Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A127773 → θ+1 → D+1
  1 · 6, 6 · 0, 36, 18 · 0, 0, 120, 40 … Σ 1, 12, 54, 160, 375, 756, 1372, 2304, …
  (−∞, 0]nonnegativerational roots
• A086645 → basis A013611 → D+1
  1 · 6, 4 · 40, 64, 16 · 224, 768, 480, 64 … Σ 1, 10, 120, 1536, 19328, 237056, …
  (−∞, 0]nonnegative
• A292219 → θ+1
  1 · 6, 2 · 60, 40, 3 · 840, 840, 126, 4 … Σ 1, 8, 103, 1810, 40109, 1069756, …
  (−∞, 0]nonnegative
• A081580 → D+1
  1 · 2, 1 · 8, 9, 1 · 14, 39, 16, 1 … Σ 1, 3, 18, 70, 303, 1197, 4756, 18396, …
  (−∞, 0]nonnegative
• A257616: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.
  1 · 2, 2 · 4, 32, 4 · 8, 312, 312, 8 … Σ 1, 4, 40, 640, 14080, 394240, 13404160, …
  (−∞, 0)nonnegative
• A152842 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 6, 8 · 1, 10, 32, 32 … Σ 1, 5, 15, 75, 225, 1125, 3375, 16875, …
  [−1, 0]nonnegativerational roots
• A274496 → D → x ↦ x+1
  2 · 4 · 16, 8 · 32, 16 … Σ 0, 0, 2, 4, 24, 48, 216, 432, …
  (−∞, −1]nonnegativerational roots
• A116412 → basis A013611 → x ↦ x/(x+1)
  1 · 4, 8 · 13, 58, 61 · 43, 297, 657, 467 … Σ 1, 12, 132, 1464, 16224, 179808, …
  [−1, 0]nonnegative
• A038271 → basis A013609 → x ↦ x/(x+1)
  1 · 12, 22 · 144, 528, 484 · 1728, 9504, 17424, 10648 … Σ 1, 34, 1156, 39304, 1336336, 45435424, …
  [−1, 0]nonnegativerational roots
• A116666 → Veronese 2 (j=1) → θ+1
  3 · 6 · 9, 14 · 12, 56 … Σ 0, 3, 6, 23, 68, 188, 496, 1264, …
  (−∞, 0]nonnegative
• A030528 → Veronese 2 (j=1) → x ↦ x+1
  1 · 2 · 2, 1 · 4, 4 … Σ 0, 1, 2, 3, 8, 16, 32, 70, …
  (−∞, −1]nonnegative
• A028326 → (θ+1)²
  2 · 2, 8 · 2, 16, 18 · 2, 24, 54, 32 … Σ 2, 10, 36, 112, 320, 864, 2240, 5632, …
  (−∞, 0]nonnegative
• narayana → basis A007318 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 14, 63, 93, 45 … Σ 1, 5, 31, 215, 1597, 12425, 99955, …
  [−1, 0]nonnegative
• A360282 → D
  1 · 3, 4 · 10, 24, 9 · 35, 120, 90, 16 … Σ 0, 1, 7, 43, 261, 1581, 9563, 57751, …
  (−∞, 0]nonnegative
• image under θ+1 → (θ+1)² → θ+1
  1 · 0, 16 · 0, 0, 81 · 0, 0, 0, 256 … Σ 1, 16, 81, 256, 625, 1296, 2401, 4096, …
  (−∞, 0]nonnegativerational roots
• A099040 → basis A013609 → D
  4 · 20, 32 · 80, 256, 192 · 288, 1376, 2112, 1024 … Σ 0, 4, 52, 528, 4800, 41040, 337392, …
  (−∞, 0]nonnegative
• A156992 → D
  2 · 12, 12 · 72, 144, 72 · 480, 1440, 1440, 480 … Σ 0, 2, 24, 288, 3840, 57600, 967680, …
  (−∞, 0]nonnegativerational roots
• A028326 → Veronese 2 → D+1
  2 · 2 · 4, 2 · 8, 6 … Σ 2, 2, 6, 14, 32, 72, 160, 352, …
  (−∞, 0]nonnegative
• A088996 → basis A013610 → θ+1
  1 · 1, 6 · 3, 30, 54 · 15, 204, 675, 648 … Σ 1, 7, 87, 1542, 35346, 991944, …
  (−∞, 0]nonnegative
• A111125 → basis A013609 → D+1
  1 · 6, 2 · 25, 22, 4 · 91, 142, 64, 8 … Σ 1, 8, 51, 305, 1752, 9787, 53565, …
  (−∞, 0]nonnegative
• A132166 → x ↦ x/(x+1)
  336, 714, 379 · 4536, 14721, 15897, 5713 … Σ 1429, 40867, 1065877, 26294227, …
  [−1, 0)nonnegative
• A059110 → D
  1 · 4, 2 · 21, 18, 3 · 136, 156, 48, 4 … Σ 0, 1, 6, 42, 344, 3240, 34512, 410032, …
  (−∞, 0]nonnegative
• A033282 → x ↦ x+1 → reverse
  1 · 2, 3 · 5, 15, 11 · 14, 63, 93, 45 … Σ 1, 5, 31, 215, 1597, 12425, 99955, …
  (−1, 0]nonnegative
• A110441 → basis A007318 → (θ+1)²
  1 · 4, 4 · 14, 32, 9 · 48, 176, 108, 16 … Σ 1, 8, 55, 348, 2087, 12056, 67735, …
  (−∞, 0]nonnegative
• A094305 → x ↦ x+1 → D+1
  1 · 9, 3 · 48, 36, 6 · 200, 240, 90, 10 … Σ 1, 12, 90, 540, 2835, 13608, 61236, …
  (−∞, 0]nonnegativerational roots
• A131436 → reverse
  1 · 3 · 7 · 15 … Σ 1, 3, 7, 15, 31, 63, 127, 255, …
  (−∞, 0]nonnegativerational roots
• A038303 → basis A013611 → reverse
  1 · 4, 11 · 16, 88, 121 · 64, 528, 1452, 1331 … Σ 1, 15, 225, 3375, 50625, 759375, …
  (−∞, 0]nonnegativerational roots
• A166960 → D
  1 · 6, 2 · 27, 42, 3 · 112, 540, 174, 4 … Σ 0, 1, 8, 72, 830, 11992, 212664, …
  (−∞, 0]nonnegative
• A143495 → x ↦ x+1
  1 · 4, 1 · 17, 9, 1 · 77, 64, 15, 1 … Σ 1, 5, 27, 157, 979, 6517, 46107, …
  (−∞, −1]nonnegative
• basis-map coefficients of A026729, shifted → (θ+1)² → basis A007318
  1 · -4, -4 · 9, 18, 9 · -16, -48, -48, -16 … Σ 1, -8, 36, -128, 400, -1152, 3136, …
  (−∞, −1]rational roots
• basis-map coefficients of A113214, shifted → basis A013609 → θ+1
  1 · -2, -4 · 4, 16, 12 · -8, -48, -72, -32 … Σ 1, -6, 32, -160, 768, -3584, 16384, …
  (−∞, 0]rational roots
• A056241 → D+1
  1 · 2, 1 · 4, 5, 1 · 7, 18, 9, 1 … Σ 1, 3, 10, 35, 123, 427, 1460, 4923, …
  (−∞, 0]nonnegative
• A049020: Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.
  1 · 1, 1 · 2, 3, 1 · 5, 10, 6, 1 … Σ 1, 2, 6, 22, 94, 454, 2430, 14214, …
  (−∞, −1]nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
  144, 288, 144 · 1728, 5184, 5184, 1728 … Σ 576, 13824, 331776, 7962624, 191102976, …
  (−∞, 0)nonnegativerational roots
• A143494 → x ↦ x+1 → reverse
  1 · 1, 3 · 1, 7, 10 · 1, 12, 40, 37 … Σ 1, 4, 18, 90, 494, 2946, 18926, 130066, …
  (−∞, 0]nonnegative
• A026374 → D+1 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 13, 10 · 5, 27, 46, 25 … Σ 1, 5, 27, 103, 473, 1661, 7139, 24079, …
  [−1, 0]nonnegative
• A371080 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 4, 5 · 0, 28, 68, 41 … Σ 1, 1, 9, 137, 2929, 80561, 2707641, …
  [−1, 0]nonnegative
• A038327 → x ↦ x+1 → reverse
  1 · 1, 13 · 1, 26, 169 · 1, 39, 507, 2197 … Σ 1, 14, 196, 2744, 38416, 537824, …
  (−∞, 0]nonnegativerational roots
• A038226 → basis A013609 → x ↦ x+1
  729, 864, 256 · 19683, 34992, 20736, 4096 … Σ 1849, 79507, 3418801, 147008443, …
  (−∞, −1]nonnegativerational roots
• A086645 → basis A007318
  1 · 2, 1 · 8, 8, 1 · 32, 48, 18, 1 … Σ 1, 3, 17, 99, 577, 3363, 19601, 114243, …
  (−∞, −1]nonnegative
• A130123 → basis A008277 → x ↦ x/(x+1)
  1 · 2, 4 · 4, 20, 20 · 8, 80, 184, 120 … Σ 1, 6, 44, 392, 4112, 49248, 659136, …
  [−1, 0]nonnegative
• A122542 → basis A013611 → reverse
  1 · 4, 1 · 16, 16, 3 · 64, 112, 52, 7 … Σ 1, 5, 35, 235, 1585, 10685, 72035, …
  (−∞, 0]nonnegative
• row-factor cofactor of A101842 → D
  1 · 4, 2 · 11, 22, 3 · 26, 132, 78, 4 … Σ 0, 1, 6, 36, 240, 1800, 15120, 141120, …
  (−∞, 0]nonnegative
• A038243 → basis A013609 → D
  2 · 24, 8 · 216, 144, 24 · 1728, 1728, 576, 64 … Σ 0, 2, 32, 384, 4096, 40960, 393216, …
  (−∞, 0]nonnegativerational roots
• A225117 → D+1
  1 · 3, 1 · 17, 15, 1 · 101, 213, 63, 1 … Σ 1, 4, 33, 378, 5508, 97200, 2012040, …
  (−∞, 0]nonnegative
• A038220 → basis A013610
  1 · 5, 6 · 25, 60, 36 · 125, 450, 540, 216 … Σ 1, 11, 121, 1331, 14641, 161051, …
  [−1, 0]nonnegativerational roots
• A143491 → basis A013610 → reverse
  9, 21, 12 · 27, 108, 141, 60 … Σ 42, 336, 3024, 30240, 332640, 3991680, …
  (−∞, 0]nonnegativerational roots
• A290319 → (θ+1)²
  1 · 1, 4 · 5, 24, 9 · 45, 236, 135, 16 … Σ 1, 5, 38, 432, 6592, 126512, 2928032, …
  (−∞, 0]nonnegative
• A161198 → basis A013609 → D
  4 · 32, 32 · 284, 480, 192 · 2976, 6592, 4608, 1024 … Σ 0, 4, 64, 956, 15200, 264036, 5029152, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → basis A013609 → D
  4, 8 · 6, 24, 24 · 8, 48, 96, 64 … Σ 12, 54, 216, 810, 2916, 10206, 34992, …
  (−∞, 0]nonnegativerational roots