Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A269944 → basis A013609 → D
  2 · 6, 8 · 34, 64, 24 · 360, 776, 432, 64 … Σ 0, 2, 14, 122, 1632, 33024, 962976, …
  (−∞, 0]nonnegative
• A103633 → Veronese 2 (j=1) → x ↦ x+1
  1 · 1 · 1, 1 · 2, 2 … Σ 0, 1, 1, 2, 4, 6, 14, 20, …
  (−∞, −1]nonnegative
• A113413 → x ↦ x+1 → reverse
  1 · 1, 3 · 1, 6, 7 · 1, 9, 23, 17 … Σ 1, 4, 14, 50, 178, 634, 2258, 8042, …
  (−∞, 0]nonnegative
• A110441 → basis A013609 → θ+1
  1 · 4, 4 · 14, 32, 12 · 48, 176, 144, 32 … Σ 1, 8, 58, 400, 2668, 17376, 111160, …
  (−∞, 0]nonnegative
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
  1 · 2, 1 · 3, 4, 1 · 4, 9, 6, 1 … Σ 1, 3, 8, 20, 48, 112, 256, 576, …
  (−∞, 0)nonnegativerational roots
• basis-map coefficients of A112227 → basis A008277 → x ↦ x+1
  5, 5, 1 · 15, 22, 9, 1 · 52, 99, 61, 14, 1 … Σ 11, 47, 227, 1215, 7107, 44959, 305091, …
  (−∞, −1]nonnegative
• A169803 → (θ+1)²
  1 · 1, 4 · 1, 8 · 1, 12, 9 … Σ 1, 5, 9, 22, 44, 91, 179, 349, …
  (−∞, 0]nonnegative
• A321331: Triangle read by rows: T(n, k) = (k+1)*S2(n+1, k+1), for n >= k >= 0, and S2 = A048993 (Stirling2).
  A008277 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 14, 18, 4 … Σ 1, 3, 10, 37, 151, 674, 3263, 17007, …
  (−∞, 0]nonnegative
• A081577: Pascal-(1,2,1) array read by antidiagonals.
  1 · 1, 1 · 1, 4, 1 · 1, 7, 7, 1 … Σ 1, 2, 6, 16, 44, 120, 328, 896, …
  (−∞, 0)nonnegative
• A063967 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 7, 6 · 3, 16, 28, 16 … Σ 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  [−1, 0]nonnegative
• A056242 → reverse
  1 · 2, 1 · 4, 5, 1 · 8, 16, 9, 1 … Σ 1, 3, 10, 34, 116, 396, 1352, 4616, …
  (−∞, 0]nonnegative
• A038301 → x ↦ x+1 → θ+1
  400, 880, 363 · 8000, 26400, 21780, 5324 … Σ 1643, 61504, 2234325, 79422806, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A112227 → basis A013609 → θ+1
  1, 8, 12 · 1, 12, 36, 32 · 1, 16, 72, 128, 80 … Σ 21, 81, 297, 1053, 3645, 12393, 41553, …
  (−∞, 0]nonnegativerational roots
• A008278: Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
  1 · 1, 1 · 1, 3, 1 · 1, 6, 7, 1 … Σ 1, 2, 5, 15, 52, 203, 877, 4140, …
  (−∞, 0)nonnegative
• A125662: A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
  1 · 3, 1 · 8, 6, 1 · 21, 25, 9, 1 … Σ 1, 4, 15, 56, 209, 780, 2911, 10864, …
  (−5, −1)nonnegative
• A257607 → x ↦ x/(x+1)
  1 · 5, 10 · 25, 110, 110 · 125, 910, 1980, 1320 … Σ 1, 15, 245, 4335, 82685, 1692375, …
  [−1, 0]nonnegative
• A038321 → basis A013609 → D
  504, 392 · 13608, 21168, 8232 … Σ 896, 43008, 1835008, 73400320, …
  (−∞, 0]nonnegativerational roots
• A249248 → (θ+1)²
  1 · 1, 12 · 1, 28, 27 · 1, 48, 198, 48 … Σ 1, 13, 56, 295, 1409, 7021, 34808, …
  (−∞, 0]nonnegative
• A038221 → basis A013610 → (θ+1)²
  1 · 6, 36 · 36, 432, 729 · 216, 3888, 13122, 11664 … Σ 1, 42, 1197, 28890, 633825, 13061250, …
  (−∞, 0]nonnegative
• A232535 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 9 · 12, 66, 81 · 48, 384, 957, 729 … Σ 1, 12, 159, 2118, 28221, 376032, …
  [−1, 0]nonnegative
• A249139 → D+1
  1 · 4, 1 · 7, 2 · 18, 9, 1 … Σ 1, 5, 9, 28, 62, 165, 391, 983, …
  (−∞, 0]nonnegative
• A038320 → basis A013611 → θ+1
  289, 1632, 1728 · 4913, 41616, 88128, 55296 … Σ 3649, 189953, 9442177, 454947521, …
  (−∞, 0]nonnegativerational roots
• A104029 → x ↦ x/(x+1)
  1 · 2, 3 · 3, 11, 9 · 4, 25, 47, 27 … Σ 1, 5, 23, 103, 457, 2021, 8927, 39415, …
  [−1, 0]nonnegative
• A038271 → (θ+1)²
  1 · 7, 20 · 49, 280, 225 · 343, 2940, 4725, 2000 … Σ 1, 27, 554, 10008, 167616, 2668032, …
  (−∞, 0]nonnegative
• A039683: Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).
  1 · -2, 1 · 8, -6, 1 · -48, 44, -12, 1 … Σ 1, -1, 3, -15, 105, -945, 10395, …
  [1, ∞)rational roots
• A013618 → basis A013610 → D
  792, 2178 · 14256, 78408, 107811 … Σ 2970, 200475, 12028500, 676603125, …
  (−∞, 0]nonnegativerational roots
• A027466 → D+1
  1 · 8, 1 · 63, 16, 1 · 490, 189, 24, 1 … Σ 1, 9, 80, 704, 6144, 53248, 458752, …
  (−∞, 0]nonnegativerational roots
• A207824 → x ↦ x+1 → reverse
  1 · 1, 6 · 1, 12, 35 · 1, 18, 106, 204 … Σ 1, 7, 48, 329, 2255, 15456, 105937, …
  (−1/4, 0]nonnegative
• A063967 → D+1
  1 · 2, 1 · 5, 5, 1 · 10, 17, 8, 1 … Σ 1, 3, 11, 36, 116, 364, 1124, 3424, …
  (−∞, 0]nonnegative
• A013612 → basis A013609 → x ↦ x/(x+1)
  36, 192, 256 · 216, 1728, 4608, 4096 … Σ 484, 10648, 234256, 5153632, 113379904, …
  [−1, 0]nonnegativerational roots
• A144697 → θ+1
  1 · 1, 6 · 1, 20, 27 · 1, 50, 201, 108 … Σ 1, 7, 48, 360, 3000, 27720, 282240, …
  (−∞, 0]nonnegative
• A038320 → basis A013609 → θ+1
  289, 816, 432 · 4913, 20808, 22032, 6912 … Σ 1537, 54665, 1877953, 62948009, …
  (−∞, 0]nonnegativerational roots
• A358735 → basis A007318 → reverse
  1 · 1, 2 · 2, 8, 7 · 6, 34, 60, 33 … Σ 1, 3, 17, 133, 1313, 15623, 217409, …
  (−1, 0]nonnegative
• Triangle of unsigned 2-Lah numbers.
  20, 10, 1 · 120, 90, 18, 1 · 840, 840, 252, 28, 1 … Σ 31, 229, 1961, 19081, 207775, 2501801, …
  (−∞, 0)nonnegative
• A126351 → x ↦ x+1
  1 · 3, 2 · 10, 13, 4 · 37, 71, 43, 8 … Σ 1, 5, 27, 159, 1025, 7221, 55307, …
  (−∞, −1]nonnegative
• A085880 → basis A013611 → (θ+1)²
  1 · 2, 16 · 8, 128, 288 · 40, 960, 4320, 5120 … Σ 1, 18, 424, 10440, 260064, 6495552, …
  (−∞, 0]nonnegative
• A161198 → basis A008277 → x ↦ x/(x+1)
  1 · 3, 5 · 15, 50, 39 · 105, 525, 819, 407 … Σ 1, 8, 104, 1856, 41920, 1142272, …
  [−1, 0]nonnegative
• A008288 → D+1
  1 · 2, 1 · 4, 5, 1 · 6, 15, 8, 1 … Σ 1, 3, 10, 30, 87, 245, 676, 1836, …
  (−∞, 0]nonnegative
• A112486 → θ+1
  1 · 1, 2 · 2, 10, 9 · 6, 52, 105, 60 … Σ 1, 3, 21, 223, 3137, 54715, 1136909, …
  (−∞, 0]nonnegative
• A261711 → D+1
  1 · 4 · 16, 1 · 64, 8 … Σ 1, 4, 17, 72, 304, 1280, 5376, 22528, …
  (−∞, 0]nonnegative
• A123519 → x ↦ x/(x+1)
  1, 8, 11 · 1, 15, 47, 41 · 1, 24, 126, 240, 153 … Σ 20, 104, 544, 2848, 14912, 78080, …
  [−1, 0)nonnegative
• A225470 → basis A013611 → D+1
  1 · 7, 4 · 54, 68, 16 · 558, 972, 480, 64 … Σ 1, 11, 138, 2074, 36824, 757896, …
  (−∞, 0]nonnegative
• A257623 → x ↦ x+1
  1 · 6, 3 · 66, 66, 9 · 1056, 1584, 582, 27 … Σ 1, 9, 141, 3249, 98421, 3691449, …
  (−∞, −1]nonnegative
• Hadamard-factor parent of A127773 → x ↦ x+1
  1 · 3, 3 · 12, 24, 12 · 60, 180, 180, 60 … Σ 1, 6, 48, 480, 5760, 80640, 1290240, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A297191, shifted → θ+1
  1 · 0, 0, 3 · 0, 0, 0, 0, 5 · 0, 0, 0, 0, 0, 0, 7 … Σ 1, 3, 5, 7, 9, 11, 13, 15, …
  [−1, 0]nonnegativerational roots
• A038328 → basis A013611 → θ+1
  196, 448, 192 · 2744, 9408, 8064, 2048 … Σ 836, 22264, 574992, 14523872, …
  (−∞, 0]nonnegativerational roots
• A013609 → θ+1
  1 · 1, 4 · 1, 8, 12 · 1, 12, 36, 32 … Σ 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  (−∞, 0]nonnegativerational roots
• A038305 → basis A013609 → reverse
  36, 156, 169 · 216, 1404, 3042, 2197 … Σ 361, 6859, 130321, 2476099, 47045881, …
  (−∞, 0]nonnegativerational roots
• A094816 → D
  1 · 3, 2 · 8, 12, 3 · 24, 58, 30, 4 … Σ 0, 1, 5, 23, 116, 669, 4429, 33375, …
  (−∞, 0]nonnegative
• A245962 → basis A007318
  1 · 1 · 5, 2 · 7, 3 … Σ 1, 1, 7, 10, 31, 61, 154, 337, …
  (−∞, −1]nonnegative