Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A167884 → (θ+1)²
  1 · 1, 4 · 1, 72, 9 · 1, 716, 1611, 16 … Σ 1, 5, 82, 2344, 87520, 4037040, …
  (−∞, 0]nonnegative
• A027467 → basis A013610 → D+1
  1 · 19, 3 · 352, 114, 9 · 6400, 3168, 513, 27 … Σ 1, 22, 475, 10108, 212629, 4430914, …
  (−∞, 0]nonnegativerational roots
• A013618 → basis A013610 → x ↦ x+1
  2025, 2970, 1089 · 91125, 200475, 147015, 35937 … Σ 6084, 474552, 37015056, 2887174368, …
  (−∞, −1]nonnegativerational roots
• A038208 → basis A013610 → D+1
  1 · 10, 6 · 64, 120, 36 · 352, 1152, 1080, 216 … Σ 1, 16, 220, 2800, 34000, 400000, …
  (−∞, 0]nonnegativerational roots
• A299989 → θ+1
  0, 2 · 0, 6, 12, 4 · 0, 18, 72, 88, 40, 6 … Σ 2, 22, 224, 2176, 20480, 188416, …
  (−∞, 0]nonnegative
• A008955 → basis A013609 → θ+1
  1 · 2, 4 · 10, 52, 48 · 100, 880, 1884, 1152 … Σ 1, 6, 110, 4016, 243376, 22063776, …
  (−∞, 0]nonnegative
• delannoy → basis A013611 → D+1
  1 · 6, 4 · 25, 52, 16 · 84, 328, 320, 64 … Σ 1, 10, 93, 796, 6509, 51582, 399665, …
  (−∞, 0]nonnegative
• A204420 → basis A007318 → x ↦ x+1
  1 · 2, 1 · 24, 18, 3 · 720, 660, 180, 15 … Σ 1, 3, 45, 1575, 99225, 9823275, …
  (−∞, −1]nonnegativerational roots
• A038255 → basis A013611 → (θ+1)²
  1 · 7, 16 · 49, 224, 144 · 343, 2352, 3024, 1024 … Σ 1, 23, 417, 6743, 101761, 1465431, …
  (−∞, 0]nonnegative
• A165253 → (θ+1)²
  1 · 1 · 1, 4 · 1, 12, 9 … Σ 1, 1, 5, 22, 86, 313, 1085, 3629, …
  (−∞, 0]nonnegative
• A038336 → x ↦ x+1 → θ+1
  484, 880, 300 · 10648, 29040, 19800, 4000 … Σ 1664, 63488, 2359296, 85983232, …
  (−∞, 0]nonnegativerational roots
• A062145 → reverse → (θ+1)²
  1 · 4, 4 · 10, 40, 9 · 20, 180, 162, 16 … Σ 1, 8, 59, 378, 2202, 12012, 62491, …
  (−∞, 0]nonnegative
• A347823 → D
  3 · 8, 10 · 15, 36, 21 · 24, 84, 96, 36 … Σ 0, 3, 18, 72, 240, 720, 2016, 5376, …
  (−∞, 0]nonnegativerational roots
• A257609 → D
  2 · 16, 8 · 88, 176, 24 · 416, 2112, 1248, 64 … Σ 0, 2, 24, 288, 3840, 57600, 967680, …
  (−∞, 0]nonnegative
• A038329 → x ↦ x+1 → (θ+1)²
  225, 360, 81 · 3375, 8100, 3645, 432 … Σ 666, 15552, 349920, 7663248, 164392416, …
  (−∞, 0]nonnegative
• A132166 → (θ+1)²
  336, 168, 9 · 4536, 4452, 567, 16 … Σ 513, 9571, 169516, 2901186, 48342979, …
  (−∞, 0]nonnegative
• narayana → γ → basis A007318
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 21, 61, 191, 603, …
  (−∞, −1]nonnegative
• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
  1 · 3 · 5 · 7 … Σ 1, 3, 5, 7, 9, 11, 13, 15, …
  (−∞, −1][−1, 0][−3−2√2, −3+2√2]+2
• A143494 → θ+1
  1 · 2, 2 · 4, 10, 3 · 8, 38, 27, 4 … Σ 1, 4, 17, 77, 372, 1915, 10481, 60814, …
  (−∞, 0]nonnegative
• A100861 → basis A013610 → x ↦ x+1
  1 · 1 · 5, 3 · 13, 9 … Σ 1, 1, 8, 22, 190, 806, 7456, 41308, …
  (−∞, −1]nonnegative
• A122542 → D+1 → θ+1
  1 · 1, 2 · 2, 8, 3 · 2, 20, 21, 4 … Σ 1, 3, 13, 47, 161, 523, 1637, 4975, …
  (−∞, 0]nonnegative
• A100861 → basis A013611 → D+1
  1 · 1 · 6, 4 · 16, 12 … Σ 1, 1, 10, 28, 250, 1066, 9436, 51640, …
  (−∞, 0]nonnegative
• A144441 → θ+1
  1 · 1, 2 · 1, 28, 3 · 1, 166, 249, 4 … Σ 1, 3, 32, 420, 7248, 154560, 3924736, …
  (−∞, 0]nonnegative
• A159854 → basis A013611 → basis A007318
  25, 40, 16 · 150, 340, 256, 64 … Σ 81, 810, 8100, 81000, 810000, 8100000, …
  (−∞, −1]nonnegativerational roots
• A132166 → θ+1
  336, 84, 3 · 4536, 2226, 189, 4 … Σ 423, 6955, 108134, 1635402, 24367069, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/4) → θ+1 → x ↦ x+1
  1 · 2 · 8, 4 · 40, 32 … Σ 1, 2, 12, 72, 560, 4960, 50624, 579712, …
  (−∞, −1]nonnegative
• A013610 → basis A013611 → D+1
  1 · 16, 12 · 112, 384, 144 · 640, 4032, 6912, 1728 … Σ 1, 28, 640, 13312, 262144, 4980736, …
  (−∞, 0]nonnegativerational roots
• A143491 → basis A007318 → D
  7, 2 · 47, 24, 3 · 342, 238, 54, 4 … Σ 9, 74, 638, 5944, 60216, 662640, …
  (−∞, 0]nonnegative
• A038291 → basis A013609
  1 · 10, 2 · 100, 40, 4 · 1000, 600, 120, 8 … Σ 1, 12, 144, 1728, 20736, 248832, …
  (−∞, −1]nonnegativerational roots
• A123160 → θ+1
  1 · 1, 2 · 1, 8, 9 · 1, 18, 54, 40 … Σ 1, 3, 18, 113, 708, 4407, 27254, …
  (−∞, 0]nonnegative
• Veronese parent of A098172 → D+1
  1 · 0, 2, 4, 1 · 0, 0, 0, 4, 11, 8, 1 … Σ 1, 7, 24, 68, 176, 432, 1024, 2368, …
  (−∞, 0]nonnegative
• A038330 → basis A013610 → θ+1
  256, 768, 432 · 4096, 18432, 20736, 6912 … Σ 1456, 50176, 1668352, 54089728, …
  (−∞, 0]nonnegativerational roots
• A116412 → θ+1
  1 · 3, 2 · 6, 12, 3 · 12, 42, 27, 4 … Σ 1, 5, 21, 85, 332, 1266, 4741, 17507, …
  (−∞, 0]nonnegative
• A136124 → x ↦ x+1
  1 · 3, 1 · 12, 7, 1 · 60, 47, 12, 1 … Σ 1, 4, 20, 120, 840, 6720, 60480, …
  (−∞, −1]nonnegativerational roots
• A391045 → (θ+1)²
  1 · 1, 4 · 1, 16, 9 · 1, 36, 90, 16 … Σ 1, 5, 26, 143, 788, 4332, 23735, …
  (−∞, 0]nonnegative
• A102547 → reverse → basis A007318
  1, 1 · 1, 2, 1 · 1, 3, 3, 1 · 2, 7, 9, 5, 1 … Σ 2, 4, 8, 24, 64, 160, 416, 1088, …
  (−∞, −1]nonnegative
• A347823 → D+1
  1 · 5, 3 · 11, 18, 5 · 19, 51, 39, 7 … Σ 1, 8, 34, 116, 352, 992, 2656, 6848, …
  (−∞, 0]nonnegative
• A132393 → basis A013610 → (θ+1)²
  1 · 1, 12 · 2, 36, 81 · 6, 132, 486, 432 … Σ 1, 13, 119, 1056, 9804, 97476, 1044756, …
  (−∞, 0]nonnegative
• A266213 → basis A013610 → D+1
  1 · 1 · 9, 6 · 31, 60, 18 … Σ 1, 1, 15, 109, 779, 5321, 35367, …
  (−∞, 0]nonnegative
• A145597 → γ → x ↦ x+1
  1 · 3 · 9, 3 · 25, 15 … Σ 1, 3, 12, 40, 141, 483, 1680, 5832, …
  (−∞, −1]nonnegative
• A156992 → basis A008277 → D
  2 · 30, 12 · 456, 432, 72 · 7800, 13200, 5040, 480 … Σ 0, 2, 42, 960, 26520, 893520, 36378720, …
  (−∞, 0]nonnegative
• A123519 → basis A013609 → D
  28, 32 · 152, 352, 192 · 744, 2592, 2880, 1024 … Σ 60, 696, 7240, 70860, 666996, 6110320, …
  (−∞, 0]nonnegative
• basis-map coefficients of A371080, shifted → basis A013609 → x ↦ x/(x+1)
  1 · 1, 3 · 4, 18, 18 · 28, 162, 288, 162 … Σ 1, 4, 40, 640, 14080, 394240, 13404160, …
  [−1, 0]nonnegativerational roots
• A131431 → D+1 → (θ+1)²
  1 · 4, 16 · 0, 56, 63 · 0, 0, 270, 160 … Σ 1, 20, 119, 430, 1157, 2576, 5035, …
  (−∞, 0]nonnegativerational roots
• A060920 → x ↦ x+1
  1 · 3, 1 · 11, 7, 1 · 43, 41, 12, 1 … Σ 1, 4, 19, 97, 508, 2683, 14209, 75316, …
  (−∞, −1]nonnegative
• A102365 → x ↦ x+1 → x ↦ x+1
  1 · 1 · 3, 1 · 15, 9, 1 … Σ 1, 1, 4, 25, 217, 2416, 32839, 527185, …
  (−∞, −1]nonnegative
• A131436: Triangle read by rows, (n-1) zeros followed by 2^n - 1.
  1 · 0, 3 · 0, 0, 7 · 0, 0, 0, 15 … Σ 1, 3, 7, 15, 31, 63, 127, 255, …
  [−1, 0]nonnegativerational roots
• A038338 → x ↦ x/(x+1)
  144, 576, 576 · 1728, 10368, 20736, 13824 … Σ 1296, 46656, 1679616, 60466176, …
  [−1, 0]nonnegativerational roots
• A257610 → x ↦ x/(x+1)
  1 · 2, 4 · 4, 28, 28 · 8, 156, 420, 280 … Σ 1, 6, 60, 864, 16200, 373896, 10248120, …
  [−1, 0]nonnegative
• A145324 → basis A013610 → D
  6 · 51, 108 · 399, 1764, 1944 … Σ 0, 6, 159, 4107, 116082, 3653892, …
  (−∞, 0]nonnegative