Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A085880 → D → basis A007318
  1 · 8, 4 · 60, 60, 15 · 448, 672, 336, 56 … Σ 0, 1, 12, 135, 1512, 17010, 192456, …
  (−∞, −1]nonnegativerational roots
• A169803 → D → x ↦ x+1
  1 · 2 · 5, 2 · 10, 6 … Σ 0, 1, 2, 7, 16, 41, 94, 219, …
  (−∞, −1]nonnegative
• A113214 → Veronese 2 → x ↦ x+1
  1 · 2 · 1, 1 · 4, 4 … Σ 1, 2, 2, 8, 14, 28, 64, 128, …
  (−∞, −1]nonnegative
• A257620 → x ↦ x+1
  1 · 6, 3 · 54, 54, 9 · 648, 972, 378, 27 … Σ 1, 9, 117, 2025, 43821, 1137969, …
  (−∞, −1]nonnegative
• A063967 → basis A013610 → reverse
  1 · 3, 2 · 9, 15, 6 · 27, 72, 60, 16 … Σ 1, 5, 30, 175, 1025, 6000, 35125, …
  (−∞, 0]nonnegative
• A038314 → basis A013609 → reverse
  576, 1056, 484 · 13824, 38016, 34848, 10648 … Σ 2116, 97336, 4477456, 205962976, …
  (−∞, 0]nonnegativerational roots
• A173018 → reverse (shift +1) → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 1, 6, 6 … Σ 1, 1, 3, 13, 75, 541, 4683, 47293, …
  [−1, 0]nonnegative
• A038330 → basis A013610 → D
  384, 288 · 9216, 13824, 5184 … Σ 672, 28224, 1053696, 36879360, …
  (−∞, 0]nonnegativerational roots
• narayana → γ → D+1
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 19, 51, 141, 393, …
  (−∞, 0]nonnegative
• A172431 → basis A013609 → D+1
  1 · 7, 4 · 28, 44, 12 · 97, 252, 184, 32 … Σ 1, 11, 84, 565, 3587, 22056, 132841, …
  (−∞, 0]nonnegative
• A143491 → basis A007318 → x ↦ x/(x+1)
  12, 31, 20 · 60, 227, 286, 120 … Σ 63, 693, 9009, 135135, 2297295, …
  [−1, 0]nonnegativerational roots
• A110162 → x ↦ x+1
  1 · -1, 1 · -1, -2, 1 · 2, 0, -3, 1 … Σ 1, 0, -2, 0, 2, 0, -2, 0, …
  (−1, 3)
• A034807 → (θ+1)²
  2 · 1 · 1, 8 · 1, 12 … Σ 2, 1, 9, 13, 35, 66, 138, 267, …
  (−∞, 0]nonnegative
• A111577 → reverse
  1 · 1, 1 · 1, 5, 1 · 1, 12, 21, 1 … Σ 1, 2, 7, 35, 214, 1523, 12349, 112052, …
  (−∞, 0]nonnegative
• A046716: Coefficients of a special case of Poisson-Charlier polynomials.
  1 · 1, 1 · 1, 3, 1 · 1, 6, 8, 1 … Σ 1, 2, 5, 16, 65, 326, 1957, 13700, …
  (−∞, 0)nonnegative
• A257613 → (θ+1)²
  1 · 4, 16 · 16, 192, 144 · 64, 1664, 3744, 1024 … Σ 1, 20, 352, 6496, 129280, 2787840, …
  (−∞, 0]nonnegative
• A143495 → θ+1 → x ↦ x/(x+1)
  1 · 3, 5 · 9, 32, 26 · 27, 155, 265, 141 … Σ 1, 8, 67, 588, 5409, 52144, 526515, …
  [−1, 0]nonnegative
• A013609 → basis A013609 → x ↦ x/(x+1)
  1 · 3, 7 · 9, 42, 49 · 27, 189, 441, 343 … Σ 1, 10, 100, 1000, 10000, 100000, …
  [−1, 0]nonnegativerational roots
• A038259 → basis A013611 → (θ+1)²
  1 · 11, 80 · 121, 1760, 3600 … Σ 1, 91, 5481, 277171, 12686161, …
  (−∞, 0]nonnegative
• type_b_eulerian → γ (scale 1/4) → x ↦ x+1 → x ↦ x+1
  1 · 1 · 3, 1 · 11, 5 … Σ 1, 1, 4, 16, 100, 724, 6496, 67024, …
  (−∞, −1]nonnegative
• A007318 → −D²+(−1+n)D+1+n → x ↦ x+1 → basis A007318
  1 · 6, 2 · 31, 20, 3 · 144, 138, 42, 4 … Σ 1, 8, 54, 328, 1856, 9984, 51712, …
  (−∞, −1]nonnegative
• A143495 → reverse → θ+1
  1 · 1, 6 · 1, 14, 27 · 1, 24, 111, 108 … Σ 1, 7, 42, 244, 1433, 8669, 54477, …
  (−∞, 0]nonnegative
• A136572 → (θ+1)² → reverse
  1 · 4 · 18 · 96 … Σ 1, 4, 18, 96, 600, 4320, 35280, 322560, …
  (−∞, 0]nonnegativerational roots
• A110552 → reverse
  1 · 2, 1 · 4, 5, 1 · 8, 17, 10, 1 … Σ 1, 3, 10, 36, 136, 528, 2080, 8256, …
  (−∞, 0]nonnegativerational roots
• A256890 → θ+1
  4, 24, 12 · 8, 104, 156, 32 · 16, 392, 1248, 784, 80 … Σ 40, 300, 2520, 23520, 241920, 2721600, …
  (−∞, 0]nonnegative
• A035342: The convolution matrix of the double factorial of odd numbers (A001147).
  1 · 3, 1 · 15, 9, 1 · 105, 87, 18, 1 … Σ 1, 4, 25, 211, 2236, 28471, 422899, …
  (−∞, 0)nonnegative
• A081580 → γ → θ+1
  1 · 1 · 1, 10 · 1, 20 … Σ 1, 1, 11, 21, 106, 266, 1001, 2811, …
  (−∞, 0]nonnegative
• A026729 → (θ+1)²
  1 · 0, 4 · 0, 4, 9 · 0, 0, 18, 16 … Σ 1, 4, 13, 34, 82, 184, 395, 818, …
  (−∞, 0]nonnegative
• A178820 → basis A007318 → D+1
  1 · 12, 4 · 80, 60, 10 · 400, 480, 180, 20 … Σ 1, 16, 150, 1080, 6615, 36288, 183708, …
  (−∞, 0]nonnegativerational roots
• Veronese parent of A098172 → Veronese 3 (j=1) → x ↦ x+1
  1 · 1, 1 · 1, 2, 1 · 2, 5, 4, 1 … Σ 1, 2, 4, 12, 48, 192, 720, 2592, …
  (−∞, −1]nonnegative
• A120910 → D → θ+1
  3 · 12, 12 · 36, 72, 27 · 96, 288, 216, 48 … Σ 0, 3, 24, 135, 648, 2835, 11664, 45927, …
  (−∞, 0]nonnegativerational roots
• A075498 → reverse
  1 · 1, 3 · 1, 9, 9 · 1, 18, 63, 27 … Σ 1, 4, 19, 109, 742, 5815, 51193, …
  (−∞, 0]nonnegative
• A322944 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 10, 9 · 6, 56, 109, 60 … Σ 1, 3, 21, 231, 3369, 60171, 1262013, …
  [−1, 0]nonnegative
• A169803 → D → D+1
  1 · 2 · 5, 2 · 10, 6 … Σ 0, 1, 2, 7, 16, 38, 82, 173, …
  (−∞, 0]nonnegative
• A137597 → x ↦ x+1 → D
  1 · 7, 2 · 43, 24, 3 · 267, 218, 54, 4 … Σ 0, 1, 9, 70, 543, 4351, 36402, 318907, …
  (−∞, 0]nonnegative
• A133080 → basis A013609 → D
  2 · 4, 8 · 10, 32, 24 · 8, 48, 96, 64 … Σ 0, 2, 12, 66, 216, 1026, 2916, 13122, …
  (−∞, 0]nonnegativerational roots
• A060920 → reverse → D
  2 · 5, 10 · 9, 40, 39 · 14, 102, 213, 136 … Σ 0, 2, 15, 88, 465, 2314, 11075, 51580, …
  (−∞, 0]nonnegative
• A391045 → θ+1 → x ↦ x+1
  1 · 3, 2 · 12, 14, 3 · 53, 90, 42, 4 … Σ 1, 5, 29, 189, 1289, 9013, 64085, …
  (−∞, −1]nonnegative
• A038337 → basis A007318 → (θ+1)²
  529, 2024, 1089 · 12167, 69828, 75141, 21296 … Σ 3642, 178432, 8202976, 361046544, …
  (−∞, 0]nonnegative
• A116412 → basis A007318 → x ↦ x/(x+1)
  1 · 4, 5 · 13, 34, 22 · 43, 171, 225, 98 … Σ 1, 9, 69, 537, 4173, 32433, 252069, …
  [−1, 0]nonnegative
• A286724 → x ↦ x/(x+1)
  1 · 2, 3 · 8, 24, 17 · 48, 216, 306, 139 … Σ 1, 5, 49, 709, 13505, 318181, 8916145, …
  [−1, 0]nonnegative
• A038327 → D
  1 · 24, 2 · 432, 72, 3 · 6912, 1728, 144, 4 … Σ 0, 1, 26, 507, 8788, 142805, 2227758, …
  (−∞, 0]nonnegativerational roots
• A269947 → D+1
  1 · 1, 1 · 1, 3, 1 · 8, 26, 12, 1 … Σ 1, 2, 5, 47, 1334, 87214, 11021724, …
  (−∞, 0]nonnegative
• A113953 → basis A013611 → reverse
  1 · 4, 1 · 16, 16, 3 · 64, 112, 44, 5 … Σ 1, 5, 35, 225, 1475, 9625, 62875, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → basis A013609
  1 · -2, -2 · 4, 8, 4 · -8, -24, -24, -8 … Σ 1, -4, 16, -64, 256, -1024, 4096, …
  (−∞, 0]rational roots
• A130123 → D → D+1
  2 · 8, 8 · 0, 48, 24 · 0, 0, 192, 64 … Σ 0, 2, 16, 72, 256, 800, 2304, 6272, …
  (−∞, 0]nonnegativerational roots
• A049403 → D
  3, 2 · 3, 12, 3 · 0, 30, 30, 4 · 0, 30, 135, 60, 5 … Σ 5, 18, 64, 230, 846, 3220, 12608, …
  (−∞, 0]nonnegative
• A038305 → x ↦ x+1 → θ+1
  169, 156, 27 · 2197, 3042, 1053, 108 … Σ 352, 6400, 114688, 2031616, 35651584, …
  (−∞, 0]nonnegativerational roots
• A156992 → basis A013610 → x ↦ x+1
  1 · 10, 6 · 150, 180, 54 · 3000, 5400, 3240, 648 … Σ 1, 16, 384, 12288, 491520, 23592960, …
  (−∞, −1]nonnegativerational roots
• A034870 → basis A013611 → D
  16, 32 · 128, 768, 1536, 1024 … Σ 0, 48, 3456, 186624, 8957952, …
  (−∞, 0]nonnegativerational roots