Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A013620 → basis A013611 → (θ+1)²
  1 · 5, 48 · 25, 480, 1296 · 125, 3600, 19440, 27648 … Σ 1, 53, 1801, 50813, 1290385, 30603077, …
  (−∞, 0]nonnegative
• A125092 → (θ+1)²
  1 · 1, 16 · 1, 32, 81 · 1, 48, 243, 256 … Σ 1, 17, 114, 548, 2200, 7872, 25984, …
  (−∞, 0]nonnegative
• A081577 → basis A013609 → x ↦ x+1
  1 · 4, 2 · 22, 20, 4 · 112, 152, 64, 8 … Σ 1, 6, 46, 336, 2476, 18216, 134056, …
  (−∞, −1]nonnegative
• A027465 → Veronese 2 (j=1) → D
  1 · 12 · 90, 2 · 540, 36 … Σ 0, 0, 0, 1, 12, 92, 576, 3216, …
  (−∞, 0]nonnegative
• A185410 → reverse → x ↦ x+1
  1 · 1, 1 · 3, 4, 1 · 15, 27, 13, 1 … Σ 1, 2, 8, 56, 560, 7232, 114368, …
  (−∞, −1]nonnegative
• A122542 → D+1 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 8, 7 · 2, 16, 33, 20 … Σ 1, 3, 17, 71, 301, 1235, 4985, 19839, …
  [−1, 0]nonnegative
• A269951 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 2, 3 · 0, 5, 15, 11 … Σ 1, 1, 5, 31, 239, 2257, 25517, 337615, …
  [−1, 0]nonnegative
• A085478 → basis A013609 → D+1
  1 · 1 · 4, 2 · 15, 18, 4 … Σ 1, 1, 6, 37, 217, 1230, 6805, 36973, …
  (−∞, 0]nonnegative
• A063007 → basis A013609 → θ+1
  1 · 3, 8 · 13, 72, 72 · 63, 528, 1080, 640 … Σ 1, 11, 157, 2311, 34121, 502387, …
  (−∞, 0]nonnegative
• A196347 → x ↦ x+1 → D
  1 · 8, 4 · 72, 72, 18 · 768, 1152, 576, 96 … Σ 0, 1, 12, 162, 2592, 48600, 1049760, …
  (−∞, 0]nonnegativerational roots
• A145596 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 4, 6 · 14, 42, 31 · 48, 216, 320, 156 … Σ 1, 10, 87, 740, 6305, 54150, 469399, …
  [−1, 0]nonnegative
• A063007 → D+1
  1 · 3, 2 · 7, 18, 6 · 13, 72, 90, 20 … Σ 1, 5, 31, 195, 1221, 7593, 46915, …
  (−∞, 0]nonnegative
• A225470 → x ↦ x+1 → D
  1 · 9, 2 · 99, 36, 3 · 1350, 630, 90, 4 … Σ 0, 1, 11, 138, 2074, 36824, 757896, …
  (−∞, 0]nonnegative
• A178820 → basis A013611
  1 · 8, 16 · 40, 160, 160 · 160, 960, 1920, 1280 … Σ 1, 24, 360, 4320, 45360, 435456, …
  [−1, 0]nonnegativerational roots
• A053117 → D
  2 · 0, 8 · -4, 0, 24 · 0, -24, 0, 64 … Σ 0, 2, 8, 20, 40, 70, 112, 168, …
  (−∞, ∞)
• A038208 → θ+1
  1 · 2, 4 · 4, 16, 12 · 8, 48, 72, 32 … Σ 1, 6, 32, 160, 768, 3584, 16384, 73728, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A113214, shifted → Veronese 2 → x ↦ x+1
  1 · -1 · 2, 1 · -4, -3 … Σ 1, -1, 3, -7, 17, -41, 99, -239, …
  (−∞, −1]
• A013620 → Veronese 2 → (θ+1)²
  1 · 2 · 4, 36 · 8, 216 … Σ 1, 2, 40, 224, 1609, 10202, 64108, …
  (−∞, 0]nonnegative
• A143683 → γ → θ+1
  1 · 1 · 1, 16 · 1, 32 … Σ 1, 1, 17, 33, 241, 641, 3281, 10209, …
  (−∞, 0]nonnegative
• A144699 → D+1
  1 · 6, 5 · 17, 66, 25 · 40, 381, 546, 125 … Σ 1, 11, 108, 1092, 11760, 136080, …
  (−∞, 0]nonnegative
• A038277 → x ↦ x+1
  1 · 18, 11 · 324, 396, 121 · 5832, 10692, 6534, 1331 … Σ 1, 29, 841, 24389, 707281, 20511149, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A371080, shifted → basis A013610 → D+1
  1 · 4, 3 · 19, 33, 9 · 145, 333, 189, 27 … Σ 1, 7, 61, 694, 9862, 168712, 3380248, …
  (−∞, 0]nonnegative
• A008288 → basis A013611 → (θ+1)²
  1 · 2, 16 · 5, 80, 144 · 12, 288, 1152, 1024 … Σ 1, 18, 229, 2476, 24381, 225654, …
  (−∞, 0]nonnegative
• A046716 → basis A013609 → reverse
  1 · 2, 2 · 4, 10, 5 · 8, 44, 50, 16 … Σ 1, 4, 19, 118, 1021, 12088, 183727, …
  (−∞, 0]nonnegative
• A166344 → θ+1
  1, 12, 3 · 1, 30, 45, 4 · 1, 68, 270, 136, 5 … Σ 16, 80, 480, 3360, 26880, 241920, …
  (−∞, 0]nonnegative
• A008288 → basis A007318 → x ↦ x+1
  1 · 3, 1 · 11, 7, 1 · 39, 37, 11, 1 … Σ 1, 4, 19, 88, 409, 1900, 8827, 41008, …
  (−∞, −1]nonnegative
• A008292 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 18, 13 · 24, 108, 158, 75 … Σ 1, 5, 37, 365, 4501, 66605, 1149877, …
  [−1, 0]nonnegative
• A235706: (I + A132440)^3: Coefficients for normalized generalized Laguerre polynomials n!*Lag(n, 3-n, -x).
  A132440 → D+1 → (1/n)D²+(2/n)D+1/n
  1 · 3, 1 · 6, 6, 1 · 6, 18, 9, 1 … Σ 1, 4, 13, 34, 73, 136, 229, 358, …
  (−∞, 0]nonnegative
• delannoy → basis A013609 → D
  2 · 10, 8 · 36, 64, 24 · 116, 320, 264, 64 … Σ 0, 2, 18, 124, 764, 4422, 24598, …
  (−∞, 0]nonnegative
• A001263 → basis A013609 → D
  2 · 10, 8 · 42, 72, 24 · 168, 448, 336, 64 … Σ 0, 2, 18, 138, 1016, 7380, 53358, …
  (−∞, 0]nonnegative
• A143491 → basis A013609 → D+1
  26, 22, 4 · 154, 190, 72, 8 … Σ 52, 424, 3812, 37668, 406920, 4778520, …
  (−∞, 0]nonnegative
• basis-map coefficients of A021012 → reverse
  1 · -1 · 2 · -6 … Σ 1, -1, 2, -6, 24, -120, 720, -5040, …
  (−∞, −1][−1, 0][−3−2√2, −3+2√2]+1
• narayana → basis A013611 → D+1
  1 · 6, 4 · 25, 52, 16 · 98, 372, 336, 64 … Σ 1, 10, 93, 870, 8266, 79636, 775765, …
  (−∞, 0]nonnegative
• A199881 → reverse
  1, 1 · 1, 1 · 1, 2, 1 · 1, 3, 2 … Σ 2, 2, 4, 6, 10, 16, 26, 42, …
  (−∞, 0]nonnegative
• A123519 → D
  6, 8 · 12, 40, 24 · 20, 120, 168, 64 … Σ 14, 76, 372, 1718, 7642, 33112, 140712, …
  (−∞, 0]nonnegative
• A132440 → D+1 → θ+1
  1 · 2, 4 · 0, 12, 9 · 0, 0, 36, 16 … Σ 0, 1, 6, 21, 52, 105, 186, 301, …
  (−∞, 0]nonnegativerational roots
• A038320 → basis A013609 → D+1
  697, 696, 144 · 15317, 25092, 12528, 1728 … Σ 1537, 54665, 1877953, 62948009, …
  (−∞, 0]nonnegativerational roots
• A094816 → basis A013610 → (θ+1)²
  1 · 2, 12 · 5, 60, 81 · 16, 276, 729, 432 … Σ 1, 14, 146, 1453, 14795, 158312, …
  (−∞, 0]nonnegative
• A136124 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 3, 4 · 12, 31, 20 · 60, 227, 286, 120 … Σ 1, 7, 63, 693, 9009, 135135, 2297295, …
  [−1, 0]nonnegativerational roots
• A114655 → reverse → x ↦ x/(x+1)
  2 · 2, 6 · 2, 16, 22 · 2, 30, 102, 90 … Σ 2, 8, 40, 224, 1344, 8448, 54912, …
  [−1, 0]nonnegative
• A127648 → Veronese 2 → D+1
  1 · 3, 3 · 0, 10, 5 · 0, 0, 21, 7 … Σ 0, 1, 0, 6, 0, 15, 0, 28, …
  (−∞, 0]nonnegativerational roots
• A253283 → (θ+1)²
  1 · 0, 4 · 0, 8, 27 · 0, 12, 108, 160 … Σ 1, 4, 35, 280, 2121, 15456, 109427, …
  (−∞, 0]nonnegative
• A391533 → reverse → D+1
  5, 12, 4 · 6, 21, 20, 4 · 7, 32, 49, 28, 4 … Σ 21, 51, 120, 276, 624, 1392, 3072, …
  (−∞, 0]nonnegative
• A306364 → (θ+1)²
  1 · 3 · 12, 12 · 60, 180 … Σ 1, 3, 24, 240, 2925, 41895, 687960, …
  (−∞, 0]nonnegative
• A094816 → x ↦ x+1 → D
  1 · 5, 2 · 23, 18, 3 · 116, 130, 42, 4 … Σ 0, 1, 7, 44, 292, 2129, 17203, 153796, …
  (−∞, −1]nonnegative
• A124932: Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).
  1 · 2, 3 · 3, 9, 6 · 4, 18, 24, 10 … Σ 1, 5, 18, 56, 160, 432, 1120, 2816, …
  (−∞, 0)nonnegativerational roots
• A062196 → reverse → D+1
  1 · 4, 1 · 14, 10, 1 · 40, 60, 18, 1 … Σ 1, 5, 25, 119, 546, 2442, 10725, 46475, …
  (−∞, 0]nonnegative
• A075505 → D
  30, 2 · 700, 120, 3 · 15000, 5000, 300, 4 … Σ 32, 823, 20304, 510105, 13427056, …
  (−∞, 0]nonnegative
• A111125 → reverse
  1 · 1, 3 · 1, 5, 5 · 1, 7, 14, 7 … Σ 1, 4, 11, 29, 76, 199, 521, 1364, …
  (−∞, 0]nonnegative
• A151775 → D
  2 · 28, 48 · 662, 2640, 2160 … Σ 0, 2, 76, 5462, 655576, 118940522, …
  (−∞, 0]nonnegative