Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A091044 → basis A013611 → D+1
  1 · 12, 8 · 80, 160, 48 · 448, 1664, 1408, 256 … Σ 1, 20, 288, 3776, 47360, 577536, …
  (−∞, 0]nonnegative
• A286718 → basis A013609 → (θ+1)²
  1 · 2, 8 · 10, 56, 36 · 80, 528, 540, 128 … Σ 1, 10, 102, 1276, 19372, 348384, …
  (−∞, 0]nonnegative
• A257617 → D
  2 · 36, 8 · 388, 776, 24 · 3676, 24832, 11028, 64 … Σ 0, 2, 44, 1188, 39600, 1584000, …
  (−∞, 0]nonnegative
• A081580 → basis A013611 → θ+1
  1 · 2, 8 · 9, 72, 48 · 28, 336, 768, 256 … Σ 1, 10, 129, 1388, 14837, 152526, …
  (−∞, 0]nonnegative
• A109692 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 8 · 1, 12, 44, 48 … Σ 1, 3, 15, 105, 945, 10395, 135135, …
  [−1, 0]nonnegativerational roots
• A054142 → Veronese 2 → D
  1 · 6 · 15, 2 · 28, 30 … Σ 0, 0, 1, 6, 17, 58, 188, 570, …
  (−∞, 0]nonnegative
• A134309 → D
  1 · 0, 4 · 0, 0, 12 · 0, 0, 0, 32 … Σ 0, 1, 4, 12, 32, 80, 192, 448, …
  (−∞, 0]nonnegativerational roots
• A199478 → D
  1 · 3 · 5, 4 · 7, 16 … Σ 1, 3, 9, 23, 57, 135, 313, 711, …
  (−∞, 0]nonnegative
• type_b_eulerian → θ+1
  1 · 1, 2 · 1, 12, 3 · 1, 46, 69, 4 … Σ 1, 3, 16, 120, 1152, 13440, 184320, …
  (−∞, 0]nonnegative
• A225476 → D+1
  1 · 2, 1 · 5, 8, 2 · 14, 49, 36, 6 … Σ 1, 3, 15, 105, 933, 10023, 126195, …
  (−∞, 0]nonnegative
• A063967 → θ+1
  1 · 1, 2 · 2, 6, 3 · 3, 14, 15, 4 … Σ 1, 3, 11, 36, 116, 364, 1124, 3424, …
  (−∞, 0]nonnegative
• A178822 → basis A013609 → D+1
  1 · 24, 12 · 252, 336, 84 · 1792, 4032, 2688, 448 … Σ 1, 36, 672, 8960, 96768, 903168, …
  (−∞, 0]nonnegativerational roots
• A125092 → reverse → x ↦ x+1
  1 · 5, 1 · 18, 10, 1 · 56, 54, 15, 1 … Σ 1, 6, 29, 126, 513, 1998, 7533, 27702, …
  (−∞, −1]nonnegative
• A178821 → basis A013611 → θ+1
  1 · 10, 40 · 60, 480, 720 · 280, 3360, 10080, 8960 … Σ 1, 50, 1260, 22680, 332640, 4245696, …
  (−∞, 0]nonnegativerational roots
• A038305 → basis A013609 → D
  156, 72 · 3042, 2808, 648 · 52728, 73008, 33696, 5184 … Σ 228, 6498, 164616, 3909630, 89139564, …
  (−∞, 0]nonnegativerational roots
• A249248 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 9, 11 · 1, 15, 49, 38 … Σ 1, 5, 21, 103, 519, 2793, 15645, 91623, …
  [−1, 0]nonnegative
• A027467 → basis A013609 → x ↦ x/(x+1)
  1 · 16, 18 · 256, 576, 324 · 4096, 13824, 15552, 5832 … Σ 1, 34, 1156, 39304, 1336336, 45435424, …
  [−1, 0]nonnegativerational roots
• A131431 → (θ+1)² → reverse
  1 · 16 · 63 · 160 … Σ 1, 16, 63, 160, 325, 576, 931, 1408, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A021012 → basis A008277 → x ↦ x/(x+1)
  1 · -1, -2 · 2, 10, 10 · -6, -60, -138, -90 … Σ 1, -3, 22, -294, 6168, -184680, …
  [−1, 0]
• A038279: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.
  1 · 8, 1 · 64, 16, 1 · 512, 192, 24, 1 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, −1]nonnegativerational roots
• A124038 → x ↦ x+1
  1 · -1, 1 · -2, 0, 1 · -1, -3, 1, 1 … Σ 1, 0, -1, -2, -3, -4, -5, -6, …
  (−3, 2)
• A074246 → x ↦ x+1
  1 · 5, 2 · 26, 18, 3 · 154, 142, 42, 4 … Σ 1, 7, 47, 342, 2754, 24552, 241128, …
  (−∞, −1]nonnegative
• A038208 → Veronese 2 (j=1) → D+1
  2 · 8 · 32, 8 · 128, 64 … Σ 0, 2, 8, 40, 192, 896, 4096, 18432, …
  (−∞, 0]nonnegative
• A001263 → basis A013611 → θ+1
  1 · 2, 8 · 5, 40, 48 · 14, 168, 432, 256 … Σ 1, 10, 93, 870, 8266, 79636, 775765, …
  (−∞, 0]nonnegative
• A143494 → x ↦ x+1
  1 · 3, 1 · 10, 7, 1 · 37, 40, 12, 1 … Σ 1, 4, 18, 90, 494, 2946, 18926, 130066, …
  (−∞, −1]nonnegative
• A243662 → θ+1
  1 · 3, 2 · 12, 16, 3 · 55, 110, 45, 4 … Σ 1, 5, 31, 214, 1570, 11969, 93659, …
  (−∞, 0]nonnegative
• A013611 → D → D+1
  4 · 40, 32 · 108, 480, 192 · 208, 1728, 3840, 1024 … Σ 0, 4, 72, 780, 6800, 52500, 375000, …
  (−∞, 0]nonnegativerational roots
• A110441 → reverse
  1 · 1, 3 · 1, 6, 7 · 1, 9, 23, 15 … Σ 1, 4, 14, 48, 164, 560, 1912, 6528, …
  (−∞, 0]nonnegative
• A038334 → x ↦ x+1 → (θ+1)²
  400, 1280, 576 · 8000, 38400, 34560, 8192 … Σ 2256, 89152, 3324160, 119067648, …
  (−∞, 0]nonnegative
• A178821 → basis A013610 → x ↦ x/(x+1)
  1 · 10, 25 · 60, 300, 375 · 280, 2100, 5250, 4375 … Σ 1, 35, 735, 12005, 168070, 2117682, …
  [−1, 0]nonnegativerational roots
• A059364 → basis A013610 → x ↦ x+1
  1 · 6, 3 · 66, 69, 18 · 1056, 1698, 909, 162 … Σ 1, 9, 153, 3825, 126225, 5175225, …
  (−∞, −1]nonnegativerational roots
• A121574 → basis A013611 → x ↦ x/(x+1)
  1 · 3, 7 · 10, 48, 54 · 33, 239, 555, 413 … Σ 1, 10, 112, 1240, 13744, 152320, …
  [−1, 0]nonnegative
• A038226 → basis A007318 → basis A007318
  361, 304, 64 · 6859, 8664, 3648, 512 … Σ 729, 19683, 531441, 14348907, …
  (−∞, −1]nonnegativerational roots
• A114655 → reverse → x ↦ x+1
  2 · 6, 4 · 22, 28, 8 · 90, 168, 96, 16 … Σ 2, 10, 58, 370, 2514, 17850, 130890, …
  (−∞, −1]nonnegative
• A132774 → Veronese 2 → (θ+1)²
  1 · 2 · 0, 20 · 0, 24 … Σ 1, 2, 20, 24, 81, 90, 208, 224, …
  (−∞, 0]nonnegativerational roots
• A185410 → reverse → D
  1 · 2, 2 · 4, 20, 3 · 8, 120, 108, 4 … Σ 0, 1, 4, 27, 240, 2625, 34020, 509355, …
  (−∞, 0]nonnegative
• A130123 → Veronese 2 (j=1) → (θ+1)²
  2 · 0, 32 · 0, 0, 288 · 0, 0, 0, 2048 … Σ 0, 2, 0, 32, 0, 288, 0, 2048, …
  (−∞, 0]nonnegativerational roots
• A172431 → basis A013610 → θ+1
  1 · 3, 12 · 8, 60, 81 · 21, 228, 594, 432 … Σ 1, 15, 149, 1275, 10177, 78183, 586349, …
  (−∞, 0]nonnegative
• A053117 → x ↦ x+1 → θ+1
  1 · 2, 4 · 3, 16, 12 · 4, 40, 72, 32 … Σ 1, 6, 31, 148, 673, 2962, 12735, 53800, …
  (−∞, 0]nonnegative
• A249248 → reverse
  1 · 3, 1 · 3, 7, 1 · 3, 22, 12, 1 … Σ 1, 4, 11, 38, 131, 490, 1897, 7714, …
  (−∞, 0]nonnegative
• A060920 → reverse → x ↦ x+1
  1 · 3, 2 · 11, 15, 5 · 43, 88, 59, 13 … Σ 1, 5, 31, 203, 1345, 8933, 59359, …
  (−∞, −1]nonnegative
• Veronese parent of A109446 → x ↦ x+1
  1, 1 · 2, 1 · 4, 8, 5, 1 · 8, 12, 6, 1 … Σ 2, 3, 18, 27, 162, 243, 1458, 2187, …
  (−∞, −1]nonnegativerational roots
• A026729 → basis A008277 → (θ+1)²
  1 · 1, 4 · 2, 16, 9 · 3, 52, 72, 16 … Σ 1, 5, 27, 143, 790, 4567, 27722, …
  (−∞, 0]nonnegative
• A334218 → θ+1
  1 · 1 · 0, 4 · 0, 6, 9 … Σ 1, 1, 4, 15, 72, 420, 2880, 22680, …
  (−∞, 0]nonnegative
• A099040 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 8 · 6, 42, 72 · 16, 168, 576, 640 … Σ 1, 10, 120, 1400, 16400, 192000, …
  [−1, 0]nonnegative
• A120910 → basis A013609 → D+1
  3 · 15, 6 · 63, 60, 12 · 243, 378, 180, 24 … Σ 3, 21, 135, 825, 4875, 28125, 159375, …
  (−∞, 0]nonnegativerational roots
• A197208 → D+1
  150, 180, 30 · 660, 1800, 780, 60 … Σ 360, 3300, 28980, 248136, 2090592, …
  (−∞, 0]nonnegative
• A046716 → basis A013611 → basis A007318
  1 · 6, 4 · 41, 52, 16 · 356, 644, 368, 64 … Σ 1, 10, 109, 1432, 26497, 754894, …
  (−∞, −1]nonnegative
• A094638 → Veronese 2 → x ↦ x+1
  1 · 1 · 3, 2 · 12, 11 … Σ 1, 1, 5, 23, 167, 1267, 12607, 132265, …
  (−∞, −1]nonnegative
• basis-map coefficients of A026729, shifted → basis A007318 → D+1
  1 · -2, -1 · 3, 4, 1 · -4, -9, -6, -1 … Σ 1, -3, 8, -20, 48, -112, 256, -576, …
  (−∞, 0]rational roots