Proof methods / preserver_image

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8,704 sequences

• A225470 → basis A013610 → basis A007318
  1 · 6, 3 · 54, 45, 9 · 648, 702, 243, 27 … Σ 1, 9, 108, 1620, 29160, 612360, …
  (−∞, −1]nonnegativerational roots
• A046716 → basis A013611 → D
  4 · 20, 32 · 100, 352, 192 · 576, 3424, 5376, 1024 … Σ 0, 4, 52, 644, 10400, 237060, 7060484, …
  (−∞, 0]nonnegative
• A200536 → basis A013610 → D
  21, 36 · 252, 1314, 2268, 1296 … Σ 0, 57, 5130, 346275, 20776500, …
  (−∞, 0]nonnegativerational roots
• A030528 → basis A013611 → D
  4 · 16, 32 · 40, 192, 192 · 88, 672, 1536, 1024 … Σ 0, 4, 48, 424, 3320, 24340, 171200, …
  (−∞, 0]nonnegative
• A038284 → basis A013610 → basis A007318
  1024, 1152, 324 · 32768, 55296, 31104, 5832 … Σ 2500, 125000, 6250000, 312500000, …
  (−∞, −1]nonnegativerational roots
• A199881 → (θ+1)²
  1, 4 · 0, 4, 9 · 0, 4, 18, 16 · 0, 0, 18, 48, 25 … Σ 5, 13, 38, 91, 209, 452, 945, 1917, …
  (−∞, 0]nonnegative
• A257624 → (θ+1)²
  1 · 5, 20 · 25, 320, 225 · 125, 3660, 8235, 2000 … Σ 1, 25, 570, 14020, 378820, 11238240, …
  (−∞, 0]nonnegative
• A225471 → basis A007318 → D
  1 · 12, 2 · 176, 48, 3 · 3200, 1120, 120, 4 … Σ 0, 1, 14, 227, 4444, 103269, 2790570, …
  (−∞, 0]nonnegative
• A227450: Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.
  2 · 2, 4 · 2, 12, 8 · 2, 28, 48, 16 … Σ 2, 6, 22, 94, 454, 2430, 14214, 89918, …
  (−∞, 0)nonnegative
• A081579 → x ↦ x+1 → basis A007318
  1 · 3, 1 · 17, 10, 1 · 75, 67, 17, 1 … Σ 1, 4, 28, 160, 976, 5824, 35008, …
  (−∞, −1]nonnegative
• A356546 → D+1
  1 · 4, 2 · 18, 24, 6 · 80, 180, 120, 20 … Σ 1, 6, 48, 400, 3360, 28224, 236544, …
  (−∞, 0]nonnegativerational roots
• A257608 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 22, 22 · 1, 222, 660, 440 … Σ 1, 3, 45, 1323, 55485, 3042603, …
  [−1, 0]nonnegative
• A113413 → basis A013609 → D+1
  1 · 5, 2 · 19, 20, 4 · 63, 118, 60, 8 … Σ 1, 7, 43, 249, 1385, 7495, 39747, …
  (−∞, 0]nonnegative
• A128100 → Veronese 2 → x ↦ x+1
  1 · 1 · 2 · 3 … Σ 1, 1, 2, 3, 7, 14, 31, 65, …
  (−∞, −1]nonnegative
• A144696 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 9, 12 · 1, 21, 72, 60 … Σ 1, 4, 22, 154, 1306, 12994, 148282, …
  [−1, 0]nonnegative
• A178820 → basis A013611 → x ↦ x/(x+1)
  1 · 8, 24 · 40, 240, 360 · 160, 1440, 4320, 4320 … Σ 1, 32, 640, 10240, 143360, 1835008, …
  [−1, 0]nonnegativerational roots
• A091042 → basis A013609 → D+1
  1 · 10, 6 · 56, 80, 20 · 288, 672, 392, 56 … Σ 1, 16, 156, 1408, 12304, 105216, …
  (−∞, 0]nonnegative
• A038305 → basis A013610 → basis A007318
  484, 396, 81 · 10648, 13068, 5346, 729 … Σ 961, 29791, 923521, 28629151, …
  (−∞, −1]nonnegativerational roots
• A172431 → basis A013609 → basis A007318
  1 · 7, 4 · 40, 44, 12 · 217, 348, 184, 32 … Σ 1, 11, 96, 781, 6191, 48576, 379561, …
  (−∞, −1]nonnegative
• A207606 → (θ+1)²
  4, 28, 18 · 5, 64, 99, 32 · 6, 120, 324, 240, 50 … Σ 50, 200, 740, 2597, 8770, 28759, 92146, …
  (−∞, 0]nonnegative
• A013613 → x ↦ x+1 → (θ+1)²
  1 · 7, 24 · 49, 336, 324 · 343, 3528, 6804, 3456 … Σ 1, 31, 709, 14131, 259753, 4523623, …
  (−∞, 0]nonnegative
• A111125 → basis A013611 → D
  4 · 28, 32 · 124, 320, 192 · 460, 1920, 2496, 1024 … Σ 0, 4, 60, 636, 5900, 51000, 421944, …
  (−∞, 0]nonnegative
• Veronese parent of A098172 → D+1
  0, 2, 1 · 0, 0, 0, 4, 6, 1 … Σ 3, 11, 32, 84, 208, 496, 1152, 2624, …
  (−∞, 0]nonnegative
• A034870 → x ↦ x/(x+1)
  1 · 1, 4, 4 · 1, 8, 24, 32, 16 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  [−1, 0]nonnegativerational roots
• A094638 → θ+1 → basis A007318
  1 · 3, 2 · 13, 18, 6 · 70, 150, 105, 24 … Σ 1, 5, 37, 349, 3981, 53241, 816873, …
  (−∞, −1]nonnegative
• A038335 → basis A013609 → x ↦ x/(x+1)
  1 · 21, 39 · 441, 1638, 1521 … Σ 1, 60, 3600, 216000, 12960000, …
  [−1, 0]nonnegativerational roots
• A038259 → basis A007318 → D+1
  1 · 16, 5 · 231, 160, 25 · 3146, 3465, 1200, 125 … Σ 1, 21, 416, 7936, 147456, 2686976, …
  (−∞, 0]nonnegativerational roots
• A114655 → reverse → θ+1
  2 · 2, 8 · 2, 24, 24 · 2, 48, 144, 64 … Σ 2, 10, 50, 258, 1362, 7306, 39650, …
  (−∞, 0]nonnegative
• A085881 → basis A007318 → D
  1 · 12, 6 · 180, 180, 45 · 3360, 5040, 2520, 420 … Σ 0, 1, 18, 405, 11340, 382725, 15155910, …
  (−∞, 0]nonnegativerational roots
• A116414 → basis A013611 → D
  4 · 40, 32 · 276, 480, 192 · 1640, 4512, 3840, 1024 … Σ 0, 4, 72, 948, 11016, 119664, 1246104, …
  (−∞, 0]nonnegative
• A136394 → D
  1 · 5 · 20, 6 · 84, 70 … Σ 0, 0, 1, 5, 26, 154, 1044, 8028, …
  (−∞, 0]nonnegative
• A137320 → D+1
  1 · 2, 2 · 2, 10, 4 · 4, 28, 36, 8 … Σ 1, 4, 16, 76, 428, 2808, 21096, 178848, …
  (−∞, 0]nonnegative
• A330381 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 9 · 1, 12, 34, 27 … Σ 1, 4, 17, 74, 325, 1432, 6317, 27878, …
  [−1, 0]nonnegative
• A075497 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 3, 4 · 11, 30, 20 · 49, 202, 272, 120 … Σ 1, 7, 61, 643, 7897, 109855, 1698517, …
  [−1, 0]nonnegative
• A038279 → basis A013611 → D
  4 · 72, 32 · 972, 864, 192 · 11664, 15552, 6912, 1024 … Σ 0, 4, 104, 2028, 35152, 571220, …
  (−∞, 0]nonnegativerational roots
• A116412: Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
  1 · 3, 1 · 6, 6, 1 · 12, 21, 9, 1 … Σ 1, 4, 13, 43, 142, 469, 1549, 5116, …
  (−∞, 0)nonnegative
• A035607 → basis A013611 → x ↦ x+1
  1 · 1 · 11, 8 · 71, 96, 32 … Σ 1, 1, 19, 199, 2161, 23401, 253459, …
  (−∞, −1]nonnegative
• A253283 → x ↦ x+1
  1 · 1, 1 · 5, 8, 3 · 25, 57, 42, 10 … Σ 1, 2, 16, 134, 1168, 10442, 94960, …
  (−∞, −1]nonnegative
• A173018 → x ↦ x+1
  1 · 1 · 2, 1 · 6, 6, 1 … Σ 1, 1, 3, 13, 75, 541, 4683, 47293, …
  (−∞, −1]nonnegative
• A033282 → basis A013609 → D+1
  1 · 7, 4 · 41, 70, 20 · 231, 690, 588, 112 … Σ 1, 11, 131, 1621, 20525, 263927, …
  (−∞, 0]nonnegative
• A094816 → basis A013609 → reverse
  1 · 2, 2 · 4, 10, 5 · 8, 36, 46, 16 … Σ 1, 4, 19, 106, 685, 5056, 42079, …
  (−∞, 0]nonnegative
• A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
  1 · 0, 2 · 0, 0, 3 · 0, 0, 0, 4 … Σ 1, 2, 3, 4, 5, 6, 7, 8, …
  [−1, 0]nonnegativerational roots
• A038314 → (θ+1)²
  100, 960, 1296 · 1000, 14400, 38880, 27648 … Σ 2356, 81928, 2603920, 77985952, …
  (−∞, 0]nonnegative
• A109983 → x ↦ x/(x+1)
  1 · 0, 1, 3 · 0, 0, 1, 8, 13 · 0, 0, 0, 1, 15, 57, 63 … Σ 1, 4, 22, 136, 886, 5944, 40636, …
  [−1, 0]nonnegative
• A136572 → Veronese 2 → D
  2 · 0, 48 · 0, 0, 2160 · 0, 0, 0, 161280 … Σ 0, 0, 2, 0, 48, 0, 2160, 0, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A371080, shifted → basis A013611 → D+1
  1 · 5, 4 · 24, 52, 16 · 184, 540, 384, 64 … Σ 1, 9, 92, 1172, 18168, 333496, 7088800, …
  (−∞, 0]nonnegative
• A038231 → Veronese 2 (j=1) → D
  1 · 16 · 160, 2 · 1280, 48 … Σ 0, 0, 0, 1, 16, 162, 1328, 9635, …
  (−∞, 0]nonnegative
• A054142 → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 1, 20, 54, 16 … Σ 1, 5, 22, 91, 349, 1260, 4349, 14509, …
  (−∞, 0]nonnegative
• A391533 → x ↦ x+1 → D+1
  15, 8, 1 · 39, 37, 11, 1 · 96, 134, 67, 14, 1 … Σ 24, 88, 312, 1080, 3672, 12312, 40824, …
  (−∞, 0]nonnegative
• A225467 → x ↦ x/(x+1)
  1 · 3, 7 · 9, 58, 65 · 27, 397, 1049, 743 … Σ 1, 10, 132, 2216, 44560, 1030304, …
  [−1, 0]nonnegative