Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A038255 → basis A013610 → reverse
  1 · 3, 7 · 9, 42, 49 · 27, 189, 441, 343 … Σ 1, 10, 100, 1000, 10000, 100000, …
  (−∞, 0]nonnegativerational roots
• A075500 → x ↦ x/(x+1)
  25, 65, 41 · 125, 550, 755, 331 … Σ 131, 1761, 27601, 506651, 10674211, …
  [−1, 0]nonnegative
• A137597 → D+1 → D+1
  1 · 4, 1 · 17, 9, 1 · 77, 64, 15, 1 … Σ 1, 5, 27, 157, 979, 6517, 46107, …
  (−∞, −1]nonnegative
• A136394 → (θ+1)²
  1 · 1 · 1, 4 · 1, 20 … Σ 1, 1, 5, 21, 108, 652, 4532, 35732, …
  (−∞, 0]nonnegative
• A081577 → x ↦ x+1 → θ+1
  1 · 2, 2 · 6, 12, 3 · 16, 48, 30, 4 … Σ 1, 4, 21, 98, 455, 2064, 9241, 40894, …
  (−∞, 0]nonnegative
• A075499 → D
  12, 2 · 112, 48, 3 · 960, 800, 120, 4 … Σ 14, 163, 1884, 22821, 295210, 4095687, …
  (−∞, 0]nonnegative
• A114656 → basis A007318 → θ+1
  1 · 3, 2 · 11, 16, 3 · 45, 102, 45, 4 … Σ 1, 5, 30, 196, 1344, 9504, 68640, …
  (−∞, 0]nonnegative
• A008278 → x ↦ x+1 → θ+1
  1 · 2, 2 · 5, 10, 3 · 15, 46, 30, 4 … Σ 1, 4, 18, 95, 579, 3966, 29944, 246141, …
  (−∞, 0]nonnegative
• A143491 → basis A013611 → θ+1
  12, 56, 48 · 60, 376, 576, 256 … Σ 116, 1268, 14696, 181816, 2403552, …
  (−∞, 0]nonnegative
• A132393 → reverse
  1 · 1 · 1, 1 · 1, 3, 2 … Σ 1, 1, 2, 6, 24, 120, 720, 5040, …
  (−∞, 0]nonnegativerational roots
• A011973 → basis A008277 → x ↦ x+1
  1 · 1 · 3, 1 · 5, 2 … Σ 1, 1, 4, 7, 21, 46, 129, 317, …
  (−∞, −1]nonnegative
• A132166 → reverse
  1, 42, 336 · 1, 63, 1113, 4536 … Σ 379, 5713, 80032, 1080325, 14354557, …
  (−∞, 0]nonnegative
• A063967 → reverse → x ↦ x+1
  1 · 2, 1 · 6, 7, 2 · 16, 28, 16, 3 … Σ 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  (−∞, −1]nonnegative
• A062264 → reverse → (θ+1)²
  1 · 5, 4 · 15, 48, 9 · 35, 252, 189, 16 … Σ 1, 9, 72, 492, 3015, 17127, 92092, …
  (−∞, 0]nonnegative
• A269944 → (θ+1)²
  1 · 0, 4 · 0, 4, 9 · 0, 16, 45, 16 … Σ 1, 4, 13, 77, 834, 14838, 397208, …
  (−∞, 0]nonnegative
• A143495 → θ+1
  1 · 3, 2 · 9, 14, 3 · 27, 74, 36, 4 … Σ 1, 5, 26, 141, 799, 4736, 29371, …
  (−∞, 0]nonnegative
• narayana → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 1, 24, 54, 16 … Σ 1, 5, 22, 95, 406, 1722, 7260, 30459, …
  (−∞, 0]nonnegative
• A204420 → basis A013609 → x ↦ x/(x+1)
  1 · 1, 3 · 9, 42, 45 · 225, 1365, 2595, 1575 … Σ 1, 4, 96, 5760, 645120, 116121600, …
  [−1, 0]nonnegativerational roots
• A191935 → θ+1
  1 · 1, 4 · 1, 16, 12 · 1, 40, 156, 32 … Σ 1, 5, 29, 229, 2317, 28585, 421913, …
  (−∞, 0]nonnegative
• A075504 → D+1
  108, 29, 1 · 1296, 675, 57, 1 … Σ 138, 2029, 34871, 684402, 14974639, …
  (−∞, 0]nonnegative
• A091042 → reverse → D+1
  1 · 4, 1 · 15, 12, 1 · 42, 77, 24, 1 … Σ 1, 5, 28, 144, 704, 3328, 15360, 69632, …
  (−∞, 0]nonnegative
• A063007 → x ↦ x+1 → basis A007318
  1 · 5, 2 · 37, 30, 6 · 305, 372, 150, 20 … Σ 1, 7, 73, 847, 10321, 129367, 1651609, …
  (−3, −2)nonnegative
• A038221 → basis A013611 → D+1
  1 · 18, 12 · 180, 432, 144 · 1512, 6480, 7776, 1728 … Σ 1, 30, 756, 17496, 384912, 8188128, …
  (−∞, 0]nonnegativerational roots
• A122542 → reverse → (θ+1)²
  1 · 1 · 1, 8 · 1, 16, 18 … Σ 1, 1, 9, 35, 129, 437, 1409, 4375, …
  (−∞, 0]nonnegative
• A143494 → reverse → x ↦ x+1
  1 · 3, 2 · 10, 13, 4 · 37, 71, 43, 8 … Σ 1, 5, 27, 159, 1025, 7221, 55307, …
  (−∞, −1]nonnegative
• A196347 → basis A013609 → basis A007318
  1 · 4, 2 · 32, 32, 8 · 384, 576, 288, 48 … Σ 1, 6, 72, 1296, 31104, 933120, …
  (−∞, −1]nonnegativerational roots
• A121574 → basis A013610 → x ↦ x+1
  1 · 6, 3 · 40, 39, 9 · 264, 384, 180, 27 … Σ 1, 9, 88, 855, 8311, 80784, 785233, …
  (−∞, −1]nonnegative
• A008292 → γ (shift +1) → x ↦ x+1 → θ+1
  1 · 1 · 3, 4 · 9, 16 … Σ 1, 1, 7, 25, 195, 1245, 11543, 104657, …
  (−∞, 0]nonnegative
• A008278 → θ+1 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 8, 10 · 1, 15, 48, 38 … Σ 1, 4, 19, 102, 601, 3828, 26111, …
  [−1, 0]nonnegative
• A371081 → D+1
  1 · 17, 1 · 452, 54, 1 · 17348, 3180, 119, 1 … Σ 1, 18, 507, 20648, 1149645, 83935406, …
  (−∞, 0]nonnegative
• narayana → basis A007318 → x ↦ x+1
  1 · 3, 1 · 11, 7, 1 · 45, 42, 12, 1 … Σ 1, 4, 19, 100, 562, 3304, 20071, …
  (−∞, −1]nonnegative
• A249139: Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 3, 1 · 5, 2 · 11, 7, 1 … Σ 1, 4, 7, 19, 40, 97, 217, 508, …
  (−∞, −2]nonnegative
• A038207 → Veronese 2 (j=1) → θ+1
  1 · 4 · 12, 2 · 32, 16 … Σ 0, 1, 4, 14, 48, 163, 548, 1824, …
  (−∞, 0]nonnegative
• A038291 → basis A013609 → basis A013611
  1 · 12, 8 · 144, 192, 64 · 1728, 3456, 2304, 512 … Σ 1, 20, 400, 8000, 160000, 3200000, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A113214, shifted → x ↦ x+1 → θ+1
  1 · -2, -2 · 4, 8, 3 · -8, -24, -18, -4 … Σ 1, -4, 15, -54, 189, -648, 2187, -7290, …
  (−∞, 0]rational roots
• A056242 → reverse → θ+1
  1 · 2, 2 · 4, 10, 3 · 8, 32, 27, 4 … Σ 1, 4, 17, 71, 288, 1140, 4428, 16948, …
  (−∞, 0]nonnegative
• A111125 → x ↦ x/(x+1)
  1 · 3, 4 · 5, 15, 11 · 7, 35, 56, 29 … Σ 1, 7, 31, 127, 511, 2047, 8191, 32767, …
  [−1, 0)nonnegative
• A116414 → basis A013610 → θ+1
  1 · 5, 6 · 22, 60, 27 · 95, 414, 405, 108 … Σ 1, 11, 109, 1022, 9241, 81449, 704389, …
  (−∞, 0]nonnegative
• A038331 → x ↦ x+1 → reverse
  25, 170, 289 · 125, 1275, 4335, 4913 … Σ 484, 10648, 234256, 5153632, 113379904, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A113214, shifted → D+1
  1 · -2, -1 · 3, 4, 1 · -4, -9, -6, -1 … Σ 1, -3, 8, -20, 48, -112, 256, -576, …
  (−∞, 0]rational roots
• A026729 → basis A013611 → θ+1
  1 · 1, 8 · 2, 24, 48 · 3, 56, 240, 256 … Σ 1, 9, 74, 555, 3965, 27400, 184925, …
  (−∞, 0]nonnegative
• A122542 → basis A013609 → basis A007318
  1 · 3, 2 · 15, 16, 4 · 69, 106, 52, 8 … Σ 1, 5, 35, 235, 1585, 10685, 72035, …
  (−∞, −1]nonnegative
• A281260 → reverse
  1 · 2 · 3, 2 · 4, 8, 2 … Σ 1, 2, 5, 14, 42, 132, 429, 1430, …
  (−∞, 0]nonnegative
• A038325 → basis A013609 → D+1
  1 · 44, 22 · 1452, 1936, 484 … Σ 1, 66, 3872, 212960, 11244288, …
  (−∞, 0]nonnegativerational roots
• A178820 → basis A013609 → D+1
  1 · 16, 8 · 120, 160, 40 · 640, 1440, 960, 160 … Σ 1, 24, 320, 3200, 26880, 200704, …
  (−∞, 0]nonnegativerational roots
• A156366 → D+1
  13, 30, 9 · 34, 231, 180, 27 … Σ 52, 472, 5152, 65656, 956224, 15667072, …
  (−∞, 0]nonnegative
• A054142 → 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
• A200536 → basis A007318 → D
  7, 4 · 84, 146, 84, 16 … Σ 0, 11, 330, 7425, 148500, 2784375, …
  (−∞, 0]nonnegativerational roots
• A099097 → θ+1
  1 · 0, 6 · 0, 2, 27 · 0, 0, 18, 108 … Σ 1, 6, 29, 126, 516, 2034, 7807, 29382, …
  (−∞, 0]nonnegative
• A013612 → basis A013609 → D
  120, 200 · 1080, 3600, 3000 … Σ 320, 7680, 163840, 3276800, 62914560, …
  (−∞, 0]nonnegativerational roots