Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A257607 → (θ+1)²
  1 · 5, 20 · 25, 240, 225 · 125, 2140, 4815, 2000 … Σ 1, 25, 490, 9080, 167860, 3164040, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → basis A008277 → θ+1
  1, 6, 3 · 1, 14, 18, 4 · 1, 30, 75, 40, 5 … Σ 10, 37, 151, 674, 3263, 17007, 94828, …
  (−∞, 0]nonnegative
• A059344 → D
  2 · 6 · 12, 24 · 20, 120 … Σ 0, 0, 2, 6, 36, 140, 750, 3402, …
  (−∞, 0]nonnegative
• A038325 → basis A013609 → (θ+1)²
  1 · 22, 88 · 484, 3872, 4356 … Σ 1, 110, 8712, 596288, 37480960, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729, shifted → basis A013609 → basis A007318
  1 · -3, -2 · 9, 12, 4 · -27, -54, -36, -8 … Σ 1, -5, 25, -125, 625, -3125, 15625, …
  (−∞, −1]rational roots
• delannoy → basis A013611 → x ↦ x/(x+1)
  1 · 2, 6 · 5, 30, 41 · 12, 108, 308, 276 … Σ 1, 8, 76, 704, 6544, 60800, 564928, …
  [−1, 0]nonnegative
• A056939 → x ↦ x+1 → x ↦ x+1
  1 · 3, 1 · 13, 8, 1 · 69, 62, 16, 1 … Σ 1, 4, 22, 148, 1132, 9484, 85066, …
  (−∞, −1]nonnegative
• A154986 → γ → basis A007318
  1 · 1 · 3, 2 · 9, 8 … Σ 1, 1, 5, 17, 137, 817, 9037, 77665, …
  (−∞, −1]nonnegative
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewher…
  1 · 1, 2 · 1, 4, 3 · 1, 6, 9, 4 … Σ 1, 3, 8, 20, 48, 112, 256, 576, …
  (−∞, 0)nonnegativerational roots
• A008278 → x ↦ x+1 (shift +1) → reverse → D
  2 · 5, 10 · 10, 46, 45 · 19, 152, 327, 208 … Σ 0, 2, 15, 101, 706, 5268, 42149, …
  (−∞, 0]nonnegative
• A225468 → D+1
  1 · 3, 1 · 11, 9, 1 · 47, 69, 18, 1 … Σ 1, 4, 21, 135, 1008, 8505, 79785, …
  (−∞, 0]nonnegative
• A116414 → basis A013609 → reverse
  1 · 2, 5 · 4, 20, 22 · 8, 60, 138, 95 … Σ 1, 7, 46, 301, 1969, 12880, 84253, …
  (−∞, 0]nonnegative
• A094638 → basis A013611 → x ↦ x/(x+1)
  1 · 2, 6 · 6, 40, 66 · 24, 256, 904, 1056 … Σ 1, 8, 112, 2240, 58240, 1863680, …
  [−1, 0]nonnegativerational roots
• A193685 → reverse
  1 · 1, 5 · 1, 11, 25 · 1, 18, 91, 125 … Σ 1, 6, 37, 235, 1540, 10427, 73013, …
  (−∞, 0]nonnegative
• A081577 → basis A013611 → x ↦ x+1
  1 · 6, 4 · 46, 56, 16 · 336, 608, 352, 64 … Σ 1, 10, 118, 1360, 15724, 181720, …
  (−∞, −1]nonnegative
• A038336 → x ↦ x+1 → reverse
  100, 440, 484 · 1000, 6600, 14520, 10648 … Σ 1024, 32768, 1048576, 33554432, …
  (−∞, 0]nonnegativerational roots
• A225471 → basis A013611 → θ+1
  1 · 4, 8 · 32, 96, 48 · 384, 1408, 1152, 256 … Σ 1, 12, 176, 3200, 70144, 1806336, …
  (−∞, 0]nonnegative
• A249252 → (θ+1)²
  1 · 1, 16 · 1, 32, 72 · 1, 48, 288, 256 … Σ 1, 17, 105, 593, 3049, 14801, 68873, …
  (−∞, 0]nonnegative
• A137597 → reverse → D
  2 · 5, 10 · 9, 44, 45 · 14, 122, 297, 208 … Σ 0, 2, 15, 98, 641, 4344, 30823, 229606, …
  (−∞, 0]nonnegative
• A038325: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.
  1 · 11, 11 · 121, 242, 121 · 1331, 3993, 3993, 1331 … Σ 1, 22, 484, 10648, 234256, 5153632, …
  (−∞, 0)nonnegativerational roots
• A075505 → x ↦ x+1
  131, 32, 1 · 1761, 823, 63, 1 … Σ 164, 2648, 50816, 1134432, 28670784, …
  (−∞, −1]nonnegative
• A081578 → basis A013610 → x ↦ x+1
  1 · 5, 3 · 37, 39, 9 · 245, 387, 189, 27 … Σ 1, 8, 85, 848, 8569, 86360, 870829, …
  (−∞, −1]nonnegative
• A122542 → x ↦ x+1 → basis A007318
  1 · 2, 1 · 8, 6, 1 · 28, 30, 10, 1 … Σ 1, 3, 15, 69, 321, 1491, 6927, 32181, …
  (−∞, −1]nonnegative
• A133087 → θ+1
  1 · 2, 2 · 1, 4, 3 · 2, 10, 12, 4 … Σ 1, 4, 8, 28, 48, 160, 256, 832, …
  (−∞, 0]nonnegative
• A111125 → basis A013609 → (θ+1)²
  1 · 4, 8 · 11, 56, 36 · 29, 248, 360, 128 … Σ 1, 12, 103, 765, 5220, 33675, 208685, …
  (−∞, 0]nonnegative
• A178822 → (θ+1)²
  1 · 6, 24 · 21, 168, 189 · 56, 672, 1512, 896 … Σ 1, 30, 378, 3136, 20160, 108864, …
  (−∞, 0]nonnegative
• A038281 → x ↦ x+1
  121, 66, 9 · 1331, 1089, 297, 27 … Σ 196, 2744, 38416, 537824, 7529536, …
  (−∞, −1]nonnegativerational roots
• A038305 → reverse
  9, 60, 100 · 27, 270, 900, 1000 … Σ 169, 2197, 28561, 371293, 4826809, …
  (−∞, 0]nonnegativerational roots
• A038334 → basis A013609 → D+1
  1040, 1152, 256 · 27200, 49920, 27648, 4096 … Σ 2448, 108864, 4665600, 194835456, …
  (−∞, 0]nonnegativerational roots
• A038220 → Veronese 2 (j=1) → D
  8 · 96 · 720, 64 · 4320, 1152 … Σ 0, 0, 0, 8, 96, 784, 5472, 35160, …
  (−∞, 0]nonnegative
• A161198 → basis A008277 → reverse
  1 · 2, 3 · 4, 20, 15 · 8, 84, 210, 105 … Σ 1, 5, 39, 407, 5281, 81597, 1459015, …
  (−∞, 0]nonnegative
• A059473 → basis A013610 → D+1
  1 · 10, 6 · 72, 126, 36 · 440, 1368, 1152, 216 … Σ 1, 16, 234, 3176, 41240, 519552, …
  (−∞, 0]nonnegative
• A038277 → x ↦ x+1 → D
  11 · 396, 242 · 10692, 13068, 3993 … Σ 0, 11, 638, 27753, 1073116, 38900455, …
  (−∞, 0]nonnegativerational roots
• A121574 → D
  1 · 5, 2 · 16, 16, 3 · 44, 74, 33, 4 … Σ 0, 1, 7, 35, 155, 642, 2550, 9841, …
  (−∞, 0]nonnegative
• A060920 → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 9, 20, 13 … Σ 1, 3, 11, 43, 171, 683, 2731, 10923, …
  (−∞, 0]nonnegative
• A028338 → basis A013609 → θ+1
  1 · 2, 4 · 8, 24, 12 · 48, 176, 144, 32 … Σ 1, 6, 44, 400, 4384, 56448, 836352, …
  (−∞, 0]nonnegative
• A038284 → basis A013611 → θ+1
  196, 1344, 1728 · 2744, 28224, 72576, 55296 … Σ 3268, 158840, 7352848, 329451488, …
  (−∞, 0]nonnegativerational roots
• A113214 → basis A007318 → θ+1
  1 · 3, 2 · 4, 10, 3 · 7, 26, 21, 4 … Σ 1, 5, 17, 58, 188, 596, 1852, 5672, …
  (−∞, 0]nonnegative
• A372873 → θ+1
  1 · 0, 4 · 0, 2, 12 · 0, 0, 18, 32 … Σ 1, 4, 14, 50, 179, 632, 2192, 7478, …
  (−∞, 0]nonnegative
• A207823 → basis A013610 → θ+1
  1 · 5, 6 · 24, 60, 27 · 115, 438, 405, 108 … Σ 1, 11, 111, 1066, 9905, 89889, 801439, …
  (−∞, 0]nonnegative
• A038291 → D
  1 · 18, 2 · 243, 54, 3 · 2916, 972, 108, 4 … Σ 0, 1, 20, 300, 4000, 50000, 600000, …
  (−∞, 0]nonnegativerational roots
• A152198 → basis A013611 → θ+1
  1 · 1 · 2, 8 · 2, 8 … Σ 1, 1, 10, 10, 84, 84, 648, 648, …
  (−∞, 0]nonnegativerational roots
• A008288: Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
  1 · 1, 1 · 1, 3, 1 · 1, 5, 5, 1 … Σ 1, 2, 5, 12, 29, 70, 169, 408, …
  [−3−2√2, −3+2√2]nonnegative
• A038329 → basis A013610 → (θ+1)²
  225, 1080, 729 · 3375, 24300, 32805, 11664 … Σ 2034, 72144, 2384640, 75147264, …
  (−∞, 0]nonnegative
• A104029 → θ+1
  1 · 2, 2 · 3, 10, 3 · 4, 26, 27, 4 … Σ 1, 4, 16, 61, 223, 790, 2734, 9295, …
  (−∞, 0]nonnegative
• A075497 → x ↦ x+1 → x ↦ x+1
  1 · 4, 1 · 20, 10, 1 · 120, 88, 18, 1 … Σ 1, 5, 31, 227, 1897, 17693, 181351, …
  (−∞, −1]nonnegative
• A249250 → reverse → x ↦ x/(x+1)
  1 · 3, 4 · 3, 12, 10 · 3, 24, 48, 28 … Σ 1, 7, 25, 103, 409, 1639, 6553, 26215, …
  [−1, 0]nonnegative
• A110441 → basis A013611 → θ+1
  1 · 4, 8 · 14, 64, 48 · 48, 352, 576, 256 … Σ 1, 12, 126, 1232, 11524, 104592, …
  (−∞, 0]nonnegative
• A104684 → D+1
  1 · 3, 1 · 12, 8, 1 · 50, 54, 15, 1 … Σ 1, 4, 21, 120, 705, 4188, 24997, …
  (−∞, 0]nonnegative
• A049020 → x ↦ x+1 → D
  1 · 5, 2 · 25, 18, 3 · 133, 134, 42, 4 … Σ 0, 1, 7, 46, 313, 2251, 17170, 138727, …
  (−∞, 0]nonnegative