Proof methods / preserver_image

Image under a real-rootedness preserver

1,387 sequences

• A126351 → reverse
  1 · 2, 1 · 4, 5, 1 · 8, 19, 9, 1 … Σ 1, 3, 10, 37, 151, 674, 3263, 17007, …
  (−∞, 0]nonnegative
• A062264 → D
  5 · 12, 30 · 21, 126, 105 · 32, 336, 672, 280 … Σ 0, 5, 42, 252, 1320, 6435, 30030, …
  (−∞, 0]nonnegative
• A114687 → D+1
  1 · 3, 2 · 7, 14, 4 · 13, 60, 48, 8 … Σ 1, 5, 25, 129, 681, 3653, 19825, …
  (−∞, 0]nonnegative
• A111125 → (θ+1)²
  1 · 3, 4 · 5, 20, 9 · 7, 56, 63, 16 … Σ 1, 7, 34, 142, 541, 1939, 6653, 22091, …
  (−∞, 0]nonnegative
• A063967 → basis A007318
  1 · 2, 1 · 6, 5, 1 · 16, 20, 8, 1 … Σ 1, 3, 12, 45, 171, 648, 2457, 9315, …
  (−∞, −1]nonnegative
• A106800 → x ↦ x+1
  1 · 1 · 2, 1 · 5, 5, 1 … Σ 1, 1, 3, 11, 49, 257, 1539, 10299, …
  (−∞, −1]nonnegative
• basis-map coefficients of A026729, shifted → reverse
  1 · -1 · 1 · -1 … Σ 1, -1, 1, -1, 1, -1, 1, -1, …
  (−∞, 0]rational roots
• A132774 → (θ+1)²
  1 · 2, 12 · 0, 16, 45 · 0, 0, 54, 112 … Σ 1, 14, 61, 166, 353, 646, 1069, 1646, …
  (−∞, 0]nonnegativerational roots
• A108838 → reverse
  2 · 2, 3 · 2, 8, 4 · 2, 15, 20, 5 … Σ 2, 5, 14, 42, 132, 429, 1430, 4862, …
  (−∞, 0]nonnegative
• A063967: Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
  1 · 1, 1 · 2, 3, 1 · 3, 7, 5, 1 … Σ 1, 2, 6, 16, 44, 120, 328, 896, …
  (−5, −1]nonnegative
• A060920 → D
  1 · 5, 2 · 20, 18, 3 · 71, 102, 42, 4 … Σ 0, 1, 7, 41, 219, 1101, 5311, 24881, …
  (−∞, ∞)nonnegative
• A122848 → (θ+1)²
  1 · 0, 4 · 0, 4, 9 · 0, 0, 27, 16 … Σ 1, 4, 13, 43, 148, 526, 1954, 7498, …
  (−∞, 0]nonnegative
• A169803 → D+1
  1 · 2, 1 · 3, 2 · 4, 5, 1 … Σ 1, 3, 5, 10, 18, 33, 59, 105, …
  (−∞, 0]nonnegative
• A116412 → D+1
  1 · 4, 1 · 12, 8, 1 · 33, 39, 12, 1 … Σ 1, 5, 21, 85, 332, 1266, 4741, 17507, …
  (−∞, 0]nonnegative
• A099093 → x ↦ x−1
  1 · -3, 3 · 6, -15, 9 · -9, 45, -63, 27 … Σ 1, 0, 0, 0, 0, 0, 0, 0, …
  (−1/3, 1]
• A049020 → reverse
  1 · 1, 1 · 1, 3, 2 · 1, 6, 10, 5 … Σ 1, 2, 6, 22, 94, 454, 2430, 14214, …
  (−∞, 0]nonnegative
• A391533 → x ↦ x/(x+1)
  4, 12, 9 · 4, 20, 33, 18 · 4, 28, 73, 84, 36 … Σ 25, 75, 225, 675, 2025, 6075, 18225, …
  [−1, 0]nonnegativerational roots
• A013609 → D+1
  1 · 3, 2 · 5, 12, 4 · 7, 30, 36, 8 … Σ 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  (−∞, 0]nonnegativerational roots
• A062190 → D+1
  1 · 7, 6 · 15, 56, 21 · 25, 192, 252, 56 … Σ 1, 13, 92, 525, 2695, 13013, 60424, …
  (−∞, 0]nonnegative
• A134082 → D+1
  1 · 3, 1 · 4, 6, 1 · 0, 12, 9, 1 … Σ 1, 4, 11, 22, 37, 56, 79, 106, …
  (−∞, 0]nonnegative
• A134082 → D
  1 · 4, 2 · 0, 12, 3 · 0, 0, 24, 4 … Σ 0, 1, 6, 15, 28, 45, 66, 91, …
  (−∞, 0]nonnegativerational roots
• A052553: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.
  1 · 1 · 1, 1 · 1, 2 … Σ 1, 1, 2, 3, 5, 8, 13, 21, …
  (−∞, 0)nonnegative
• A007318: Binomial
  1 · 1, 1 · 1, 2, 1 · 1, 3, 3, 1 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, −1]nonnegativerational roots
• A056939 → D
  1 · 4, 2 · 10, 20, 3 · 20, 100, 60, 4 … Σ 0, 1, 6, 33, 184, 1055, 6222, 37639, …
  (−∞, 0]nonnegative
• A026729 → D
  1 · 1, 2 · 0, 4, 3 · 0, 2, 9, 4 … Σ 0, 1, 3, 7, 15, 30, 58, 109, …
  (−∞, 0]nonnegative
• A008277 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 10, 23, 15 … Σ 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  [−1, 0]nonnegative
• A143494 → reverse
  1 · 1, 2 · 1, 5, 4 · 1, 9, 19, 8 … Σ 1, 3, 10, 37, 151, 674, 3263, 17007, …
  (−∞, 0]nonnegative
• A173018 → reverse (shift +1) → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 1, 6, 6 … Σ 1, 1, 3, 13, 75, 541, 4683, 47293, …
  [−1, 0]nonnegative
• A046716: Coefficients of a special case of Poisson-Charlier polynomials.
  1 · 1, 1 · 1, 3, 1 · 1, 6, 8, 1 … Σ 1, 2, 5, 16, 65, 326, 1957, 13700, …
  (−∞, 0)nonnegative
• A026729 → (θ+1)²
  1 · 0, 4 · 0, 4, 9 · 0, 0, 18, 16 … Σ 1, 4, 13, 34, 82, 184, 395, 818, …
  (−∞, 0]nonnegative
• A102547 → x ↦ (1/27)x → x ↦ x+1
  1 · 3 · 9 · 28, 1 … Σ 1, 3, 9, 29, 93, 297, 949, 3033, …
  (−∞, −1]nonnegative
• A391533 → θ+1
  4, 8, 3 · 4, 16, 15, 4 · 4, 24, 39, 24, 5 … Σ 15, 39, 96, 228, 528, 1200, 2688, 5952, …
  (−∞, 0]nonnegative
• A111125 → θ+1
  1 · 3, 2 · 5, 10, 3 · 7, 28, 21, 4 … Σ 1, 5, 18, 60, 191, 589, 1775, 5257, …
  (−∞, 0]nonnegative
• A196347: Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
  1 · 1, 1 · 2, 4, 2 · 6, 18, 18, 6 … Σ 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  (−∞, −1]nonnegativerational roots
• A063967 → x ↦ x−15
  1 · -14, 1 · 182, -27, 1 · -2352, 532, -40, 1 … Σ 1, -13, 156, -1859, 22139, -263640, …
  (10, 14]
• A113214 → D+1
  1 · 3, 1 · 3, 5, 1 · 2, 10, 7, 1 … Σ 1, 4, 9, 20, 40, 78, 147, 272, …
  (−∞, 0]nonnegative
• A130123 → D+1
  1 · 2, 2 · 0, 8, 4 · 0, 0, 24, 8 … Σ 1, 4, 12, 32, 80, 192, 448, 1024, …
  (−∞, 0]nonnegativerational roots
• A063967 → x ↦ x−6
  1 · -5, 1 · 20, -9, 1 · -75, 55, -13, 1 … Σ 1, -4, 12, -32, 80, -192, 448, -1024, …
  (1, 5]
• A038220 → D+1
  1 · 5, 2 · 21, 20, 4 · 81, 126, 60, 8 … Σ 1, 7, 45, 275, 1625, 9375, 53125, …
  (−∞, 0]nonnegativerational roots
• A113413 → D+1
  1 · 3, 1 · 6, 6, 1 · 10, 20, 9, 1 … Σ 1, 4, 13, 40, 117, 332, 921, 2512, …
  (−∞, 0]nonnegative
• A075497 → (θ+1)²
  1 · 2, 4 · 4, 24, 9 · 8, 112, 108, 16 … Σ 1, 6, 37, 244, 1741, 13442, 111833, …
  (−∞, 0]nonnegative
• A030528 → x ↦ x−7
  1 · -6, 1 · 35, -12, 1 · -203, 106, -18, 1 … Σ 1, -5, 24, -114, 540, -2556, 12096, …
  (3, 7]
• A123519 → D+1
  7, 14, 4 · 13, 52, 44, 8 · 21, 140, 228, 120, 16 … Σ 25, 117, 525, 2289, 9773, 41065, …
  (−∞, 0]nonnegative
• A232535 → x ↦ x+1
  1 · 3, 2 · 12, 14, 3 · 48, 80, 37, 4 … Σ 1, 5, 29, 169, 985, 5741, 33461, …
  (−∞, −1]nonnegative
• A131431: 3n + 1 preceded by n zeros.
  1 · 0, 4 · 0, 0, 7 · 0, 0, 0, 10 … Σ 1, 4, 7, 10, 13, 16, 19, 22, …
  [−1, 0]nonnegativerational roots
• A129533: Array read by antidiagonals: T(n,k) = binomial(n+1,2)*binomial(n+k,n+1) for 0 <= k <= n.
  0, 1 · 0, 3, 3 · 0, 6, 12, 6 · 0, 10, 30, 30, 10 … Σ 0, 0, 1, 6, 24, 80, 240, 672, …
  (−∞, 0]nonnegativerational roots
• A123519 → reverse
  4, 6, 1 · 8, 20, 12, 1 · 16, 56, 60, 20, 1 … Σ 11, 41, 153, 571, 2131, 7953, 29681, …
  (−∞, 0]nonnegative
• A391533 → (θ+1)²
  4, 16, 9 · 4, 32, 45, 16 · 4, 48, 117, 96, 25 … Σ 29, 97, 290, 808, 2144, 5488, 13664, …
  (−∞, 0]nonnegative
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x…
  1 · 0, 0, 1 · 0, 0, 0, 0, 1 · 0, 0, 0, 0, 0, 0, 1 … Σ 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0][0, ∞)nonnegative+1
• A102365 → θ+1
  1 · 1 · 1, 2 · 1, 10, 3 … Σ 1, 1, 3, 14, 86, 657, 5995, 63541, …
  (−∞, 0]nonnegative