Proof methods / preserver_image

Image under a real-rootedness preserver

1,387 sequences

• A106800 → D+1
  1 · 1 · 2, 1 · 4, 5, 1 … Σ 1, 1, 3, 10, 38, 161, 747, 3753, …
  (−∞, 0]nonnegative
• A178822: Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
  1 · 6, 6 · 21, 42, 21 · 56, 168, 168, 56 … Σ 1, 12, 84, 448, 2016, 8064, 29568, …
  (−∞, 0)nonnegativerational roots
• A053123 → D
  -2 · -4, 6 · -6, 20, -12 · -8, 42, -60, 20 … Σ 0, -2, 2, 2, -6, 4, 4, -10, …
  (−∞, ∞)
• A091044 → D
  2 · 10, 6 · 28, 56, 12 · 60, 252, 180, 20 … Σ 0, 2, 16, 96, 512, 2560, 12288, 57344, …
  (−∞, 0]nonnegative
• A100861: Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
  1 · 1 · 1, 1 · 1, 3 … Σ 1, 1, 2, 4, 10, 26, 76, 232, …
  (−∞, 0)nonnegativeA144299
• A038231 → D+1
  1 · 5, 1 · 24, 10, 1 · 112, 72, 15, 1 … Σ 1, 6, 35, 200, 1125, 6250, 34375, …
  (−∞, 0]nonnegativerational roots
• A132812: Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
  1 · 2, 2 · 3, 9, 3 · 4, 24, 24, 4 … Σ 1, 4, 15, 56, 210, 792, 3003, 11440, …
  (−∞, 0)nonnegative
• A054142 → D+1
  1 · 2, 1 · 4, 5, 1 · 6, 17, 9, 1 … Σ 1, 3, 10, 33, 105, 324, 977, 2895, …
  (−∞, 0]nonnegative
• A062264 → x ↦ x/(x+1)
  1 · 1, 6 · 1, 14, 28 · 1, 24, 108, 120 … Σ 1, 7, 43, 253, 1462, 8378, 47818, …
  [−1, 0]nonnegative
• A094638 → θ+1
  1 · 1, 2 · 1, 6, 6 · 1, 12, 33, 24 … Σ 1, 3, 13, 70, 446, 3276, 27252, 253296, …
  (−∞, 0]nonnegative
• A113953 → x ↦ x−8
  1 · -8, 1 · 48, -14, 1 · -256, 128, -20, 1 … Σ 1, -7, 35, -147, 539, -1715, 4459, …
  (0, 8]
• A391533 → Pₙ + Pₙ₊₁
  8, 12, 6, 1 · 8, 20, 18, 7, 1 · 8, 28, 38, 25, 8, 1 … Σ 27, 54, 108, 216, 432, 864, 1728, 3456, …
  (−∞, −1]nonnegativerational roots
• A125662 → D+1
  1 · 4, 1 · 14, 8, 1 · 46, 43, 12, 1 … Σ 1, 5, 23, 102, 441, 1871, 7823, 32332, …
  (−∞, 0]nonnegative
• A249250 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 8, 10 · 1, 12, 36, 28 … Σ 1, 5, 19, 77, 307, 1229, 4915, 19661, …
  [−1, 0]nonnegative
• A053123 → x ↦ x+1
  1 · -1, -2 · 0, 2, 3 · 1, 2, -2, -4 … Σ 1, -3, 5, -3, -11, 45, -91, 93, …
  (−∞, ∞)
• basis-map coefficients of A112227 → θ+1
  0, 0, 3 · 0, 0, 0, 4 · 0, 0, 0, 0, 5 … Σ 3, 4, 5, 6, 7, 8, 9, 10, …
  (−∞, 0]nonnegativerational roots
• A114656 → θ+1
  1 · 2, 2 · 4, 12, 3 · 8, 48, 36, 4 … Σ 1, 4, 19, 96, 501, 2668, 14407, 78592, …
  (−∞, 0]nonnegative
• A137320 → θ+1
  1 · 0, 4 · 0, 4, 12 · 0, 8, 36, 32 … Σ 1, 4, 16, 76, 428, 2808, 21096, 178848, …
  (−∞, 0]nonnegative
• A367270 → (θ+1)²
  1 · 1 · 1, 8 · 1, 24, 27 … Σ 1, 1, 9, 52, 275, 1386, 6762, 32208, …
  (−∞, 0]nonnegative
• A062196 → θ+1
  1 · 1, 6 · 1, 16, 18 · 1, 30, 90, 40 … Σ 1, 7, 35, 161, 714, 3102, 13299, 56485, …
  (−∞, 0]nonnegative
• A055151 → D+1
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 19, 51, 141, 393, …
  (−∞, 0]nonnegative
• A081582 → x ↦ x+1
  1 · 2, 1 · 11, 11, 1 · 36, 54, 20, 1 … Σ 1, 3, 23, 111, 655, 3519, 19727, …
  (−∞, −1]nonnegative
• A249250 → (θ+1)²
  1 · 1, 12 · 1, 24, 27 · 1, 36, 135, 48 … Σ 1, 13, 52, 220, 832, 3028, 10600, …
  (−∞, 0]nonnegative
• A133080 → θ+1
  1 · 1, 2 · 0, 0, 3 · 0, 0, 3, 4 … Σ 1, 3, 3, 7, 5, 11, 7, 15, …
  (−∞, 0]nonnegativerational roots
• A126351 → (θ+1)²
  1 · 1, 8 · 1, 20, 36 · 1, 36, 171, 128 … Σ 1, 9, 57, 336, 1992, 12193, 77725, …
  (−∞, 0]nonnegative
• A344565 → D+1
  3 · 30, 15 · 135, 180, 45 · 420, 945, 630, 105 … Σ 3, 45, 360, 2100, 10080, 42336, 161280, …
  (−∞, 0]nonnegativerational roots
• A391045 → D+1
  1 · 2, 1 · 5, 6, 1 · 10, 29, 13, 1 … Σ 1, 3, 12, 53, 244, 1152, 5531, 26875, …
  (−∞, 0]nonnegative
• A100862 → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 1, 24, 54, 16 … Σ 1, 5, 22, 95, 415, 1847, 8410, 39209, …
  (−∞, 0]nonnegative
• A116412 → x ↦ x/(x+1)
  1 · 3, 4 · 6, 18, 13 · 12, 57, 87, 43 … Σ 1, 7, 37, 199, 1069, 5743, 30853, …
  [−1, 0]nonnegative
• A121574 → reverse
  1 · 1, 2 · 1, 5, 4 · 1, 8, 16, 8 … Σ 1, 3, 10, 33, 109, 360, 1189, 3927, …
  (−∞, 0]nonnegative
• A237765: Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
  A007318 → ×(−(1/2)n+(1/2)n²)
  1, 2, 1 · 3, 9, 9, 3 · 6, 24, 36, 24, 6 … Σ 0, 0, 4, 24, 96, 320, 960, 2688, …
  (−∞, 0]nonnegativerational roots
• A046716 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 12, 24, 4 … Σ 1, 3, 10, 41, 209, 1287, 9270, 76225, …
  (−∞, 0]nonnegative
• A120910 → basis A007318
  3 · 9, 3 · 27, 18, 3 · 81, 81, 27, 3 … Σ 3, 12, 48, 192, 768, 3072, 12288, …
  (−∞, −1]nonnegativerational roots
• A023531 → (θ+1)² → D
  4 · 0, 18 · 0, 0, 48 · 0, 0, 0, 100 … Σ 0, 4, 18, 48, 100, 180, 294, 448, …
  (−∞, 0]nonnegativerational roots
• A046716 → D+1
  1 · 2, 1 · 4, 5, 1 · 7, 22, 11, 1 … Σ 1, 3, 10, 41, 209, 1287, 9270, 76225, …
  (−∞, 0]nonnegative
• A143495: Triangle read by rows: 3-Stirling numbers of the second kind.
  1 · 3, 1 · 9, 7, 1 · 27, 37, 12, 1 … Σ 1, 4, 17, 77, 372, 1915, 10481, 60814, …
  (−∞, 0)nonnegative
• A121574 → θ+1
  1 · 2, 2 · 4, 10, 3 · 8, 32, 24, 4 … Σ 1, 4, 17, 68, 264, 1002, 3739, 13768, …
  (−∞, 0]nonnegative
• chebyshev_u → x ↦ x+1
  1 · 2, 2 · 3, 8, 4 · 4, 20, 24, 8 … Σ 1, 4, 15, 56, 209, 780, 2911, 10864, …
  (−2, 0)nonnegative
• A143685 → D+1
  1 · 2, 1 · 12, 13, 1 · 22, 63, 24, 1 … Σ 1, 3, 26, 110, 615, 2821, 13828, 63756, …
  (−∞, 0]nonnegative
• A130534 → D
  1 · 3, 2 · 11, 12, 3 · 50, 70, 30, 4 … Σ 0, 1, 5, 26, 154, 1044, 8028, 69264, …
  (−∞, 0]nonnegative
• A007318 → −D²+(−1+n)D+1+n → D
  2 · 8, 6 · 18, 36, 12 · 32, 108, 96, 20 … Σ 0, 2, 14, 66, 256, 880, 2784, 8288, …
  (−∞, 0]nonnegative
• A127648 → D
  2 · 0, 6 · 0, 0, 12 · 0, 0, 0, 20 … Σ 0, 0, 2, 6, 12, 20, 30, 42, …
  (−∞, 0]nonnegativerational roots
• A102547 → reverse → D+1
  1, 1 · 0, 2, 1 · 0, 0, 3, 1 · 0, 0, 3, 5, 1 … Σ 2, 3, 4, 9, 16, 25, 42, 70, …
  (−∞, 0]nonnegative
• A075497 → θ+1
  1 · 2, 2 · 4, 12, 3 · 8, 56, 36, 4 … Σ 1, 4, 19, 104, 641, 4380, 32803, …
  (−∞, 0]nonnegative
• A111125 → D
  1 · 5, 2 · 14, 14, 3 · 30, 54, 27, 4 … Σ 0, 1, 7, 31, 115, 390, 1254, 3893, …
  (−∞, 0]nonnegative
• A116666 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 8, 12 · 1, 12, 36, 32 … Σ 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  [−1, 0]nonnegativerational roots
• A048994: Triangle of Stirling numbers of first kind, s(n,k), n >= 0, 0 <= k <= n.
  1 · 0, 1 · 0, -1, 1 · 0, 2, -3, 1 … Σ 1, 1, 0, 0, 0, 0, 0, 0, …
  [0, ∞)rational roots
• A122431 → (θ+1)²
  1, 12, 27, 16 · 0, 4, 27, 48, 25 … Σ 56, 104, 168, 248, 344, 456, 584, 728, …
  (−∞, 0]nonnegative
• A059473 → γ
  1 · 2 · 4, 2 · 8, 8 … Σ 1, 2, 6, 16, 44, 120, 328, 896, …
  (−∞, 0]nonnegative
• A317497: Triangle T(n,k) = 3*T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.
  A102547 → x ↦ (1/27)x
  1 · 3 · 9 · 27, 1 … Σ 1, 3, 9, 28, 87, 270, 838, 2601, …
  (−∞, 0]nonnegative