Proof methods / preserver_image

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8,704 sequences

• A059364 → basis A013611 → reverse
  1 · 4, 3 · 32, 44, 15 · 384, 752, 488, 105 … Σ 1, 7, 91, 1729, 43225, 1339975, …
  (−∞, 0]nonnegativerational roots
• A225476 → D
  1 · 4, 4 · 13, 36, 18 · 40, 232, 288, 96 … Σ 0, 1, 8, 67, 656, 7501, 98648, 1470127, …
  (−∞, 0]nonnegative
• A128813 → basis A013610 → D
  3 · 30, 54 · 354, 1458, 1458 … Σ 0, 3, 84, 3270, 182820, 14150145, …
  (−∞, 0]nonnegative
• A008459 → basis A013609 → reverse
  1 · 2, 2 · 4, 12, 6 · 8, 48, 60, 20 … Σ 1, 4, 22, 136, 886, 5944, 40636, …
  (−∞, 0]nonnegative
• A207607 → θ+1
  1 · 1, 4 · 1, 10, 6 · 1, 18, 27, 8 … Σ 1, 5, 17, 54, 166, 499, 1475, 4303, …
  (−∞, 0]nonnegative
• A249251 → (θ+1)²
  1 · 1, 12 · 1, 24, 54 · 1, 36, 189, 192 … Σ 1, 13, 79, 418, 2014, 9127, 39583, …
  (−∞, 0]nonnegative
• A038323 → basis A013610 → basis A007318
  1 · 47, 27 · 2209, 2538, 729 … Σ 1, 74, 5476, 405224, 29986576, …
  (−∞, −1]nonnegativerational roots
• A290316 → D+1
  1 · 7, 6 · 49, 228, 90 · 235, 4662, 8694, 2160 … Σ 1, 13, 367, 15751, 899041, 63773269, …
  (−∞, 0]nonnegative
• A143491 → basis A013609 → θ+1
  12, 28, 12 · 60, 188, 144, 32 … Σ 52, 424, 3812, 37668, 406920, 4778520, …
  (−∞, 0]nonnegative
• A257611 → x ↦ x+1
  1 · 6, 3 · 48, 48, 9 · 480, 720, 294, 27 … Σ 1, 9, 105, 1521, 26385, 533529, …
  (−∞, −1]nonnegative
• A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
  1 · 2, 2 · 6, 12, 6 · 24, 72, 72, 24 … Σ 1, 4, 24, 192, 1920, 23040, 322560, …
  (−∞, −1]nonnegativerational roots
• A178820 → basis A007318 → reverse
  1 · 4, 8 · 10, 40, 40 · 20, 120, 240, 160 … Σ 1, 12, 90, 540, 2835, 13608, 61236, …
  (−∞, 0]nonnegativerational roots
• A103633 → basis A013609 → θ+1
  1 · 1, 4 · 2, 12, 12 · 2, 20, 48, 32 … Σ 1, 5, 26, 102, 480, 1728, 7776, 26784, …
  (−∞, 0]nonnegative
• A144699 → reverse
  1 · 5, 1 · 25, 16, 1 · 125, 171, 39, 1 … Σ 1, 6, 42, 336, 3024, 30240, 332640, …
  (−∞, 0]nonnegative
• A133087 → reverse → x ↦ x/(x+1)
  1 · 1, 3 · 1, 4, 4 · 1, 7, 16, 12 … Σ 1, 4, 9, 36, 81, 324, 729, 2916, …
  [−1, 0]nonnegativerational roots
• A039756 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 18, 13 · 24, 110, 164, 79 … Σ 1, 5, 37, 377, 4729, 68957, 1140157, …
  [−1, 0]nonnegative
• A079621 → x ↦ x+1
  1 · 5, 1 · 37, 14, 1 · 361, 195, 27, 1 … Σ 1, 6, 52, 584, 8016, 129632, 2408512, …
  (−∞, −1]nonnegative
• A152198 → basis A013609 → (θ+1)²
  1 · 1 · 2, 8 · 2, 8 … Σ 1, 1, 10, 10, 72, 72, 448, 448, …
  (−∞, 0]nonnegative
• A011973 → basis A013610 → D
  3 · 6 · 15, 18 · 30, 54 … Σ 0, 0, 3, 6, 33, 84, 303, 834, …
  (−∞, 0]nonnegative
• A090582 → D+1
  1 · 3, 1 · 12, 8, 1 · 60, 64, 17, 1 … Σ 1, 4, 21, 142, 1175, 11476, 129073, …
  (−∞, 0]nonnegative
• A038327 → basis A013609 → θ+1
  1 · 13, 4 · 169, 104, 12 · 2197, 2028, 468, 32 … Σ 1, 17, 285, 4725, 77625, 1265625, …
  (−∞, 0]nonnegativerational roots
• A038334 → reverse
  64, 192, 144 · 512, 2304, 3456, 1728 … Σ 400, 8000, 160000, 3200000, 64000000, …
  (−∞, 0]nonnegativerational roots
• A225471 → x ↦ x/(x+1)
  1 · 3, 4 · 21, 52, 32 · 231, 824, 976, 384 … Σ 1, 7, 105, 2415, 74865, 2919735, …
  [−1, 0]nonnegativerational roots
• A371080 → (θ+1)²
  1 · 0, 4 · 0, 16, 9 · 0, 112, 108, 16 … Σ 1, 4, 25, 236, 2969, 46596, 875569, …
  (−∞, 0]nonnegative
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k…
  3 · 3, 6 · 3, 12, 12 · 3, 18, 36, 24 … Σ 3, 9, 27, 81, 243, 729, 2187, 6561, …
  (−∞, 0)nonnegativerational roots
• A131431 → x ↦ x+1 → D+1
  1 · 8, 4 · 21, 28, 7 · 40, 90, 60, 10 … Σ 1, 12, 56, 200, 624, 1792, 4864, 12672, …
  (−∞, 0]nonnegativerational roots
• A038327 → basis A013610 → reverse
  1 · 3, 13 · 9, 78, 169 · 27, 351, 1521, 2197 … Σ 1, 16, 256, 4096, 65536, 1048576, …
  (−∞, 0]nonnegativerational roots
• A038281 → x ↦ x+1 → θ+1
  121, 132, 27 · 1331, 2178, 891, 108 … Σ 280, 4508, 71344, 1114064, 17210368, …
  (−∞, 0]nonnegativerational roots
• A122848 → basis A013610 → x ↦ x/(x+1)
  1 · 1, 4 · 2, 13, 20 · 4, 39, 120, 112 … Σ 1, 5, 35, 275, 2425, 23125, 236875, …
  [−1, 0]nonnegative
• A099093 → x ↦ x−9
  1 · -27, 3 · 702, -159, 9 · -18225, 6237, -711, 27 … Σ 1, -24, 552, -12672, 290880, -6676992, …
  (23/3, 9]
• A130123 → basis A013610 → (θ+1)²
  1 · 2, 24 · 4, 96, 324 · 8, 288, 1944, 3456 … Σ 1, 26, 424, 5696, 68608, 770048, …
  (−∞, 0]nonnegative
• A304236 → basis A013609 → θ+1
  1 · 1 · 4, 12 · 7, 24 … Σ 1, 1, 16, 31, 235, 628, 3397, 10729, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → basis A013610 → θ+1
  1, 12, 27 · 1, 18, 81, 108 · 1, 24, 162, 432, 405 … Σ 40, 208, 1024, 4864, 22528, 102400, …
  (−∞, 0]nonnegativerational roots
• A026729 → θ+1
  1 · 0, 2 · 0, 2, 3 · 0, 0, 6, 4 … Σ 1, 2, 5, 10, 20, 38, 71, 130, …
  (−∞, 0]nonnegative
• A011973 → basis A013610 → D+1
  1 · 1 · 5, 3 · 9, 6 … Σ 1, 1, 8, 15, 62, 149, 484, 1275, …
  (−∞, 0]nonnegative
• A038303 → D
  1 · 20, 2 · 300, 60, 3 · 4000, 1200, 120, 4 … Σ 0, 1, 22, 363, 5324, 73205, 966306, …
  (−∞, 0]nonnegativerational roots
• A370258 → x ↦ x+1
  26, 40, 15 · 190, 441, 336, 84 … Σ 81, 1051, 14353, 201807, 2891409, …
  (−∞, −1]nonnegative
• A008292 → γ (scale 1/4) (shift +1) → θ+1 → x ↦ x+1
  1 · 2 · 8, 4 · 40, 32 … Σ 1, 2, 12, 72, 560, 4960, 50624, 579712, …
  (−∞, −1]nonnegative
• A027465 → basis A013611 → D+1
  1 · 8, 4 · 48, 64, 16 · 256, 576, 384, 64 … Σ 1, 12, 128, 1280, 12288, 114688, …
  (−∞, 0]nonnegativerational roots
• A245962 → (θ+1)²
  1 · 1 · 3, 8 · 4, 12 … Σ 1, 1, 11, 16, 57, 116, 305, 680, …
  (−∞, 0]nonnegative
• A085880 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 8, 40, 50 · 40, 300, 750, 625 … Σ 1, 7, 98, 1715, 33614, 705894, …
  [−1, 0]nonnegativerational roots
• A114655 → x ↦ x+1 → reverse
  2 · 2, 6 · 2, 16, 22 · 2, 30, 102, 90 … Σ 2, 8, 40, 224, 1344, 8448, 54912, …
  (−∞, 0]nonnegative
• A081582 → θ+1
  1 · 1, 2 · 1, 18, 3 · 1, 34, 51, 4 … Σ 1, 3, 22, 90, 447, 1925, 8572, 36612, …
  (−∞, 0]nonnegative
• A008292 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 24, 36, 14, 1 … Σ 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  (−∞, −1]nonnegative
• A038284 → basis A013611 → reverse
  576, 672, 196 · 13824, 24192, 14112, 2744 … Σ 1444, 54872, 2085136, 79235168, …
  (−∞, 0]nonnegativerational roots
• A003506 → x ↦ x+1 → θ+1
  1 · 4, 4 · 12, 24, 9 · 32, 96, 72, 16 … Σ 0, 1, 8, 45, 216, 945, 3888, 15309, …
  (−∞, 0]nonnegativerational roots
• A290595 → x ↦ x/(x+1)
  1 · 1, 3 · 4, 27, 27 · 28, 306, 675, 405 … Σ 1, 4, 58, 1414, 48334, 2125246, …
  [−1, 0]nonnegative
• A038220 → basis A007318 → reverse
  1 · 2, 5 · 4, 20, 25 · 8, 60, 150, 125 … Σ 1, 7, 49, 343, 2401, 16807, 117649, …
  (−∞, 0]nonnegativerational roots
• A008292 → x ↦ x+1 → x ↦ x+1
  1 · 3, 1 · 13, 8, 1 · 75, 67, 17, 1 … Σ 1, 4, 22, 160, 1456, 15904, 202672, …
  (−∞, −1]nonnegative
• A113413 → basis A013611 → x ↦ x+1
  1 · 7, 4 · 47, 56, 16 · 317, 572, 336, 64 … Σ 1, 11, 119, 1289, 13961, 151211, …
  (−∞, −1]nonnegative