Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A143685 → γ → D
  9 · 18 · 27, 162 · 36, 486 … Σ 0, 0, 9, 18, 189, 522, 3204, 10422, …
  (−∞, 0]nonnegative
• A111006 → basis A007318
  1 · 1, 1 · 3, 5, 2 · 5, 13, 11, 3 … Σ 1, 2, 10, 32, 124, 440, 1624, 5888, …
  (−∞, −1]nonnegative
• A013610 → Veronese 2 (j=1) → x ↦ x+1
  3 · 6 · 36, 27 · 120, 108 … Σ 0, 3, 6, 63, 228, 1527, 6930, 39819, …
  (−∞, −1]nonnegative
• A145597 → γ → D+1
  1 · 3 · 9, 3 · 25, 15 … Σ 1, 3, 12, 40, 132, 420, 1316, 4068, …
  (−∞, 0]nonnegative
• A115951 → basis A007318
  1 · 2, 2 · 8, 14, 6 · 32, 84, 72, 20 … Σ 1, 4, 28, 208, 1624, 13024, 106336, …
  (−∞, −1]nonnegative
• A108426 → D
  1 · 5, 4 · 28, 42, 15 · 165, 360, 252, 56 … Σ 0, 1, 9, 85, 833, 8361, 85305, 880685, …
  (−∞, 0]nonnegative
• A038325 → basis A013609 → θ+1
  1 · 22, 44 · 484, 1936, 1452 … Σ 1, 66, 3872, 212960, 11244288, …
  (−∞, 0]nonnegativerational roots
• A133214 → D
  8, 8 · 18, 72, 24 · 32, 288, 384, 64 … Σ 16, 114, 768, 5010, 32016, 201698, …
  (−∞, 0]nonnegative
• A027467 → D
  1 · 30, 2 · 675, 90, 3 · 13500, 2700, 180, 4 … Σ 0, 1, 32, 768, 16384, 327680, 6291456, …
  (−∞, 0]nonnegativerational roots
• A027467 → x ↦ x+1 → (θ+1)²
  1 · 16, 4 · 256, 128, 9 · 4096, 3072, 432, 16 … Σ 1, 20, 393, 7616, 145945, 2770932, …
  (−∞, 0]nonnegative
• A049460: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 4.
  1 · -5, 1 · 30, -11, 1 · -210, 107, -18, 1 … Σ 1, -4, 20, -120, 840, -6720, 60480, …
  [9/2, ∞)rational roots
• A062145 → x ↦ x/(x+1)
  1 · 1, 5 · 1, 12, 21 · 1, 21, 84, 84 … Σ 1, 6, 34, 190, 1059, 5908, 33028, …
  [−1, 0]nonnegative
• A038284 → basis A013611 → D
  672, 1152 · 14112, 48384, 41472 … Σ 1824, 103968, 5267712, 250216320, …
  (−∞, 0]nonnegativerational roots
• A048993 → θ+1 → D+1
  1 · 2, 2 · 2, 8, 3 · 2, 20, 21, 4 … Σ 1, 4, 13, 47, 188, 825, 3937, 20270, …
  (−∞, 0]nonnegative
• A286785 → reverse
  2, 14, 14 · 2, 27, 72, 42 · 2, 44, 220, 330, 132 … Σ 30, 143, 728, 3876, 21318, 120175, …
  (−∞, 0]nonnegative
• A136572 → Veronese 2 (j=1) → D+1
  1 · 6, 6 · 0, 240, 120 · 0, 0, 15120, 5040 … Σ 0, 1, 0, 12, 0, 360, 0, 20160, …
  (−∞, 0]nonnegativerational roots
• A103406 → Veronese 2 → D
  3 · 9 · 18, 10 · 30, 50 … Σ 0, 0, 3, 9, 28, 80, 216, 560, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → θ+1 → x ↦ x+1
  1 · 3, 6, 3 · 5, 20, 30, 20, 5 … Σ 1, 12, 80, 448, 2304, 11264, 53248, …
  (−∞, −1]nonnegativerational roots
• A027465 → Veronese 2 (j=1) → θ+1
  1 · 6 · 27, 2 · 108, 24 … Σ 0, 1, 6, 29, 132, 588, 2592, 11344, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A013609 → x ↦ x+1
  1 · 9, 12, 4 · 81, 216, 216, 96, 16 … Σ 1, 25, 625, 15625, 390625, 9765625, …
  (−∞, −1]nonnegativerational roots
• A075497 → D+1 → D
  1 · 8, 2 · 52, 30, 3 · 320, 320, 72, 4 … Σ 0, 1, 10, 85, 716, 6221, 56526, 539785, …
  (−∞, 0]nonnegative
• Veronese parent of A098172 → (θ+1)²
  0, 0, 9 · 0, 0, 0, 0, 25, 36 … Σ 9, 61, 258, 888, 2720, 7728, 20832, …
  (−∞, 0]nonnegative
• A159854 → basis A013609 → D+1
  5, 12, 4 · 12, 42, 40, 8 · 28, 128, 196, 112, 16 … Σ 21, 102, 480, 2208, 9984, 44544, …
  (−∞, 0]nonnegative
• A274498 → D → D+1
  6 · 18 · 108, 72 · 324, 216 … Σ 0, 0, 6, 18, 180, 540, 3402, 10206, …
  (−∞, 0]nonnegativerational roots
• A391533 → basis A013609 → (θ+1)²
  9, 48, 36 · 18, 168, 288, 128 … Σ 93, 602, 3528, 19392, 101888, 517632, …
  (−∞, 0]nonnegative
• A013609 → Veronese 2 → D+1
  1 · 1 · 5, 4 · 13, 12 … Σ 1, 1, 9, 25, 97, 321, 1097, 3641, …
  (−∞, 0]nonnegative
• A178821 → basis A007318 → x ↦ x+1
  1 · 15, 5 · 135, 90, 15 · 945, 945, 315, 35 … Σ 1, 20, 240, 2240, 17920, 129024, …
  (−∞, −1]nonnegativerational roots
• A200536 → θ+1
  1 · 1, 6, 6 · 1, 12, 39, 48, 20 … Σ 1, 13, 120, 972, 7344, 53136, 373248, …
  (−∞, 0]nonnegative
• A038321 → θ+1
  121, 308, 147 · 1331, 5082, 4851, 1372 … Σ 576, 12636, 268272, 5563728, 113374080, …
  (−∞, 0]nonnegativerational roots
• A144084 → θ+1
  1 · 1, 2 · 1, 8, 6 · 1, 18, 54, 24 … Σ 1, 3, 15, 97, 753, 6771, 68983, 783945, …
  (−∞, 0]nonnegative
• A038331 → basis A013611 → reverse
  400, 680, 289 · 8000, 20400, 17340, 4913 … Σ 1369, 50653, 1874161, 69343957, …
  (−∞, 0]nonnegativerational roots
• A272866 → basis A013610 → reverse
  1 · 9, 15, 5 · 81, 270, 315, 150, 25 … Σ 1, 29, 841, 24389, 707281, 20511149, …
  (−∞, 0]nonnegative
• A257609 → D (shift +2) → x ↦ x+1
  36, 28, 3 · 240, 300, 90, 4 … Σ 67, 634, 6793, 81780, 1095967, …
  (−∞, −1]nonnegative
• A059473 → basis A013610 → reverse
  1 · 6, 4 · 36, 54, 18 · 216, 504, 360, 80 … Σ 1, 10, 108, 1160, 12464, 133920, …
  (−∞, 0]nonnegative
• A086645 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 8, 8 · 1, 18, 48, 32 … Σ 1, 3, 17, 99, 577, 3363, 19601, 114243, …
  [−1, 0]nonnegative
• A122076 → x ↦ x+1
  2 · 5, 2 · 17, 12, 2 · 65, 66, 21, 2 … Σ 2, 7, 31, 154, 799, 4207, 22258, …
  (−∞, −1]nonnegative
• A063967 → x ↦ x−10
  1 · -9, 1 · 72, -17, 1 · -567, 207, -25, 1 … Σ 1, -8, 56, -384, 2624, -17920, 122368, …
  (5, 9]
• A249139 → basis A013611 → x ↦ x+1
  1 · 8, 4 · 15, 8 · 71, 68, 16 … Σ 1, 12, 23, 155, 408, 2113, 6601, 29844, …
  (−∞, −1]nonnegative
• A034867 → D+1
  1 · 2 · 4, 1 · 8, 4 … Σ 1, 2, 5, 12, 28, 64, 144, 320, …
  (−∞, 0]nonnegative
• A013611 → D → basis A007318
  4 · 40, 32 · 300, 480, 192 · 2000, 4800, 3840, 1024 … Σ 0, 4, 72, 972, 11664, 131220, 1417176, …
  (−∞, −1]nonnegativerational roots
• A143491 → basis A008277 → (θ+1)²
  12, 32, 9 · 60, 240, 135, 16 … Σ 53, 451, 4309, 45851, 538537, 6924583, …
  (−∞, 0]nonnegative
• A081579 → basis A013610 → x ↦ x+1
  1 · 5, 3 · 41, 42, 9 · 285, 441, 207, 27 … Σ 1, 8, 92, 960, 10256, 108928, 1158592, …
  (−∞, −1]nonnegative
• A269939 → x ↦ x+1
  1 · 1, 1 · 4, 7, 3 · 26, 66, 55, 15 … Σ 1, 2, 14, 162, 2622, 54546, 1386702, …
  (−2, −1]nonnegative
• A130534 → D+1 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 17, 74, 106, 50 … Σ 1, 5, 31, 247, 2433, 28653, 393279, …
  [−1, 0]nonnegative
• A122542 → D+1 → reverse
  1 · 1, 1 · 1, 4, 2 · 1, 7, 10, 2 … Σ 1, 2, 7, 20, 57, 158, 431, 1160, …
  (−∞, 0]nonnegative
• A091042 → basis A007318 → x ↦ x/(x+1)
  1 · 4, 7 · 16, 52, 41 · 64, 304, 472, 239 … Σ 1, 11, 109, 1079, 10681, 105731, …
  [−1, 0]nonnegative
• A081580 → basis A007318 → x ↦ x+1
  1 · 3, 1 · 19, 11, 1 · 87, 77, 19, 1 … Σ 1, 4, 31, 184, 1201, 7564, 48271, …
  (−∞, −1]nonnegative
• A081579 → basis A013609 → θ+1
  1 · 2, 4 · 8, 32, 12 · 24, 144, 168, 32 … Σ 1, 6, 52, 368, 2640, 18208, 123968, …
  (−∞, 0]nonnegative
• A062190 → x ↦ x/(x+1)
  1 · 1, 7 · 1, 16, 36 · 1, 27, 135, 165 … Σ 1, 8, 53, 328, 1966, 11584, 67582, …
  [−1, 0]nonnegative
• A038208 → D
  2 · 8, 8 · 24, 48, 24 · 64, 192, 192, 64 … Σ 0, 2, 16, 96, 512, 2560, 12288, 57344, …
  (−∞, −1]nonnegativerational roots