Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A038284 → basis A007318 → x ↦ x/(x+1)
  196, 560, 400 · 2744, 11760, 16800, 8000 … Σ 1156, 39304, 1336336, 45435424, …
  [−1, 0]nonnegativerational roots
• A038329 → basis A013609 → x ↦ x/(x+1)
  225, 630, 441 · 3375, 14175, 19845, 9261 … Σ 1296, 46656, 1679616, 60466176, …
  [−1, 0]nonnegativerational roots
• A008955 → basis A007318 → θ+1
  1 · 2, 2 · 10, 26, 12 · 100, 440, 471, 144 … Σ 1, 4, 48, 1155, 46323, 2785698, …
  (−∞, 0]nonnegative
• A114656 → basis A013609 → θ+1
  1 · 3, 4 · 11, 32, 12 · 45, 204, 180, 32 … Σ 1, 7, 55, 461, 4013, 35795, 324715, …
  (−∞, 0]nonnegative
• A127773 → θ+1 → D
  6 · 0, 36 · 0, 0, 120 · 0, 0, 0, 300 … Σ 0, 6, 36, 120, 300, 630, 1176, 2016, …
  (−∞, 0]nonnegativerational roots
• A088617 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 6 · 1, 9, 25, 22 … Σ 1, 3, 12, 57, 300, 1686, 9912, 60213, …
  [−1, 0]nonnegative
• narayana → basis A013610 → D+1
  1 · 5, 3 · 20, 33, 9 · 77, 225, 162, 27 … Σ 1, 8, 62, 491, 3975, 32730, 272940, …
  (−∞, 0]nonnegative
• A081580 → basis A007318 → D+1
  1 · 3, 1 · 18, 11, 1 · 70, 74, 19, 1 … Σ 1, 4, 30, 164, 974, 5448, 30562, …
  (−∞, 0]nonnegative
• A008517: Second-order Eulerian triangle T(n,k), 1 <= k <= n.
  1 · 1, 2 · 1, 8, 6 · 1, 22, 58, 24 … Σ 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, 0)nonnegative
• A130534 → D+1 → D+1
  1 · 3, 1 · 10, 7, 1 · 40, 41, 12, 1 … Σ 1, 4, 18, 94, 570, 3968, 31304, 276568, …
  (−∞, 0]nonnegative
• A391533 → reverse → (θ+1)²
  1, 16, 36 · 1, 20, 72, 64 · 1, 24, 117, 192, 100 … Σ 53, 157, 434, 1144, 2912, 7216, 17504, …
  (−∞, 0]nonnegative
• A038329 → x ↦ x+1 → θ+1
  225, 180, 27 · 3375, 4050, 1215, 108 … Σ 432, 8748, 174960, 3464208, 68024448, …
  (−∞, 0]nonnegativerational roots
• A122542 → θ+1 → D+1
  1 · 2, 2 · 4, 10, 3 · 4, 28, 24, 4 … Σ 1, 4, 17, 60, 201, 640, 1969, 5896, …
  (−∞, 0]nonnegative
• A142461 → θ+1
  1 · 1, 2 · 1, 28, 3 · 1, 222, 333, 4 … Σ 1, 3, 32, 560, 13440, 407680, 14909440, …
  (−∞, 0]nonnegative
• A034870 → basis A013609 → x ↦ x+1
  1 · 16, 16, 4 · 256, 512, 384, 128, 16 … Σ 1, 36, 1296, 46656, 1679616, 60466176, …
  (−∞, −1]nonnegativerational roots
• A046716 → x ↦ x+1 → D+1
  1 · 3, 1 · 10, 7, 1 · 41, 47, 14, 1 … Σ 1, 4, 18, 103, 791, 7930, 98052, …
  (−∞, 0]nonnegative
• A130123 → basis A013610
  1 · 2, 6 · 4, 24, 36 · 8, 72, 216, 216 … Σ 1, 8, 64, 512, 4096, 32768, 262144, …
  [−1, 0]nonnegativerational roots
• A196347 → (θ+1)²
  1 · 1, 4 · 2, 16, 18 · 6, 72, 162, 96 … Σ 1, 5, 36, 336, 3840, 51840, 806400, …
  (−∞, 0]nonnegative
• A069466 → basis A007318 → reverse
  1 · 2, 4 · 6, 36, 36 · 20, 240, 600, 400 … Σ 1, 6, 78, 1260, 22470, 424116, 8305836, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/2) (shift +1) → x ↦ x+1 → θ+1
  1 · 1 · 2, 2 · 5, 8 … Σ 1, 1, 4, 13, 66, 351, 2292, 16425, …
  (−∞, 0]nonnegative
• A356546 → θ+1
  1 · 2, 4 · 6, 24, 18 · 20, 120, 180, 80 … Σ 1, 6, 48, 400, 3360, 28224, 236544, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A112227 → basis A013610 → (θ+1)²
  1, 24, 81 · 1, 36, 243, 432 · 1, 48, 486, 1728, 2025 … Σ 106, 712, 4288, 24064, 128512, 661504, …
  (−∞, 0]nonnegative
• A081577 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 6, 30, 33 · 16, 120, 282, 205 … Σ 1, 7, 69, 623, 5741, 52647, 483349, …
  [−1, 0]nonnegative
• A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
  image under θ+1 → θ+1
  1 · 0, 4 · 0, 0, 9 · 0, 0, 0, 16 … Σ 1, 4, 9, 16, 25, 36, 49, 64, …
  [−1, 0]nonnegativerational roots
• A116666 → Veronese 2 (j=1) → D
  7 · 28 · 70, 22 · 140, 132 … Σ 0, 0, 0, 7, 28, 92, 272, 752, …
  (−∞, 0]nonnegative
• A172431 → Veronese 2 → D+1
  1 · 1 · 4, 3 · 11, 10 … Σ 1, 1, 7, 21, 58, 178, 517, 1483, …
  (−∞, 0]nonnegative
• A344565 → D
  15 · 90, 90 · 315, 630, 315 · 840, 2520, 2520, 840 … Σ 0, 15, 180, 1260, 6720, 30240, 120960, …
  (−∞, 0]nonnegativerational roots
• A027465 → D+1
  1 · 4, 1 · 15, 8, 1 · 54, 45, 12, 1 … Σ 1, 5, 24, 112, 512, 2304, 10240, 45056, …
  (−∞, 0]nonnegativerational roots
• A352880 → x ↦ x+1
  8 · 30, 18 · 112, 128, 32 · 420, 700, 350, 50 … Σ 8, 48, 272, 1520, 8472, 47264, 264224, …
  (−∞, −1]nonnegative
• A089231: Triangular array A066667 or A008297 unsigned and transposed.
  1 · 1, 2 · 1, 6, 6 · 1, 12, 36, 24 … Σ 1, 3, 13, 73, 501, 4051, 37633, 394353, …
  (−∞, 0)nonnegative
• A367270 → θ+1
  1 · 1 · 1, 4 · 1, 12, 9 … Σ 1, 1, 5, 22, 95, 406, 1722, 7260, …
  (−∞, 0]nonnegative
• A039755: Triangle of B-analogs of Stirling numbers of the second kind.
  1 · 1, 1 · 1, 4, 1 · 1, 13, 9, 1 … Σ 1, 2, 6, 24, 116, 648, 4088, 28640, …
  (−∞, 0)nonnegative
• A113214 → basis A013610 → reverse
  1 · 3, 3 · 9, 15, 4 · 27, 63, 39, 7 … Σ 1, 6, 28, 136, 656, 3168, 15296, 73856, …
  (−∞, 0]nonnegative
• A261711 → x ↦ x+1
  1 · 4 · 16, 1 · 64, 8 … Σ 1, 4, 17, 72, 305, 1292, 5473, 23184, …
  (−∞, −1]nonnegative
• A038281 → basis A013611 → θ+1
  121, 528, 432 · 1331, 8712, 14256, 6912 … Σ 1081, 31211, 863857, 23226803, …
  (−∞, 0]nonnegativerational roots
• A131689 → x ↦ x+1 → D+1
  1 · 2, 1 · 8, 9, 2 · 44, 79, 42, 6 … Σ 1, 3, 19, 171, 1987, 28203, 472099, …
  (−∞, −1]nonnegative
• A085880 → (θ+1)²
  1 · 1, 4 · 2, 16, 18 · 5, 60, 135, 80 … Σ 1, 5, 36, 280, 2240, 18144, 147840, …
  (−∞, 0]nonnegative
• A122076 → θ+1
  2 · 3, 4 · 7, 16, 6 · 18, 60, 45, 8 … Σ 2, 7, 29, 131, 601, 2735, 12293, 54619, …
  (−∞, 0]nonnegative
• A028338 → basis A013610 → (θ+1)²
  1 · 2, 12 · 8, 72, 81 · 48, 528, 972, 432 … Σ 1, 14, 161, 1980, 27189, 418626, …
  (−∞, 0]nonnegative
• A028326 → basis A013610 → D
  6 · 24, 36 · 72, 216, 162 · 192, 864, 1296, 648 … Σ 0, 6, 60, 450, 3000, 18750, 112500, …
  (−∞, 0]nonnegativerational roots
• A038243 → basis A013609 → x ↦ x/(x+1)
  1 · 6, 8 · 36, 96, 64 · 216, 864, 1152, 512 … Σ 1, 14, 196, 2744, 38416, 537824, …
  [−1, 0]nonnegativerational roots
• A109692 → basis A013609 → D
  2 · 20, 24 · 200, 544, 360 · 2272, 9952, 14304, 6720 … Σ 0, 2, 44, 1104, 33248, 1184384, …
  (−∞, 0]nonnegative
• A121574: Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
  1 · 2, 1 · 4, 5, 1 · 8, 16, 8, 1 … Σ 1, 3, 10, 33, 109, 360, 1189, 3927, …
  (−∞, 0)nonnegative
• A052553 → basis A008277 → x ↦ x+1
  1 · 1 · 3, 1 · 5, 2 … Σ 1, 1, 4, 7, 21, 46, 129, 317, …
  (−∞, −1]nonnegative
• A008277 → θ+1 → reverse → D+1
  1 · 3, 1 · 9, 8, 1 · 22, 46, 17, 1 … Σ 1, 4, 18, 86, 435, 2332, 13237, 79382, …
  (−∞, 0]nonnegative
• A290319 → D
  1 · 6, 2 · 59, 30, 3 · 812, 508, 84, 4 … Σ 0, 1, 8, 92, 1408, 27024, 624768, …
  (−∞, 0]nonnegative
• A257622 → D
  4 · 56, 32 · 552, 1104, 192 … Σ 0, 4, 88, 1848, 41888, 1047200, …
  (−∞, 0]nonnegative
• A038221 → basis A013610 → x ↦ x/(x+1)
  1 · 6, 15 · 36, 180, 225 · 216, 1620, 4050, 3375 … Σ 1, 21, 441, 9261, 194481, 4084101, …
  [−1, 0]nonnegativerational roots
• A038331 → basis A013610 → x ↦ x+1
  1024, 960, 225 · 32768, 46080, 21600, 3375 … Σ 2209, 103823, 4879681, 229345007, …
  (−∞, −1]nonnegativerational roots
• A106246 → (θ+1)²
  1 · 2, 4 · 1, 16, 9 · 0, 12, 54, 16 … Σ 1, 6, 26, 82, 207, 446, 856, 1506, …
  (−∞, 0]nonnegative