Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A013622 → basis A013610 → reverse
  1 · 15, 8 · 225, 240, 64 · 3375, 5400, 2880, 512 … Σ 1, 23, 529, 12167, 279841, 6436343, …
  (−∞, 0]nonnegativerational roots
• A081577 → x ↦ x+1 → (θ+1)²
  1 · 2, 4 · 6, 24, 9 · 16, 96, 90, 16 … Σ 1, 6, 39, 218, 1167, 5982, 29791, …
  (−∞, 0]nonnegative
• A344565: Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
  3 · 15, 15 · 45, 90, 45 · 105, 315, 315, 105 … Σ 3, 30, 180, 840, 3360, 12096, 40320, …
  (−∞, 0)nonnegativerational roots
• A257626 → (θ+1)²
  1 · 6, 24 · 36, 432, 324 · 216, 5616, 12636, 3456 … Σ 1, 30, 792, 21924, 654480, 21170160, …
  (−∞, 0]nonnegative
• A121574 → x ↦ x+1 → (θ+1)²
  1 · 3, 4 · 10, 28, 9 · 33, 140, 99, 16 … Σ 1, 7, 47, 288, 1678, 9428, 51560, …
  (−∞, 0]nonnegative
• A123202 → D+1
  1 · -1, -2 · -6, 6, 7 · -30, 90, -39, -34 … Σ 1, -3, 7, -13, 21, -31, 43, -57, …
  (−∞, ∞)
• basis-map coefficients of A112227 → x ↦ x+1
  1, 2, 1 · 1, 3, 3, 1 · 1, 4, 6, 4, 1 … Σ 4, 8, 16, 32, 64, 128, 256, 512, …
  (−∞, −1]nonnegativerational roots
• A207606 → x ↦ x+1
  13, 11, 2 · 34, 44, 17, 2 · 89, 155, 93, 23, 2 … Σ 26, 97, 362, 1351, 5042, 18817, 70226, …
  (−5, −1)nonnegative
• A356546: Triangle read by rows. T(n, k) = RisingFactorial(n + 1, n) / (k! * (n - k)!).
  1 · 2, 2 · 6, 12, 6 · 20, 60, 60, 20 … Σ 1, 4, 24, 160, 1120, 8064, 59136, …
  (−∞, 0)nonnegativerational roots
• row-factor cofactor of A101842 → (θ+1)²
  1 · 1, 4 · 1, 16, 9 · 1, 44, 99, 16 … Σ 1, 5, 26, 160, 1140, 9240, 84000, …
  (−∞, 0]nonnegative
• A038334 → x ↦ x+1 → x ↦ x/(x+1)
  400, 1120, 784 · 8000, 33600, 47040, 21952 … Σ 2304, 110592, 5308416, 254803968, …
  [−1, 0]nonnegativerational roots
• A178821 → Veronese 2 (j=1) → θ+1
  5 · 30 · 105, 70 · 280, 560 … Σ 0, 5, 30, 175, 840, 3528, 13440, 47520, …
  (−∞, 0]nonnegative
• A052553 → D → θ+1
  1 · 2 · 3, 4 · 4, 12 … Σ 0, 0, 1, 2, 7, 16, 38, 82, …
  (−∞, 0]nonnegative
• A062145 → (θ+1)²
  1 · 1, 16 · 1, 40, 90 · 1, 72, 405, 320 … Σ 1, 17, 131, 798, 4362, 22407, 110539, …
  (−∞, 0]nonnegative
• A142458: Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).
  1 · 1, 1 · 1, 8, 1 · 1, 39, 39, 1 … Σ 1, 2, 10, 80, 880, 12320, 209440, …
  (−∞, 0)nonnegative
• A120909 → Veronese 2 (j=1) → (θ+1)²
  6 · 12 · 18, 96 · 24, 384 … Σ 0, 6, 12, 114, 408, 1854, 7140, 27690, …
  (−∞, 0]nonnegative
• A088996 → basis A013611 → x ↦ x+1
  1 · 5, 4 · 55, 84, 32 · 935, 2088, 1552, 384 … Σ 1, 9, 171, 4959, 193401, 9476649, …
  (−∞, −1]nonnegativerational roots
• A094305 → x ↦ x+1 → x ↦ x+1
  1 · 9, 3 · 54, 36, 6 · 270, 270, 90, 10 … Σ 1, 12, 96, 640, 3840, 21504, 114688, …
  (−∞, −1]nonnegativerational roots
• A130123 → Veronese 2 → x ↦ x+1
  1 · 4, 4 · 16, 32, 16 · 64, 192, 192, 64 … Σ 1, 0, 8, 0, 64, 0, 512, 0, …
  (−∞, −1]nonnegativerational roots
• A056939 → θ+1 → reverse
  1 · 2, 1 · 3, 8, 1 · 4, 30, 20, 1 … Σ 1, 3, 12, 55, 276, 1477, 8296, 48393, …
  (−∞, 0]nonnegative
• A001263 → basis A013610 → θ+1
  1 · 2, 6 · 5, 30, 27 · 14, 126, 243, 108 … Σ 1, 8, 62, 491, 3975, 32730, 272940, …
  (−∞, 0]nonnegative
• A094638 → basis A013609 → D+1
  1 · 4, 2 · 20, 30, 8 · 116, 324, 260, 48 … Σ 1, 6, 58, 748, 11964, 227824, 5027536, …
  (−∞, 0]nonnegative
• A013610 → basis A007318 → x ↦ x/(x+1)
  1 · 4, 7 · 16, 56, 49 · 64, 336, 588, 343 … Σ 1, 11, 121, 1331, 14641, 161051, …
  [−1, 0]nonnegativerational roots
• A281000 → θ+1
  1 · 3, 2 · 10, 20, 3 · 35, 140, 63, 4 … Σ 1, 5, 33, 242, 1871, 14933, 121739, …
  (−∞, 0]nonnegative
• A091042 → basis A013610 → x ↦ x/(x+1)
  1 · 4, 13 · 16, 92, 121 · 64, 528, 1368, 1093 … Σ 1, 17, 229, 3053, 40681, 542057, …
  [−1, 0]nonnegative
• A079510 → D
  2 · 3, 12 · 0, 40, 72 · 0, 30, 390, 480 … Σ 0, 2, 15, 112, 900, 7926, 76615, …
  (−∞, 0]nonnegative
• A159854 → basis A008277 → D
  3, 2 · 10, 14, 3 · 32, 76, 36, 4 … Σ 5, 27, 148, 839, 4957, 30612, 197715, …
  (−∞, 0]nonnegative
• A147716 → basis A013609 → θ+1
  1 · 15, 4 · 225, 120, 12 · 3375, 2700, 540, 32 … Σ 1, 19, 357, 6647, 122825, 2255067, …
  (−∞, 0]nonnegativerational roots
• A111125 → basis A013609 → D
  2 · 14, 8 · 62, 80, 24 · 230, 480, 312, 64 … Σ 0, 2, 22, 166, 1086, 6596, 38276, …
  (−∞, 0]nonnegative
• basis-map coefficients of A030528, shifted → D
  -1 · 0, 2 · 0, 0, -3 · 0, 0, 0, 4 … Σ 0, -1, 2, -3, 4, -5, 6, -7, …
  (−∞, 0]rational roots
• A114687 → basis A013609 → x ↦ x/(x+1)
  1 · 3, 7 · 11, 50, 55 · 45, 303, 663, 469 … Σ 1, 10, 116, 1480, 20112, 285600, …
  [−1, 0]nonnegative
• A181543 → D
  1 · 8, 2 · 27, 54, 3 · 64, 432, 192, 4 … Σ 0, 1, 10, 84, 692, 5630, 45552, 367360, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → x ↦ x+1 → D+1
  1 · -3, -1 · 8, 6, 1 · -20, -24, -9, -1 … Σ 1, -4, 15, -54, 189, -648, 2187, -7290, …
  (−∞, 0]rational roots
• A053117 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 4 · 3, 14, 15 · 4, 32, 76, 56 … Σ 1, 6, 32, 168, 880, 4608, 24128, …
  [−1, 0)nonnegative
• A034870 → basis A013611 → D+1
  1 · 20, 48, 16 · 144, 896, 1920, 1536, 256 … Σ 1, 84, 4752, 233280, 10637568, …
  (−∞, 0]nonnegativerational roots
• A356546 → γ
  1 · 2 · 6 · 20 … Σ 1, 2, 6, 20, 70, 252, 924, 3432, …
  (−∞, −1][−1, 0][−3−2√2, −3+2√2]+2
• A178821 → basis A013611 → D+1
  1 · 30, 20 · 300, 720, 240 · 1960, 8400, 10080, 2240 … Σ 1, 50, 1260, 22680, 332640, 4245696, …
  (−∞, 0]nonnegativerational roots
• A357367 → (θ+1)²
  1 · 0, 8 · 0, 24, 108 · 0, 96, 1080, 1920 … Σ 1, 8, 132, 3096, 92520, 3339360, …
  (−∞, 0]nonnegative
• A038284 → basis A007318 → (θ+1)²
  196, 672, 324 · 2744, 14112, 13608, 3456 … Σ 1192, 33920, 908800, 23360000, …
  (−∞, 0]nonnegative
• A269944 → basis A013609 → x ↦ x+1
  1 · 3, 2 · 12, 14, 4 · 84, 122, 56, 8 … Σ 1, 5, 30, 270, 3780, 79380, 2381400, …
  (−∞, −1]nonnegativerational roots
• A112493 → x ↦ x+1
  1 · 2, 1 · 8, 10, 3 · 52, 106, 70, 15 … Σ 1, 3, 21, 243, 3933, 81819, 2080053, …
  [−2, −1)nonnegative
• A033282 → basis A013609 → (θ+1)²
  1 · 3, 16 · 11, 120, 180 · 45, 744, 2268, 1792 … Σ 1, 19, 311, 4849, 73781, 1105927, …
  (−∞, 0]nonnegative
• A196347 → basis A013611 → θ+1
  1 · 2, 8 · 8, 64, 96 · 48, 576, 1728, 1536 … Σ 1, 10, 168, 3888, 114048, 4043520, …
  (−∞, 0]nonnegativerational roots
• A145596 → x ↦ x+1 → x ↦ x+1
  1 · 6, 2 · 31, 20, 3 · 156, 148, 44, 4 … Σ 1, 8, 54, 352, 2285, 14904, 97930, …
  (−∞, −1]nonnegative
• A209599 → (θ+1)²
  1 · 2 · 3, 4 · 5, 12 … Σ 1, 2, 7, 17, 45, 109, 265, 625, …
  (−∞, 0]nonnegative
• A038330 → basis A007318 → θ+1
  256, 256, 48 · 4096, 6144, 2304, 256 … Σ 560, 12800, 288000, 6400000, 140800000, …
  (−∞, 0]nonnegativerational roots
• A075497 → θ+1 → x ↦ x+1
  1 · 4, 2 · 19, 18, 3 · 104, 140, 48, 4 … Σ 1, 6, 40, 296, 2416, 21568, 208832, …
  (−∞, −1]nonnegative
• A169803 → D → D
  2 · 6 · 12, 6 · 20, 24 … Σ 0, 0, 0, 2, 6, 18, 44, 102, …
  (−∞, 0]nonnegative
• basis-map coefficients of A030528, shifted → D+1
  1 · -1, -1 · 0, 2, 1 · 0, 0, -3, -1 … Σ 1, -2, 3, -4, 5, -6, 7, -8, …
  (−∞, 0]rational roots
• A114656 → D
  1 · 6, 2 · 24, 24, 3 · 80, 160, 60, 4 … Σ 0, 1, 8, 51, 304, 1765, 10128, 57799, …
  (−∞, 0]nonnegative