Proof methods / preserver_image

Image under a real-rootedness preserver

1,387 sequences

• A111910: Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).
  A091044 → ⊙ C(n+2,k+1)
  1 · 1, 1 · 1, 5, 1 · 1, 14, 14, 1 … Σ 1, 2, 7, 30, 146, 772, 4331, 25398, …
  (−∞, 0]nonnegative
• A060920 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 13, 59, 88, 43 … Σ 1, 5, 31, 203, 1345, 8933, 59359, …
  (−∞, ∞)nonnegative
• A094305 → x ↦ x+1
  1 · 6, 3 · 24, 24, 6 · 80, 120, 60, 10 … Σ 1, 9, 54, 270, 1215, 5103, 20412, …
  (−∞, −1]nonnegativerational roots
• A094645: Triangle of generalized Stirling numbers of the first kind read by rows: T(n, k) = (-1)^(n+k)*Sum_{m=0..n} Stirling1(n, m) * binomial(m, k), where Sti…
  1 · -1, 1 · 0, -1, 1 · 0, -1, 0, 1 … Σ 1, 0, 0, 0, 0, 0, 0, 0, …
  (−∞, 3/2]rational roots
• A120910 → D
  3 · 12, 6 · 36, 36, 9 · 96, 144, 72, 12 … Σ 0, 3, 18, 81, 324, 1215, 4374, 15309, …
  (−∞, 0]nonnegativerational roots
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
  1 · 2, 1 · 4, 4, 1 · 8, 12, 6, 1 … Σ 1, 3, 9, 27, 81, 243, 729, 2187, …
  (−∞, −1]nonnegativerational roots
• A056242 → D
  2 · 5, 8 · 9, 32, 24 · 14, 82, 132, 64 … Σ 0, 2, 13, 65, 292, 1236, 5036, 19980, …
  (−∞, 0]nonnegative
• A099040 → x ↦ x−5
  1 · -10, 2 · 90, -38, 4 · -800, 520, -112, 8 … Σ 1, -8, 56, -384, 2624, -17920, 122368, …
  (3, 5]
• A108299 → D+1
  1 · 0, -1 · 0, -3, -1 · 0, -5, 1, 1 … Σ 1, -1, -4, -3, 3, 8, 5, -5, …
  (−∞, ∞)
• A064861 → D+1
  4, 7, 2 · 6, 21, 20, 4 · 7, 32, 49, 28, 4 … Σ 13, 51, 120, 432, 972, 3348, 7344, …
  (−∞, 0]nonnegative
• narayana → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 12, 18, 4 … Σ 1, 3, 10, 35, 126, 462, 1716, 6435, …
  (−∞, 0]nonnegative
• A007318 → −D²+(−1+n)D+1+n → x ↦ x+1
  1 · 4, 2 · 14, 14, 3 · 44, 66, 30, 4 … Σ 1, 6, 31, 144, 621, 2538, 9963, 37908, …
  (−∞, −1]nonnegative
• A008459 → D
  1 · 4, 2 · 9, 18, 3 · 16, 72, 48, 4 … Σ 0, 1, 6, 30, 140, 630, 2772, 12012, …
  (−∞, 0]nonnegative
• A086645 → ⊙ C(n+1,k) (shift +2)
  1, 12, 1 · 1, 45, 45, 1 · 1, 112, 420, 112, 1 … Σ 14, 92, 646, 4652, 34124, 253528, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/2) (shift +1) → D
  1 · 4 · 11, 8 · 26, 68 … Σ 0, 0, 1, 4, 19, 94, 519, 3144, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729 → D+1
  1 · 1, 1 · 0, 2, 1 · 0, 0, 3, 1 … Σ 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 0]nonnegativerational roots
• A003506 → x ↦ x/(x+1)
  1 · 2, 4 · 3, 12, 12 · 4, 24, 48, 32 … Σ 1, 6, 27, 108, 405, 1458, 5103, 17496, …
  [−1, 0]nonnegativerational roots
• A054142 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 10, 18, 4 … Σ 1, 3, 10, 33, 105, 324, 977, 2895, …
  (−∞, 0]nonnegative
• A066325: Hermite he
  1 · 0, 1 · -1, 0, 1 · 0, -3, 0, 1 … Σ 1, 1, 0, -2, -2, 6, 16, -20, …
  (−∞, ∞)
• A099097 → x ↦ x−1
  1 · -3, 3 · 8, -17, 9 · -21, 69, -75, 27 … Σ 1, 0, 0, 0, 0, 0, 0, 0, …
  (5/9, 1]
• A008459: Square the entries of Pascal's triangle.
  1 · 1, 1 · 1, 4, 1 · 1, 9, 9, 1 … Σ 1, 2, 6, 20, 70, 252, 924, 3432, …
  (−∞, 0)nonnegative
• basis-map coefficients of A030528, shifted → (θ+1)²
  1 · 0, -4 · 0, 0, 9 · 0, 0, 0, -16 … Σ 1, -4, 9, -16, 25, -36, 49, -64, …
  (−∞, 0]rational roots
• A062145 → D
  4 · 10, 20 · 18, 90, 60 · 28, 252, 420, 140 … Σ 0, 4, 30, 168, 840, 3960, 18018, 80080, …
  (−∞, 0]nonnegative
• A053117 → 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
• A118931 → basis A007318
  1 · 1 · 1 · 3, 2 … Σ 1, 1, 1, 5, 17, 41, 241, 1261, …
  (−∞, −1]nonnegative
• A116666 → D+1
  1 · 4, 3 · 7, 16, 5 · 10, 39, 36, 7 … Σ 1, 7, 28, 92, 272, 752, 1984, 5056, …
  (−∞, 0]nonnegative
• A207823 → D+1
  1 · 5, 1 · 23, 10, 1 · 102, 70, 15, 1 … Σ 1, 6, 34, 188, 1021, 5468, 28959, …
  (−∞, 0]nonnegative
• A099093 → x ↦ x−8
  1 · -24, 3 · 552, -141, 9 · -12672, 4896, -630, 27 … Σ 1, -21, 420, -8379, 167139, -3333960, …
  (20/3, 8]
• A132812 → x ↦ x/(x+1)
  1 · 2, 4 · 3, 15, 15 · 4, 36, 84, 56 … Σ 1, 6, 33, 180, 985, 5418, 29953, …
  [−1, 0]nonnegative
• A272866: Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
  1 · 1, 3, 1 · 1, 6, 11, 6, 1 · 1, 9, 30, 45, 30, 9, 1 … Σ 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  (−∞, 0]nonnegative
• A100862 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 9, 21, 14 … Σ 1, 3, 11, 45, 201, 963, 4899, 26253, …
  [−1, 0]nonnegative
• A317496 → x ↦ x+1
  1 · 1 · 1 · 4, 3 … Σ 1, 1, 1, 7, 13, 19, 61, 139, …
  (−∞, −1]nonnegative
• A099040 → θ+1
  1 · 0, 4 · 0, 4, 12 · 0, 0, 24, 32 … Σ 1, 4, 16, 56, 188, 608, 1920, 5952, …
  (−∞, 0]nonnegative
• A188440 → D
  2 · 2 · 4, 8 · 4, 8 … Σ 0, 0, 2, 2, 12, 12, 54, 54, …
  (−∞, 0]nonnegativerational roots
• A137597 → D
  1 · 5, 2 · 22, 18, 3 · 99, 122, 42, 4 … Σ 0, 1, 7, 43, 267, 1731, 11819, 85107, …
  (−∞, 0]nonnegative
• A099093 → x ↦ x−7
  1 · -21, 3 · 420, -123, 9 · -8379, 3717, -549, 27 … Σ 1, -18, 306, -5184, 87804, -1487160, …
  (17/3, 7]
• A103633 → basis A007318
  1 · 1, 1 · 2, 3, 1 · 2, 5, 4, 1 … Σ 1, 2, 6, 12, 36, 72, 216, 432, …
  (−∞, −1]nonnegativerational roots
• A103633 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 0, 1, 2 … Σ 1, 1, 3, 3, 9, 9, 27, 27, …
  [−1, 0]nonnegativerational roots
• A099040 → x ↦ x−3
  1 · -6, 2 · 30, -22, 4 · -144, 168, -64, 8 … Σ 1, -4, 12, -32, 80, -192, 448, -1024, …
  (1, 3]
• A272866 → D
  3, 2 · 6, 22, 18, 4 · 9, 60, 135, 120, 45, 6 … Σ 0, 5, 50, 375, 2500, 15625, 93750, …
  (−∞, 0]nonnegative
• A143683 → x ↦ x+1
  1 · 2, 1 · 12, 12, 1 · 40, 60, 22, 1 … Σ 1, 3, 25, 123, 769, 4275, 25129, …
  (−∞, −1]nonnegative
• A122848 → D
  1 · 1, 2 · 0, 6, 3 · 0, 6, 18, 4 … Σ 0, 1, 3, 9, 28, 90, 306, 1078, …
  (−∞, 0]nonnegative
• A114687 → basis A007318
  1 · 3, 2 · 11, 14, 4 · 45, 84, 48, 8 … Σ 1, 5, 29, 185, 1257, 8925, 65445, …
  (−∞, −1]nonnegative
• A152198 → θ+1
  1 · 1 · 1, 2 · 1, 2 … Σ 1, 1, 3, 3, 8, 8, 20, 20, …
  (−∞, 0]nonnegativerational roots
• A124932 → D
  3 · 9, 12 · 18, 48, 30 · 30, 120, 150, 60 … Σ 0, 3, 21, 96, 360, 1200, 3696, 10752, …
  (−∞, 0]nonnegativerational roots
• A137338 → x ↦ x+1
  1 · 0, 1 · -2, -1, 1 · 4, -3, -3, 1 … Σ 1, 1, -2, -1, 10, -25, 40, -25, …
  (−∞, ∞)
• A086645 → θ+1
  1 · 1, 2 · 1, 12, 3 · 1, 30, 45, 4 … Σ 1, 3, 16, 80, 384, 1792, 8192, 36864, …
  (−∞, 0]nonnegative
• A008277 → θ+2 (shift +1) → D+1
  2 · 5, 3 · 11, 17, 4 · 23, 69, 39, 5 … Σ 2, 8, 32, 136, 622, 3060, 16130, 90688, …
  (−∞, 0]nonnegative
• A091042 → θ+1
  1 · 1, 6 · 1, 20, 15 · 1, 42, 105, 28 … Σ 1, 7, 36, 176, 832, 3840, 17408, 77824, …
  (−∞, 0]nonnegative
• A013611 → x ↦ x/(x+1)
  1 · 1, 5 · 1, 10, 25 · 1, 15, 75, 125 … Σ 1, 6, 36, 216, 1296, 7776, 46656, …
  [−1, 0]nonnegativerational roots