Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A227450 → reverse → x ↦ x+1
  2 · 6, 2 · 22, 16, 2 · 94, 110, 34, 2 … Σ 2, 8, 40, 240, 1664, 12992, 112256, …
  (−∞, −1]nonnegative
• A232535 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 10, 12 · 1, 21, 64, 48 … Σ 1, 4, 23, 134, 781, 4552, 26531, …
  [−1, 0]nonnegative
• A290319 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 12, 48, 45 · 120, 636, 1074, 585 … Σ 1, 7, 105, 2415, 74865, 2919735, …
  [−1, 0]nonnegativerational roots
• A038271 → x ↦ x/(x+1) → (θ+1)²
  1 · 7, 48 · 49, 672, 1296 · 343, 7056, 27216, 27648 … Σ 1, 55, 2017, 62263, 1741825, 45687799, …
  (−∞, 0]nonnegative
• A102662 → D+1
  1 · 4, 3 · 6, 11, 3 · 8, 29, 20, 3 … Σ 1, 7, 20, 60, 171, 477, 1306, 3526, …
  (−∞, 0]nonnegative
• A156992 → basis A007318
  1 · 4, 2 · 24, 24, 6 · 192, 288, 144, 24 … Σ 1, 6, 54, 648, 9720, 174960, 3674160, …
  (−∞, −1]nonnegativerational roots
• A081577 → basis A013609 → reverse
  1 · 2, 2 · 4, 12, 6 · 8, 40, 48, 16 … Σ 1, 4, 22, 112, 580, 2992, 15448, 79744, …
  (−∞, 0]nonnegative
• A038279 → D+1
  1 · 9, 1 · 80, 18, 1 · 704, 240, 27, 1 … Σ 1, 10, 99, 972, 9477, 91854, 885735, …
  (−∞, 0]nonnegativerational roots
• A038303 → basis A013611 → (θ+1)²
  1 · 11, 16 · 121, 352, 144 · 1331, 5808, 4752, 1024 … Σ 1, 27, 617, 12915, 255825, 4876875, …
  (−∞, 0]nonnegative
• A085881 → basis A013609 → (θ+1)²
  1 · 2, 8 · 12, 96, 108 · 120, 1440, 3240, 1920 … Σ 1, 10, 216, 6720, 268800, 13063680, …
  (−∞, 0]nonnegative
• A110608 → reverse
  1, 8, 6 · 1, 18, 45, 20 · 1, 32, 168, 224, 70 … Σ 15, 84, 495, 3003, 18564, 116280, …
  (−∞, 0]nonnegative
• A075497 → θ+1 → reverse
  1 · 2, 2 · 3, 12, 4 · 4, 36, 56, 8 … Σ 1, 4, 19, 104, 641, 4380, 32803, …
  (−∞, 0]nonnegative
• A088617 → reverse
  1 · 1, 1 · 2, 3, 1 · 5, 10, 6, 1 … Σ 1, 2, 6, 22, 90, 394, 1806, 8558, …
  (−∞, 0]nonnegative
• A075497 → D+1
  1 · 3, 1 · 10, 8, 1 · 36, 52, 15, 1 … Σ 1, 4, 19, 104, 641, 4380, 32803, …
  (−∞, 0]nonnegative
• A108838 → x ↦ x+1 → x ↦ x/(x+1)
  2 · 5, 7 · 14, 40, 28 · 42, 182, 259, 121 … Σ 2, 12, 82, 604, 4674, 37460, 308210, …
  [−1, 0]nonnegative
• A156992 → basis A007318 → D
  2 · 24, 12 · 288, 288, 72 · 3840, 5760, 2880, 480 … Σ 0, 2, 36, 648, 12960, 291600, 7348320, …
  (−∞, 0]nonnegativerational roots
• A204420 → D
  1 · 6, 6 · 120, 180, 45 · 5040, 9240, 3780, 420 … Σ 0, 1, 12, 345, 18480, 1596105, …
  (−∞, 0]nonnegative
• A111577 → D+1
  1 · 2, 1 · 6, 7, 1 · 22, 45, 15, 1 … Σ 1, 3, 14, 83, 579, 4624, 41467, 410967, …
  (−∞, 0]nonnegative
• A207606: Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.
  4, 7, 2 · 5, 16, 11, 2 · 6, 30, 36, 15, 2 … Σ 13, 34, 89, 233, 610, 1597, 4181, …
  (−4, 0)nonnegative
• A013610 → Veronese 2 (j=1) → θ+1
  3 · 6 · 9, 54 · 12, 216 … Σ 0, 3, 6, 63, 228, 1284, 5472, 25968, …
  (−∞, 0]nonnegative
• A038334 → x ↦ x+1 → D
  320, 128 · 9600, 7680, 1536 … Σ 448, 18816, 702464, 24586240, …
  (−∞, 0]nonnegativerational roots
• A038291 → D+1
  1 · 10, 1 · 99, 20, 1 · 972, 297, 30, 1 … Σ 1, 11, 120, 1300, 14000, 150000, …
  (−∞, 0]nonnegativerational roots
• A123519 → basis A013609 → θ+1
  11, 56, 48 · 41, 304, 528, 256 … Σ 115, 1129, 10649, 97699, 878299, …
  (−∞, 0]nonnegative
• A099093 → reverse → D
  3 · 18 · 81, 18 · 324, 162 … Σ 0, 0, 3, 18, 99, 486, 2268, 10206, …
  (−∞, 0]nonnegative
• A099040 → basis A013609 → D+1
  1 · 6, 4 · 26, 52, 16 · 96, 336, 320, 64 … Σ 1, 10, 94, 816, 6780, 54648, 430920, …
  (−∞, 0]nonnegative
• A257620: Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*…
  1 · 3, 3 · 9, 36, 9 · 27, 297, 297, 27 … Σ 1, 6, 54, 648, 9720, 174960, 3674160, …
  (−∞, 0)nonnegative
• A013620 → basis A013610 → D
  9 · 90, 162 · 675, 2430, 2187 … Σ 0, 9, 252, 5292, 98784, 1728720, …
  (−∞, 0]nonnegativerational roots
• A013622 → basis A007318 → D
  5 · 80, 50 · 960, 1200, 375 … Σ 0, 5, 130, 2535, 43940, 714025, …
  (−∞, 0]nonnegativerational roots
• A038325 → γ
  1 · 11 · 121 · 1331 … Σ 1, 11, 121, 1331, 14641, 161051, …
  (−∞, −1][−1, 0][−3−2√2, −3+2√2]+2
• delannoy
  1 · 1, 1 · 1, 3, 1 · 1, 5, 5, 1 … Σ 1, 2, 5, 12, 29, 70, 169, 408, …
  [−3−2√2, −3+2√2]nonnegative
• A075497 → D+1 → θ+1
  1 · 3, 2 · 10, 16, 3 · 36, 104, 45, 4 … Σ 1, 5, 29, 189, 1357, 10601, 89329, …
  (−∞, 0]nonnegative
• A038321 → basis A007318 → D
  252, 98 · 6804, 5292, 1029 … Σ 350, 13125, 437500, 13671875, …
  (−∞, 0]nonnegativerational roots
• A137320 → Veronese 2 (j=1) → θ+1
  2 · 2 · 4, 16 · 12, 96 … Σ 0, 2, 2, 20, 108, 704, 5280, 44736, …
  (−∞, 0]nonnegative
• A038243 → basis A013609 → θ+1
  1 · 6, 4 · 36, 48, 12 · 216, 432, 216, 32 … Σ 1, 10, 96, 896, 8192, 73728, 655360, …
  (−∞, 0]nonnegativerational roots
• A038255 → basis A013609 → θ+1
  1 · 7, 4 · 49, 56, 12 · 343, 588, 252, 32 … Σ 1, 11, 117, 1215, 12393, 124659, …
  (−∞, 0]nonnegativerational roots
• A038303 → basis A013611
  1 · 11, 4 · 121, 88, 16 · 1331, 1452, 528, 64 … Σ 1, 15, 225, 3375, 50625, 759375, …
  (−∞, −1]nonnegativerational roots
• A144442 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 19, 19 · 1, 121, 357, 238 … Σ 1, 3, 39, 717, 18031, 580173, 22797479, …
  [−1, 0]nonnegative
• A356546 → x ↦ x/(x+1)
  1 · 2, 4 · 6, 24, 24 · 20, 120, 240, 160 … Σ 1, 6, 54, 540, 5670, 61236, 673596, …
  [−1, 0]nonnegativerational roots
• A038335 → basis A013610 → x ↦ x/(x+1)
  1 · 21, 48 · 441, 2016, 2304 … Σ 1, 69, 4761, 328509, 22667121, …
  [−1, 0]nonnegativerational roots
• A391533 → Veronese 2 (j=1) → D+1
  4 · 9, 1 · 18, 6 · 35, 21, 1 … Σ 4, 10, 24, 57, 132, 300, 672, 1488, …
  (−∞, 0]nonnegative
• A114687 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 9 · 11, 64, 89 · 45, 387, 1071, 945 … Σ 1, 12, 164, 2448, 38816, 642432, …
  [−1, 0]nonnegative
• A091042 → Veronese 2 → D
  5 · 35 · 126, 18 · 330, 330 … Σ 0, 0, 5, 35, 144, 660, 3328, 15600, …
  (−∞, 0]nonnegative
• A035607: Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).
  1 · 1, 2 · 1, 4, 2 · 1, 6, 8, 2 … Σ 1, 3, 7, 17, 41, 99, 239, 577, …
  (−∞, 0)nonnegative
• A091042 → basis A013609 → reverse
  1 · 6, 4 · 20, 40, 16 · 56, 224, 224, 64 … Σ 1, 10, 76, 568, 4240, 31648, 236224, …
  (−∞, 0]nonnegative
• A008278 → x ↦ x+1 (shift +1) → x ↦ x+1 → D+1
  1 · 4, 1 · 18, 9, 1 · 95, 72, 16, 1 … Σ 1, 5, 28, 184, 1415, 12403, 120985, …
  (−∞, 0]nonnegative
• A121574 → basis A013610 → D+1
  1 · 6, 3 · 31, 39, 9 · 138, 303, 180, 27 … Σ 1, 9, 79, 648, 5116, 39312, 296128, …
  (−∞, 0]nonnegative
• A026729 → basis A013610 → x ↦ x+1
  1 · 4, 3 · 20, 27, 9 · 96, 192, 126, 27 … Σ 1, 7, 56, 441, 3479, 27440, 216433, …
  (−∞, −1]nonnegative
• basis-map coefficients of A021012 → θ+1 → basis A007318
  1 · -2, -2 · 6, 12, 6 · -24, -72, -72, -24 … Σ 1, -4, 24, -192, 1920, -23040, 322560, …
  (−∞, −1]rational roots
• A008459 → basis A013609 → D
  2 · 12, 8 · 60, 96, 24 · 280, 720, 480, 64 … Σ 0, 2, 20, 180, 1544, 12860, 105048, …
  (−∞, 0]nonnegative
• A114687 → reverse → D+1
  1 · 3, 1 · 10, 8, 1 · 32, 48, 15, 1 … Σ 1, 4, 19, 96, 501, 2668, 14407, 78592, …
  (−∞, 0]nonnegative