Proof methods / preserver_image

Image under a real-rootedness preserver

1,387 sequences

• A099093 → x ↦ x−5
  1 · -15, 3 · 210, -87, 9 · -2925, 1845, -387, 27 … Σ 1, -12, 132, -1440, 15696, -171072, …
  (11/3, 5]
• A144438 → θ+1
  1 · 1, 2 · 1, 10, 3 · 1, 28, 42, 4 … Σ 1, 3, 14, 75, 471, 3402, 27844, 254970, …
  (−∞, 0]nonnegative
• A081582 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 11, 11 · 1, 20, 54, 36 … Σ 1, 3, 23, 111, 655, 3519, 19727, …
  [−1, 0]nonnegative
• basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
  1 · 0, -1 · 0, 0, 2 · 0, 0, 0, -6 … Σ 1, -1, 2, -6, 24, -120, 720, -5040, …
  [−1, 0]rational roots
• A110441 → x ↦ x/(x+1)
  1 · 3, 4 · 7, 20, 14 · 15, 68, 100, 48 … Σ 1, 7, 41, 231, 1289, 7175, 39913, …
  [−1, 0)nonnegative
• A229032: Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).
  1 · 2 · 3, 4 · 4, 16 … Σ 1, 2, 7, 20, 61, 182, 547, 1640, …
  (−∞, 0)nonnegative
• A062196 → D+1
  1 · 4, 3 · 9, 20, 6 · 16, 75, 60, 10 … Σ 1, 7, 35, 161, 714, 3102, 13299, 56485, …
  (−∞, 0]nonnegative
• A125662 → θ+1
  1 · 3, 2 · 8, 12, 3 · 21, 50, 27, 4 … Σ 1, 5, 23, 102, 441, 1871, 7823, 32332, …
  (−∞, 0]nonnegative
• A013611 → basis A007318
  1 · 5, 4 · 25, 40, 16 · 125, 300, 240, 64 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, −1]nonnegativerational roots
• A178820 → D
  4 · 20, 20 · 60, 120, 60 · 140, 420, 420, 140 … Σ 0, 4, 40, 240, 1120, 4480, 16128, …
  (−∞, 0]nonnegativerational roots
• A085478 → x ↦ x+1
  1 · 1 · 2, 1 · 5, 5, 1 … Σ 1, 1, 3, 11, 41, 153, 571, 2131, …
  (−5, −1)nonnegative
• A144438 → (θ+1)²
  1 · 1, 4 · 1, 20, 9 · 1, 56, 126, 16 … Σ 1, 5, 30, 199, 1487, 12450, 115866, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/4) → θ+1
  1 · 2 · 4, 4 · 8, 32 … Σ 1, 2, 8, 40, 240, 1680, 13440, 120960, …
  (−∞, 0]nonnegative
• A132440 → D+1
  1 · 2, 2 · 0, 6, 3 · 0, 0, 12, 4 … Σ 0, 1, 4, 9, 16, 25, 36, 49, …
  (−∞, 0]nonnegativerational roots
• A113214 → x ↦ x−12
  1 · -10, 1 · 108, -21, 1 · -1176, 338, -32, 1 … Σ 1, -9, 88, -869, 8591, -84942, 839861, …
  (8, 12]
• A113953 → D+1
  1 · 1, 1 · 2, 4, 1 · 0, 8, 7, 1 … Σ 1, 2, 7, 16, 41, 94, 219, 492, …
  (−∞, 0]nonnegative
• A063967 → x ↦ x−5
  1 · -4, 1 · 12, -7, 1 · -32, 32, -10, 1 … Σ 1, -3, 6, -9, 9, 0, -27, 81, …
  (0, 4]
• A391403 → θ+1
  1 · 1 · 1, 4 · 1, 12 … Σ 1, 1, 5, 13, 43, 131, 411, 1275, …
  (−∞, 0]nonnegative
• A249252 → D+1
  1 · 5, 4 · 9, 24, 8 · 13, 76, 80, 16 … Σ 1, 9, 41, 185, 793, 3305, 13481, 54105, …
  (−∞, 0]nonnegative
• A038208: Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
  1 · 2, 2 · 4, 8, 4 · 8, 24, 24, 8 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  (−∞, −1]nonnegativerational roots
• A039756 → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 18, 39, 4 … Σ 1, 3, 12, 62, 372, 2492, 18384, 147800, …
  (−∞, 0]nonnegative
• A126216 → (θ+1)²
  1 · 2, 4 · 5, 20, 9 · 14, 84, 81, 16 … Σ 1, 6, 34, 195, 1131, 6608, 38788, …
  (−∞, 0]nonnegative
• A113214 → x ↦ x−1
  1 · 1, 1 · -2, 1, 1 · 1, -3, 1, 1 … Σ 1, 2, 0, 0, 0, 0, 0, 0, …
  (−3, 1]
• A136572 → (θ+1)²
  1 · 0, 4 · 0, 0, 18 · 0, 0, 0, 96 … Σ 1, 4, 18, 96, 600, 4320, 35280, 322560, …
  (−∞, 0]nonnegativerational roots
• A011973 → D
  1 · 2 · 3, 2 · 4, 6 … Σ 0, 0, 1, 2, 5, 10, 20, 38, …
  (−∞, 0]nonnegative
• A274496 → x ↦ x+1
  1 · 2 · 4, 2 · 8, 4 … Σ 1, 2, 6, 12, 36, 72, 216, 432, …
  (−∞, −1]nonnegativerational roots
• A145596 → x ↦ x/(x+1)
  1 · 2, 4 · 3, 14, 14 · 4, 32, 72, 48 … Σ 1, 6, 31, 156, 785, 3978, 20335, …
  [−1, 0]nonnegative
• A110441 → θ+1
  1 · 3, 2 · 7, 12, 3 · 15, 46, 27, 4 … Σ 1, 5, 22, 92, 372, 1468, 5688, 21728, …
  (−∞, 0]nonnegative
• A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
  A023531 → (θ+1)²
  1 · 0, 4 · 0, 0, 9 · 0, 0, 0, 16 … Σ 1, 4, 9, 16, 25, 36, 49, 64, …
  [−1, 0]nonnegativerational roots
• A344565 → D
  15 · 90, 90 · 315, 630, 315 · 840, 2520, 2520, 840 … Σ 0, 15, 180, 1260, 6720, 30240, 120960, …
  (−∞, 0]nonnegativerational roots
• A027465 → D+1
  1 · 4, 1 · 15, 8, 1 · 54, 45, 12, 1 … Σ 1, 5, 24, 112, 512, 2304, 10240, 45056, …
  (−∞, 0]nonnegativerational roots
• A352880 → x ↦ x+1
  8 · 30, 18 · 112, 128, 32 · 420, 700, 350, 50 … Σ 8, 48, 272, 1520, 8472, 47264, 264224, …
  (−∞, −1]nonnegative
• A367270 → θ+1
  1 · 1 · 1, 4 · 1, 12, 9 … Σ 1, 1, 5, 22, 95, 406, 1722, 7260, …
  (−∞, 0]nonnegative
• A085880 → (θ+1)²
  1 · 1, 4 · 2, 16, 18 · 5, 60, 135, 80 … Σ 1, 5, 36, 280, 2240, 18144, 147840, …
  (−∞, 0]nonnegative
• A106246 → (θ+1)²
  1 · 2, 4 · 1, 16, 9 · 0, 12, 54, 16 … Σ 1, 6, 26, 82, 207, 446, 856, 1506, …
  (−∞, 0]nonnegative
• A207606 → (θ+1)²
  4, 28, 18 · 5, 64, 99, 32 · 6, 120, 324, 240, 50 … Σ 50, 200, 740, 2597, 8770, 28759, 92146, …
  (−∞, 0]nonnegative
• A034870 → x ↦ x/(x+1)
  1 · 1, 4, 4 · 1, 8, 24, 32, 16 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  [−1, 0]nonnegativerational roots
• A137320 → D+1
  1 · 2, 2 · 2, 10, 4 · 4, 28, 36, 8 … Σ 1, 4, 16, 76, 428, 2808, 21096, 178848, …
  (−∞, 0]nonnegative
• A116412: Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
  1 · 3, 1 · 6, 6, 1 · 12, 21, 9, 1 … Σ 1, 4, 13, 43, 142, 469, 1549, 5116, …
  (−∞, 0)nonnegative
• A131689 → reverse
  1 · 1 · 2, 1 · 6, 6, 1 … Σ 1, 1, 3, 13, 75, 541, 4683, 47293, …
  (−∞, −1]nonnegative
• A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
  1 · 0, 2 · 0, 0, 3 · 0, 0, 0, 4 … Σ 1, 2, 3, 4, 5, 6, 7, 8, …
  [−1, 0]nonnegativerational roots
• A054142 → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 1, 20, 54, 16 … Σ 1, 5, 22, 91, 349, 1260, 4349, 14509, …
  (−∞, 0]nonnegative
• A133087: A133080 * A007318.
  1 · 2, 1 · 1, 2, 1 · 2, 5, 4, 1 … Σ 1, 3, 4, 12, 16, 48, 64, 192, …
  (−∞, −1]nonnegativerational roots
• A055151 → D
  1 · 3 · 6, 4 · 10, 20 … Σ 0, 0, 1, 3, 10, 30, 90, 266, …
  (−∞, 0]nonnegative
• A038231 → (θ+1)²
  1 · 4, 4 · 16, 32, 9 · 64, 192, 108, 16 … Σ 1, 8, 57, 380, 2425, 15000, 90625, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → (θ+1)²
  1 · -1, -4 · 1, 8, 9 · -1, -12, -27, -16 … Σ 1, -5, 18, -56, 160, -432, 1120, -2816, …
  (−∞, 0]
• A216973 → θ+1
  1 · 2, 4 · 3, 12, 9 · 4, 24, 36, 16 … Σ 0, 1, 6, 24, 80, 240, 672, 1792, …
  (−∞, 0]nonnegativerational roots
• A008278 → x ↦ x+1 (shift +1) → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 15, 68, 101, 49 … Σ 1, 5, 31, 233, 2047, 20417, 226711, …
  [−1, 0]nonnegative
• A116666 → x ↦ x+1
  1 · 4, 3 · 12, 16, 5 · 32, 60, 36, 7 … Σ 1, 7, 33, 135, 513, 1863, 6561, 22599, …
  (−∞, −1]nonnegativerational roots
• A085478 → reverse
  1 · 1, 1 · 1, 3, 1 · 1, 5, 6, 1 … Σ 1, 2, 5, 13, 34, 89, 233, 610, …
  (−∞, 0]nonnegative