, 1089, 5324 … Σ 408, 6480, 96768, 1389312, 19408896, …
  (−∞, 0]nonnegativerational roots
• A093375 → Veronese 2 → D+1
  1 · 2 · 4, 1 · 10, 6 … Σ 1, 2, 5, 16, 44, 116, 296, 736, …
  (−∞, 0]nonnegative
• A122848 → D+1
  1 · 1, 1 · 1, 3, 1 · 0, 6, 6, 1 … Σ 1, 2, 5, 13, 38, 116, 382, 1310, …
  (−∞, 0]nonnegative
• A114687 → basis A013610 → D+1
  1 · 9, 6 · 53, 114, 36 · 297, 1116, 1080, 216 … Σ 1, 15, 203, 2709, 36365, 492819, …
  (−∞, 0]nonnegative
• A038327 → basis A013610 → x ↦ x/(x+1)
  1 · 13, 16 · 169, 416, 256 · 2197, 8112, 9984, 4096 … Σ 1, 29, 841, 24389, 707281, 20511149, …
  [−1, 0]nonnegativerational roots