Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A030528 → basis A013611 → θ+1
  1 · 2, 8 · 3, 32, 48 · 5, 80, 288, 256 … Σ 1, 10, 83, 629, 4520, 31365, 212325, …
  (−∞, 0]nonnegative
• A156289 → x ↦ x+1
  1 · 4, 3 · 31, 45, 15 · 379, 798, 525, 105 … Σ 1, 7, 91, 1807, 49651, 1777447, …
  (−∞, −1]nonnegative
• A391533 → Veronese 2 (j=1) → D
  1 · 6 · 19, 2 · 44, 16 … Σ 0, 1, 6, 21, 60, 156, 384, 912, …
  (−∞, 0]nonnegative
• A130123 → Veronese 2 (j=1) → θ+1
  2 · 0, 16 · 0, 0, 96 · 0, 0, 0, 512 … Σ 0, 2, 0, 16, 0, 96, 0, 512, …
  (−∞, 0]nonnegativerational roots
• A094638 → D+1 → D+1
  1 · 3, 1 · 11, 11, 2 · 35, 86, 47, 6 … Σ 1, 4, 24, 174, 1430, 13040, 130592, …
  (−∞, 0]nonnegative
• A152830 → basis A013609 → basis A007318
  1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 6, 6, 6, 6, …
  (−∞, −1]nonnegativerational roots
• A257607 → θ+1
  1 · 5, 10 · 25, 120, 75 · 125, 1070, 1605, 500 … Σ 1, 15, 220, 3300, 51480, 840840, …
  (−∞, 0]nonnegative
• A094638 → θ+1
  1 · 1, 2 · 1, 6, 6 · 1, 12, 33, 24 … Σ 1, 3, 13, 70, 446, 3276, 27252, 253296, …
  (−∞, 0]nonnegative
• A143685 → γ → basis A007318
  1 · 1 · 10, 9 · 19, 18 … Σ 1, 1, 19, 37, 379, 1045, 7867, 26677, …
  (−∞, −1]nonnegative
• A113953 → x ↦ x−8
  1 · -8, 1 · 48, -14, 1 · -256, 128, -20, 1 … Σ 1, -7, 35, -147, 539, -1715, 4459, …
  (0, 8]
• A038321 → basis A007318 → (θ+1)²
  324, 1008, 441 · 5832, 27216, 23814, 5488 … Σ 1773, 62350, 2070625, 66093750, …
  (−∞, 0]nonnegative
• A086645 → basis A013610 → reverse
  1 · 3, 2 · 9, 24, 8 · 27, 162, 144, 32 … Σ 1, 5, 41, 365, 3281, 29525, 265721, …
  (−∞, 0]nonnegative
• A038243 → basis A013609 → D+1
  1 · 8, 2 · 60, 32, 4 · 432, 360, 96, 8 … Σ 1, 10, 96, 896, 8192, 73728, 655360, …
  (−∞, 0]nonnegativerational roots
• A038208 → basis A013611 → θ+1
  1 · 4, 16 · 16, 128, 192 · 64, 768, 2304, 2048 … Σ 1, 20, 336, 5184, 76032, 1078272, …
  (−∞, 0]nonnegativerational roots
• A099040 → basis A013610 → θ+1
  1 · 2, 12 · 6, 60, 108 · 16, 240, 864, 864 … Σ 1, 14, 174, 1984, 21536, 226176, …
  (−∞, 0]nonnegative
• A038336 → x ↦ x/(x+1) → (θ+1)²
  144, 2112, 4356 · 1728, 38016, 156816, 170368 … Σ 6612, 366928, 18426640, 866889024, …
  (−∞, 0]nonnegative
• A122848 → basis A013609 → reverse
  1 · 2, 1 · 4, 6, 2 · 8, 24, 18, 4 … Σ 1, 3, 12, 54, 270, 1458, 8424, 51516, …
  (−∞, 0]nonnegative
• A038324 → basis A007318 → x ↦ x/(x+1)
  1 · 21, 31 · 441, 1302, 961 … Σ 1, 52, 2704, 140608, 7311616, …
  [−1, 0]nonnegativerational roots
• A038255 → basis A013610 → D+1
  1 · 10, 3 · 91, 60, 9 · 784, 819, 270, 27 … Σ 1, 13, 160, 1900, 22000, 250000, …
  (−∞, 0]nonnegativerational roots
• A391045 → θ+1 → reverse
  1 · 2, 1 · 3, 8, 1 · 4, 30, 18, 1 … Σ 1, 3, 12, 53, 244, 1152, 5531, 26875, …
  (−∞, 0]nonnegative
• A131689 → D → x ↦ x+1
  1 · 5, 4 · 31, 48, 18 · 233, 532, 396, 96 … Σ 0, 1, 9, 97, 1257, 19201, 338889, …
  (−∞, −1]nonnegative
• A013609 → basis A008288 → x ↦ x/(x+1)
  1 · 3, 5 · 9, 34, 29 · 27, 163, 297, 169 … Σ 1, 8, 72, 656, 5984, 54592, 498048, …
  [−1, 0]nonnegative
• basis-map coefficients of A026729, shifted → basis A013609 → θ+1
  1 · -1, -4 · 1, 8, 12 · -1, -12, -36, -32 … Σ 1, -5, 21, -81, 297, -1053, 3645, …
  (−∞, 0]rational roots
• A178822 → basis A013611 → θ+1
  1 · 12, 48 · 84, 672, 1008 · 448, 5376, 16128, 14336 … Σ 1, 60, 1764, 36288, 598752, 8491392, …
  (−∞, 0]nonnegativerational roots
• A039755 → reverse → (θ+1)²
  1 · 1, 4 · 1, 16, 9 · 1, 36, 117, 16 … Σ 1, 5, 26, 170, 1252, 9972, 85448, …
  (−∞, 0]nonnegative
• A038331 → basis A013609 → D+1
  629, 540, 100 · 13583, 18870, 8100, 1000 … Σ 1269, 41553, 1318761, 40920957, …
  (−∞, 0]nonnegativerational roots
• A139548 → x ↦ x+1 → θ+1
  1 · 8, 16, 6 · 64, 256, 288, 128, 20 … Σ 1, 30, 756, 17496, 384912, 8188128, …
  (−∞, 0]nonnegativerational roots
• A143494 → θ+1 → D
  2 · 10, 6 · 38, 54, 12 · 130, 330, 168, 20 … Σ 0, 2, 16, 104, 648, 4062, 26108, …
  (−∞, 0]nonnegative
• A269944 → x ↦ x+1 → D+1
  1 · 2, 1 · 5, 5, 1 · 27, 33, 11, 1 … Σ 1, 3, 11, 72, 828, 15300, 420228, …
  (−∞, 0]nonnegative
• A107131 → basis A007318
  1 · 1, 1 · 2, 3, 1 · 4, 9, 6, 1 … Σ 1, 2, 6, 20, 72, 272, 1064, 4272, …
  (−∞, −1]nonnegative
• A145324 → basis A007318 → D
  2 · 17, 12 · 133, 196, 72 · 1098, 2506, 1902, 480 … Σ 0, 2, 29, 401, 5986, 98608, 1794435, …
  (−∞, 0]nonnegative
• A143494 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 3, 4 · 10, 27, 18 · 37, 151, 203, 90 … Σ 1, 7, 55, 481, 4633, 48655, 552271, …
  [−1, 0]nonnegative
• A120910 → basis A013609 → D
  6 · 36, 24 · 162, 216, 72 · 648, 1296, 864, 192 … Σ 0, 6, 60, 450, 3000, 18750, 112500, …
  (−∞, 0]nonnegativerational roots
• A130850 → D+1
  1 · 2, 1 · 5, 5, 1 · 18, 26, 10, 1 … Σ 1, 3, 11, 55, 359, 2891, 27635, 305439, …
  (−∞, 0]nonnegative
• A132393 → Veronese 2 (j=1) → basis A007318
  1 · 1 · 3, 1 · 12, 6 … Σ 0, 1, 1, 4, 18, 98, 630, 4676, …
  (−∞, −1]nonnegative
• A064861 → Veronese 2 (j=1) → D+1
  3 · 9, 4 · 18, 12 · 46, 54, 8 … Σ 3, 13, 30, 108, 243, 837, 1836, 6156, …
  (−∞, 0]nonnegative
• A013609 → basis A008288 → basis A007318
  1 · 5, 2 · 29, 24, 4 · 169, 210, 76, 8 … Σ 1, 7, 57, 463, 3761, 30551, 248169, …
  (−∞, −1]nonnegative
• A075497 → reverse → θ+1
  1 · 1, 4 · 1, 12, 12 · 1, 24, 84, 32 … Σ 1, 5, 25, 141, 901, 6393, 49589, …
  (−∞, 0]nonnegative
• Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(…
  1, 6, 1 · 1, 15, 15, 1 · 1, 34, 90, 34, 1 … Σ 8, 32, 160, 960, 6720, 53760, 483840, …
  (−∞, 0)nonnegative
• A035469 → (θ+1)²
  1 · 4, 4 · 28, 48, 9 · 280, 640, 216, 16 … Σ 1, 8, 85, 1152, 19065, 373096, 8431773, …
  (−∞, 0]nonnegative
• A193685 → x ↦ x+1
  1 · 6, 1 · 37, 13, 1 · 235, 130, 21, 1 … Σ 1, 7, 51, 387, 3059, 25179, 215675, …
  (−∞, −1]nonnegative
• A345013 → D+1
  1 · 7, 3 · 35, 32, 6 · 161, 225, 90, 10 … Σ 1, 10, 73, 486, 3113, 19538, 121105, …
  (−∞, 0]nonnegative
• A046716 → basis A013610 → x ↦ x+1
  1 · 5, 3 · 29, 33, 9 · 217, 354, 180, 27 … Σ 1, 8, 71, 778, 12125, 284012, 9241891, …
  (−∞, −1]nonnegative
• A026729 → basis A008277 → θ+1
  1 · 1, 2 · 2, 8, 3 · 3, 26, 24, 4 … Σ 1, 3, 13, 57, 272, 1389, 7576, 43929, …
  (−∞, 0]nonnegative
• A063967 → Veronese 2 → x ↦ x+1
  1 · 1 · 3, 1 · 8, 5 … Σ 1, 1, 4, 13, 41, 130, 413, 1311, …
  (−∞, −1]nonnegative
• A035342 → x ↦ x/(x+1)
  1 · 3, 4 · 15, 39, 25 · 105, 402, 507, 211 … Σ 1, 7, 79, 1225, 24121, 575311, …
  [−1, 0]nonnegative
• A062196 → D → D+1
  3 · 20, 12 · 75, 120, 30 · 204, 660, 420, 60 … Σ 0, 3, 32, 225, 1344, 7350, 38016, …
  (−∞, 0]nonnegative
• A274498 → x ↦ x+1 → (θ+1)²
  1 · 3 · 9, 24 · 27, 72 … Σ 1, 3, 33, 99, 837, 2511, 18765, 56295, …
  (−∞, 0]nonnegative
• A102413 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 9, 21, 14 … Σ 1, 3, 13, 45, 161, 573, 2041, 7269, …
  [−1, 0]nonnegative
• A085880 → D → θ+1
  1 · 4, 8 · 15, 60, 45 · 56, 336, 504, 224 … Σ 0, 1, 12, 120, 1120, 10080, 88704, …
  (−∞, 0]nonnegativerational roots