Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A062196 → reverse → x ↦ x/(x+1)
  1 · 3, 4 · 6, 20, 15 · 10, 60, 105, 56 … Σ 1, 7, 41, 231, 1289, 7183, 40081, …
  [−1, 0]nonnegative
• A124860 → D
  1 · 6, 6 · 15, 30, 15 · 44, 132, 132, 44 … Σ 0, 1, 12, 60, 352, 1680, 8256, 38080, …
  (−∞, 0]nonnegativerational roots
• A008299 → (θ+1)²
  1 · 1 · 1, 12 · 1, 40 … Σ 1, 1, 13, 41, 236, 1170, 6567, 38407, …
  (−∞, 0]nonnegative
• A038322 → basis A013610 → θ+1
  1 · 19, 48 · 361, 1824, 1728 … Σ 1, 67, 3913, 212635, 11051473, …
  (−∞, 0]nonnegativerational roots
• A128099 → x ↦ x+1 → x ↦ x+1
  1 · 1 · 5, 2 · 9, 4 … Σ 1, 1, 7, 13, 55, 133, 463, 1261, …
  (−∞, −1]nonnegative
• A133080 → x ↦ x/(x+1)
  1 · 1, 2 · 0, 0, 1 · 0, 0, 1, 2 … Σ 1, 3, 1, 3, 1, 3, 1, 3, …
  [−1, 0]nonnegativerational roots
• A038303 → D+1
  1 · 11, 1 · 120, 22, 1 · 1300, 360, 33, 1 … Σ 1, 12, 143, 1694, 19965, 234256, …
  (−∞, 0]nonnegativerational roots
• A113214 → x ↦ x−8
  1 · -6, 1 · 40, -13, 1 · -272, 130, -20, 1 … Σ 1, -5, 28, -161, 931, -5390, 31213, …
  (4, 8]
• A099093 → x ↦ x−4
  1 · -12, 3 · 132, -69, 9 · -1440, 1152, -306, 27 … Σ 1, -9, 72, -567, 4455, -34992, 274833, …
  (8/3, 4]
• A375853 → θ+1
  2 · 8, 16 · 20, 112, 60 · 40, 432, 648, 160 … Σ 2, 24, 192, 1280, 7680, 43008, 229376, …
  (−∞, 0]nonnegative
• A144084 → D
  1 · 4, 4 · 9, 36, 18 · 16, 144, 288, 96 … Σ 0, 1, 8, 63, 544, 5225, 55656, 653023, …
  (−∞, 0]nonnegative
• A272867 → basis A013609 → D+1
  1 · 18, 20, 4 · 180, 528, 480, 160, 16 … Σ 1, 42, 1364, 39688, 1086096, 28579232, …
  (−∞, 0]nonnegative
• A094816 → D → D
  2 · 12, 6 · 58, 60, 12 · 290, 450, 180, 20 … Σ 0, 0, 2, 18, 130, 940, 7268, 61362, …
  (−∞, 0]nonnegative
• A257625 → D
  3 · 54, 18 · 621, 1242, 81 · 6156, 37260, 18468, 324 … Σ 0, 3, 72, 1944, 62208, 2332800, …
  (−∞, 0]nonnegative
• A229032 → x ↦ x+1
  1 · 2 · 7, 4 · 20, 16 … Σ 1, 2, 11, 36, 149, 550, 2143, 8136, …
  (−∞, −1]nonnegative
• A028246 → θ+1
  1 · 1, 2 · 1, 6, 6 · 1, 14, 36, 24 … Σ 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729 → basis A007318 → (θ+1)²
  1 · 1, 4 · 1, 8, 9 · 1, 12, 27, 16 … Σ 1, 5, 18, 56, 160, 432, 1120, 2816, …
  (−∞, 0]nonnegative
• A008278 → x ↦ x+1 (shift +1) → θ+1
  1 · 2, 2 · 5, 10, 3 · 15, 46, 30, 4 … Σ 1, 4, 18, 95, 579, 3966, 29944, 246141, …
  (−∞, 0]nonnegative
• A038330 → basis A013609 → x ↦ x/(x+1)
  256, 768, 576 · 4096, 18432, 27648, 13824 … Σ 1600, 64000, 2560000, 102400000, …
  [−1, 0]nonnegativerational roots
• basis-map coefficients of A026729 → (θ+1)² → basis A007318
  1 · 4, 4 · 9, 18, 9 · 16, 48, 48, 16 … Σ 1, 8, 36, 128, 400, 1152, 3136, 8192, …
  (−∞, −1]nonnegativerational roots
• A133087 → x ↦ x+1
  1 · 3, 1 · 4, 4, 1 · 12, 16, 7, 1 … Σ 1, 4, 9, 36, 81, 324, 729, 2916, …
  (−∞, −1]nonnegativerational roots
• A160562 → D
  1 · 10, 2 · 91, 70, 3 · 820, 1932, 252, 4 … Σ 0, 1, 12, 164, 3008, 73232, 2226880, …
  (−∞, 0]nonnegative
• A207606 → x ↦ x+1 → D
  11, 4 · 44, 34, 6 · 155, 186, 69, 8 … Σ 15, 84, 418, 1950, 8733, 38024, 162180, …
  (−∞, 0]nonnegative
• A008278 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 15, 68, 101, 49 … Σ 1, 5, 31, 233, 2047, 20417, 226711, …
  [−1, 0]nonnegative
• A063967 → basis A007318 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 17, 12 · 16, 68, 96, 45 … Σ 1, 5, 35, 225, 1475, 9625, 62875, …
  [−1, 0]nonnegative
• A161198 → basis A013610 → θ+1
  1 · 3, 12 · 15, 96, 108 · 105, 852, 1620, 864 … Σ 1, 15, 219, 3441, 59337, 1124559, …
  (−∞, 0]nonnegative
• A391533 → D+1
  8, 6, 1 · 12, 18, 8, 1 · 16, 38, 31, 10, 1 … Σ 15, 39, 96, 228, 528, 1200, 2688, 5952, …
  (−∞, 0]nonnegative
• A085478 → D → θ+1
  1 · 3, 4 · 6, 20, 9 · 10, 60, 63, 16 … Σ 0, 0, 1, 7, 35, 149, 576, 2088, …
  (−∞, 0]nonnegative
• A026729 → θ+1 → D+1
  1 · 2, 2 · 2, 8, 3 · 0, 12, 18, 4 … Σ 1, 4, 13, 34, 82, 184, 395, 818, …
  (−∞, 0]nonnegative
• basis-map coefficients of A127647 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 8, 8 · 3, 18, 36, 24 … Σ 1, 3, 18, 81, 405, 1944, 9477, 45927, …
  [−1, 0]nonnegativerational roots
• A371080 → θ+1
  1 · 0, 2 · 0, 8, 3 · 0, 56, 36, 4 … Σ 1, 2, 11, 96, 1141, 17126, 310207, …
  (−∞, 0]nonnegative
• A257620 → x ↦ x/(x+1)
  1 · 3, 6 · 9, 54, 54 · 27, 378, 972, 648 … Σ 1, 9, 117, 2025, 43821, 1137969, …
  [−1, 0]nonnegative
• A360282 → D+1
  1 · 1, 1 · 3, 7, 2 · 10, 34, 21, 3 … Σ 1, 2, 12, 68, 390, 2262, 13216, 77576, …
  (−∞, 0]nonnegative
• A038327 → basis A013610
  1 · 13, 3 · 169, 78, 9 · 2197, 1521, 351, 27 … Σ 1, 16, 256, 4096, 65536, 1048576, …
  (−∞, −1]nonnegativerational roots
• A038335 → D+1
  1 · 21, 9 · 360, 378, 81 · 5616, 9720, 5103, 729 … Σ 1, 30, 819, 21168, 527877, 12835746, …
  (−∞, 0]nonnegativerational roots
• A156992 → basis A013609 → (θ+1)²
  1 · 4, 16 · 24, 192, 216 · 192, 2304, 5184, 3072 … Σ 1, 20, 432, 10752, 307200, 9953280, …
  (−∞, 0]nonnegative
• A327997 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 4, 5 · 17, 43, 27 · 78, 299, 379, 159 … Σ 1, 9, 87, 915, 10593, 136161, 1951479, …
  [−1, 0]nonnegative
• A249252: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 4 · 1, 8, 8 · 1, 12, 32, 16 … Σ 1, 5, 17, 61, 217, 773, 2753, 9805, …
  (−∞, 0)nonnegative
• A059344 → (θ+1)²
  1 · 1 · 1, 8 · 1, 24 … Σ 1, 1, 9, 25, 157, 621, 3661, 17389, …
  (−∞, 0]nonnegative
• A105278 → θ+1
  1 · 2, 2 · 6, 12, 3 · 24, 72, 36, 4 … Σ 1, 4, 21, 136, 1045, 9276, 93289, …
  (−∞, 0]nonnegative
• A038333 → basis A013609 → (θ+1)²
  361, 2128, 1764 · 6859, 60648, 100548, 43904 … Σ 4253, 211959, 9784665, 429051843, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/4) → x ↦ x+1 → D+1
  1 · 2 · 8, 2 · 40, 16 … Σ 1, 2, 10, 56, 408, 3456, 34016, 378496, …
  (−∞, 0]nonnegative
• A048993: Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
  1 · 0, 1 · 0, 1, 1 · 0, 1, 3, 1 … Σ 1, 1, 2, 5, 15, 52, 203, 877, …
  (−∞, 0]nonnegative
• A081579 → γ → θ+1
  1 · 1 · 1, 8 · 1, 16 … Σ 1, 1, 9, 17, 73, 177, 585, 1553, …
  (−∞, 0]nonnegative
• derivative of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and …
  1 · 3, 6, 3 · 7, 32, 48, 28, 5 … Σ 1, 12, 120, 1344, 17280, 253440, …
  (−∞, 0]nonnegative
• A216973 → θ+1 → basis A007318
  1 · 6, 4 · 24, 30, 9 · 80, 144, 84, 16 … Σ 0, 1, 10, 63, 324, 1485, 6318, 25515, …
  (−∞, −1]nonnegativerational roots
• A133080 → basis A013609 → D+1
  1 · 4, 2 · 5, 12, 4 · 12, 42, 40, 8 … Σ 1, 6, 21, 102, 297, 1350, 3645, 16038, …
  (−∞, 0]nonnegative
• A085880 → basis A013611
  1 · 2, 4 · 8, 32, 32 · 40, 240, 480, 320 … Σ 1, 6, 72, 1080, 18144, 326592, 6158592, …
  [−1, 0]nonnegativerational roots
• A038326 → x ↦ x+1 → D+1
  1 · 35, 12 · 1081, 840, 144 … Σ 1, 47, 2065, 86975, 3558625, 142559375, …
  (−∞, 0]nonnegativerational roots
• A285061 → D+1
  1 · 5, 4 · 25, 56, 16 · 125, 604, 432, 64 … Σ 1, 9, 97, 1225, 17665, 284873, 5059553, …
  (−∞, 0]nonnegative