Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A099040 → basis A007318 → D
  2 · 10, 8 · 40, 64, 24 · 144, 344, 264, 64 … Σ 0, 2, 18, 128, 816, 4896, 28256, …
  (−∞, 0]nonnegative
• A207606 → basis A013610 → D+1
  46, 69, 18 · 166, 438, 315, 54 … Σ 133, 973, 6824, 46493, 310147, 2035945, …
  (−∞, 0]nonnegative
• A001263 → basis A013609 → θ+1
  1 · 2, 4 · 5, 20, 12 · 14, 84, 108, 32 … Σ 1, 6, 37, 238, 1578, 10684, 73429, …
  (−∞, 0]nonnegative
• A114655: Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a la…
  2 · 4, 2 · 8, 12, 2 · 16, 48, 24, 2 … Σ 2, 6, 22, 90, 394, 1806, 8558, 41586, …
  (−∞, 0)nonnegative
• A028326 → basis A013609 → x ↦ x+1
  2 · 8, 4 · 32, 32, 8 · 128, 192, 96, 16 … Σ 2, 12, 72, 432, 2592, 15552, 93312, …
  (−∞, −1]nonnegativerational roots
• A272866 → basis A013610 → basis A007318
  1 · 29, 33, 9 · 841, 1914, 1611, 594, 81 … Σ 1, 71, 5041, 357911, 25411681, …
  (−∞, −1]nonnegative
• A391045 → reverse → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 20, 27, 4 … Σ 1, 3, 12, 52, 236, 1102, 5245, 25307, …
  (−∞, 0]nonnegative
• A359363 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 1, 6, 6 … Σ 1, 1, 3, 13, 69, 417, 2763, 19609, …
  [−1, 0]nonnegative
• A257623 → x ↦ x/(x+1)
  1 · 3, 6 · 9, 66, 66 · 27, 582, 1584, 1056 … Σ 1, 9, 141, 3249, 98421, 3691449, …
  [−1, 0]nonnegative
• A063967 → basis A013611 → θ+1
  1 · 2, 8 · 6, 40, 48 · 16, 160, 384, 256 … Σ 1, 10, 94, 816, 6780, 54648, 430920, …
  (−∞, 0]nonnegative
• A038255 → basis A013609
  1 · 7, 2 · 49, 28, 4 · 343, 294, 84, 8 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, −1]nonnegativerational roots
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
  1 · 1, 3 · 1, 6, 9 · 1, 9, 27, 27 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  (−∞, 0)[−1, 0]nonnegative+1
• A086645 → basis A013609 → x ↦ x+1
  1 · 4, 2 · 28, 24, 4 · 208, 264, 96, 8 … Σ 1, 6, 56, 576, 6016, 62976, 659456, …
  (−∞, −1]nonnegative
• A038291 → basis A013610 → D
  3 · 60, 18 · 900, 540, 81 · 12000, 10800, 3240, 324 … Σ 0, 3, 78, 1521, 26364, 428415, 6683274, …
  (−∞, 0]nonnegativerational roots
• A066325 → x ↦ x+1
  1 · 1, 1 · 0, 2, 1 · -2, 0, 3, 1 … Σ 1, 2, 3, 2, -5, -18, -11, 86, …
  (−∞, ∞)
• A257619: 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) = 9*x + 2.
  1 · 2, 2 · 4, 44, 4 · 8, 564, 564, 8 … Σ 1, 4, 52, 1144, 35464, 1418560, …
  (−∞, 0)nonnegative
• A013620 → basis A007318 → D
  3 · 30, 18 · 225, 270, 81 · 1500, 2700, 1620, 324 … Σ 0, 3, 48, 576, 6144, 61440, 589824, …
  (−∞, 0]nonnegativerational roots
• A027467 → basis A013609 → reverse
  1 · 2, 16 · 4, 64, 256 · 8, 192, 1536, 4096 … Σ 1, 18, 324, 5832, 104976, 1889568, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A030528, shifted → basis A013610 → (θ+1)²
  1 · -1, -12 · 1, 24, 81 · -1, -36, -243, -432 … Σ 1, -13, 106, -712, 4288, -24064, …
  (−∞, 0]
• A038335 → basis A013610 → (θ+1)²
  1 · 21, 108 · 441, 4536, 6561 … Σ 1, 129, 11538, 880416, 61295616, …
  (−∞, 0]nonnegative
• A008459 → x ↦ x+1 → (θ+1)²
  1 · 2, 4 · 6, 24, 9 · 20, 120, 108, 16 … Σ 1, 6, 39, 264, 1785, 11958, 79303, …
  (−∞, 0]nonnegative
• A063007 → x ↦ x+1 → D
  2 · 18, 12 · 132, 180, 60 · 900, 1860, 1260, 280 … Σ 0, 2, 30, 372, 4300, 47910, 521850, …
  (−∞, 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, 10, 1 · 1, 23, 23, 1 · 1, 50, 138, 50, 1 … Σ 12, 48, 240, 1440, 10080, 80640, …
  (−∞, 0)nonnegative
• A108838 → reverse → (θ+1)²
  2 · 2, 12 · 2, 32, 36 · 2, 60, 180, 80 … Σ 2, 14, 70, 322, 1428, 6204, 26598, …
  (−∞, 0]nonnegative
• A049020 → reverse → θ+1
  1 · 1, 2 · 1, 6, 6 · 1, 12, 30, 20 … Σ 1, 3, 13, 63, 337, 1963, 12333, 82967, …
  (−∞, 0]nonnegative
• A081579 → basis A013609 → reverse
  1 · 2, 2 · 4, 16, 8 · 8, 56, 72, 24 … Σ 1, 4, 28, 160, 976, 5824, 35008, …
  (−∞, 0]nonnegative
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
  1 · 2, 1 · 4, 5, 1 · 8, 19, 9, 1 … Σ 1, 3, 10, 37, 151, 674, 3263, 17007, …
  (−∞, 0)nonnegative
• A059473 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 4, 6 · 18, 54, 40 · 80, 360, 536, 264 … Σ 1, 10, 112, 1240, 13744, 152320, …
  [−1, 0]nonnegative
• A086645 → D
  1 · 6, 2 · 15, 30, 3 · 28, 140, 84, 4 … Σ 0, 1, 8, 48, 256, 1280, 6144, 28672, …
  (−∞, 0]nonnegative
• A091044 → ⊙ C(n+2,k+1) → x ↦ x/(x+1)
  1 · 1, 2 · 1, 7, 7 · 1, 17, 45, 30 … Σ 1, 3, 15, 93, 653, 4983, 40339, 341337, …
  [−1, 0]nonnegative
• A113953 → θ+1 → (θ+1)²
  1 · 0, 8 · 0, 16, 27 · 0, 0, 108, 64 … Σ 1, 8, 43, 172, 617, 1984, 6015, 17268, …
  (−∞, 0]nonnegative
• A081580: Pascal-(1,5,1) array.
  1 · 1, 1 · 1, 7, 1 · 1, 13, 13, 1 … Σ 1, 2, 9, 28, 101, 342, 1189, 4088, …
  (−∞, 0)nonnegative
• A114656 → basis A007318
  1 · 3, 1 · 11, 8, 1 · 45, 51, 15, 1 … Σ 1, 4, 20, 112, 672, 4224, 27456, …
  (−∞, −1]nonnegative
• A207606 → D+1
  11, 11, 2 · 21, 38, 17, 2 · 36, 102, 81, 23, 2 … Σ 24, 78, 244, 743, 2218, 6521, 18942, …
  (−∞, 0]nonnegative
• A196347 → basis A013610 → D+1
  1 · 5, 3 · 32, 60, 18 · 264, 864, 810, 162 … Σ 1, 8, 110, 2100, 51000, 1500000, …
  (−∞, 0]nonnegativerational roots
• A008278 → x ↦ x+1 (shift +1) → D+1
  1 · 3, 1 · 10, 7, 1 · 38, 43, 13, 1 … Σ 1, 4, 18, 95, 579, 3966, 29944, 246141, …
  (−∞, 0]nonnegative
• A075500 → D
  15, 2 · 175, 60, 3 · 1875, 1250, 150, 4 … Σ 17, 238, 3279, 47055, 718906, 11770347, …
  (−∞, 0]nonnegative
• A038323 → D+1
  1 · 20, 9 · 319, 360, 81 · 4598, 8613, 4860, 729 … Σ 1, 29, 760, 18800, 448000, 10400000, …
  (−∞, 0]nonnegativerational roots
• A272867 → θ+1
  1 · 1, 8, 3 · 1, 16, 54, 32, 5 … Σ 1, 12, 108, 864, 6480, 46656, 326592, …
  (−∞, 0]nonnegative
• A062145 → x ↦ x+1 → θ+1
  1 · 5, 8 · 21, 60, 30 · 84, 336, 315, 80 … Σ 1, 13, 111, 815, 5573, 36633, 234979, …
  (−∞, 0]nonnegative
• A008288 → basis A007318 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 12, 54, 80, 39 … Σ 1, 5, 31, 185, 1111, 6665, 39991, …
  [−1, 0]nonnegative
• A038330 → basis A013610 → (θ+1)²
  256, 1536, 1296 · 4096, 36864, 62208, 27648 … Σ 3088, 130816, 5130496, 191070208, …
  (−∞, 0]nonnegative
• A132774 → Veronese 2 → x ↦ x+1
  1 · 2 · 5, 5 · 6, 6 … Σ 1, 2, 10, 12, 36, 40, 104, 112, …
  (−∞, −1]nonnegativerational roots
• A132440 → θ+1
  1 · 0, 4 · 0, 0, 9 · 0, 0, 0, 16 … Σ 0, 1, 4, 9, 16, 25, 36, 49, …
  (−∞, 0]nonnegativerational roots
• A075503 → D
  24, 2 · 448, 96, 3 · 7680, 3200, 240, 4 … Σ 26, 547, 11124, 232101, 5104462, …
  (−∞, 0]nonnegative
• A089627 → x ↦ x+1 → (θ+1)²
  1 · 1 · 3, 8 · 7, 24 … Σ 1, 1, 11, 31, 169, 641, 2891, 11831, …
  (−∞, 0]nonnegative
• A124932 → D+1
  1 · 5, 3 · 12, 21, 6 · 22, 66, 54, 10 … Σ 1, 8, 39, 152, 520, 1632, 4816, 13568, …
  (−∞, 0]nonnegative
• A026729 → basis A013610 → D
  3 · 9, 18 · 21, 90, 81 · 45, 288, 567, 324 … Σ 0, 3, 27, 192, 1224, 7344, 42384, …
  (−∞, 0]nonnegative
• A128813 → basis A007318
  1 · 2, 1 · 8, 10, 3 · 56, 118, 81, 18 … Σ 1, 3, 21, 273, 5733, 177723, 7642089, …
  (−∞, −1]nonnegativerational roots
• A099097 → x ↦ x/(x+1)
  1 · 0, 3 · 0, 1, 10 · 0, 0, 6, 33 … Σ 1, 3, 11, 39, 139, 495, 1763, 6279, …
  [−1, 0]nonnegative