Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A059344 → basis A007318 → θ+1
  1 · 1 · 3, 4 · 7, 12 … Σ 1, 1, 7, 19, 133, 541, 3931, 20287, …
  (−∞, 0]nonnegative
• A094816 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 16, 71, 103, 49 … Σ 1, 5, 31, 239, 2257, 25517, 337615, …
  [−1, 0]nonnegative
• A008277 → (θ+1)² → x ↦ x+1
  1 · 5, 4 · 22, 30, 9 · 99, 184, 102, 16 … Σ 1, 9, 61, 401, 2701, 18961, 139469, …
  (−∞, −1]nonnegative
• A003506 → Veronese 2 (j=1) → D+1
  2 · 6 · 16, 4 · 40, 20 … Σ 0, 0, 2, 6, 20, 60, 168, 448, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → Veronese 2 → D
  1 · -3 · 6, 2 · -10, -10 … Σ 0, 0, 1, -3, 8, -20, 48, -112, …
  (−∞, 0]
• A059110 → x ↦ x+1
  1 · 2, 1 · 8, 6, 1 · 44, 42, 12, 1 … Σ 1, 3, 15, 99, 801, 7623, 83079, …
  (−∞, −1]nonnegative
• A172431 → basis A007318 → D+1
  1 · 5, 2 · 18, 16, 3 · 59, 82, 34, 4 … Σ 1, 7, 37, 179, 833, 3791, 16989, 75243, …
  (−∞, 0]nonnegative
• A114656 → reverse → (θ+1)²
  1 · 1, 8 · 1, 24, 36 · 1, 48, 216, 128 … Σ 1, 9, 61, 393, 2481, 15473, 95677, …
  (−∞, 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
• A143491 → basis A007318 → D+1
  19, 9, 1 · 107, 71, 15, 1 · 702, 580, 173, 22, 1 … Σ 29, 194, 1478, 12664, 120696, 1267440, …
  (−∞, 0]nonnegative
• A257625 → x ↦ x/(x+1)
  1 · 3, 6 · 9, 72, 72 · 27, 702, 1944, 1296 … Σ 1, 9, 153, 3969, 137457, 5949369, …
  [−1, 0]nonnegative
• A131431 → θ+1 → basis A007318
  1 · 8, 8 · 21, 42, 21 · 40, 120, 120, 40 … Σ 1, 16, 84, 320, 1040, 3072, 8512, …
  (−∞, −1]nonnegativerational roots
• A064861 → Veronese 2 (j=1) → θ+1
  3 · 5, 8 · 6, 24 · 8, 76, 24 … Σ 3, 13, 30, 108, 243, 837, 1836, 6156, …
  (−∞, 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]
• A123519 → basis A013610 → θ+1
  11, 84, 108 · 41, 456, 1188, 864 … Σ 203, 2549, 30573, 355507, 4042471, …
  (−∞, 0]nonnegative
• A028326 → Veronese 2 (j=1) → basis A007318
  2 · 4 · 8, 2 · 16, 8 … Σ 0, 2, 4, 10, 24, 58, 140, 338, …
  (−∞, −1]nonnegative
• A038328 → basis A013609 → D+1
  308, 144, 16 · 5096, 3696, 864, 64 … Σ 468, 9720, 198288, 3989088, 79361856, …
  (−∞, 0]nonnegativerational roots
• A094638 → θ+1 → D
  2 · 6, 12 · 12, 66, 72 · 20, 210, 600, 480 … Σ 0, 2, 18, 150, 1310, 12320, 125552, …
  (−∞, 0]nonnegative
• A028338 → θ+1
  1 · 1, 2 · 3, 8, 3 · 15, 46, 27, 4 … Σ 1, 3, 14, 92, 784, 8224, 102528, …
  (−∞, 0]nonnegative
• A038277 → basis A013609 → basis A007318
  1 · 40, 22 · 1600, 1760, 484 … Σ 1, 62, 3844, 238328, 14776336, …
  (−∞, −1]nonnegativerational roots
• 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
• A062145 → D → x ↦ x+1
  4 · 30, 20 · 168, 210, 60 · 840, 1512, 840, 140 … Σ 0, 4, 50, 438, 3332, 23560, 159462, …
  (−∞, −1]nonnegative
• A253283 → D
  1 · 2, 6 · 3, 24, 30 · 4, 60, 180, 140 … Σ 0, 1, 8, 57, 384, 2505, 16008, 100849, …
  (−∞, 0]nonnegative
• A102547 → x ↦ (1/27)x → basis A007318 → x ↦ x+1
  1 · 3 · 9 · 29, 1 … Σ 1, 3, 9, 30, 99, 324, 1062, 3483, …
  (−∞, −1]nonnegative
• A008556: Triangle of coefficients of Legendre polynomials 2^n P_n (x).
  1 · 2 · 6, 2 · 20, 12 … Σ 1, 2, 8, 32, 136, 592, 2624, 11776, …
  (−∞, 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
• A143497 → x ↦ x+1
  31, 12, 1 · 229, 129, 21, 1 · 1961, 1432, 342, 32, 1 … Σ 44, 380, 3768, 42112, 523072, 7141248, …
  (−∞, −1]nonnegative
• A165891: Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1/x)*(1-x)^n*(1+x)^(n+2)*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.
  1 · 1, 2, 1 · 1, 6, 10, 6, 1 … Σ 1, 4, 24, 192, 1920, 23040, 322560, …
  (−∞, 0]nonnegative
• A286718 → basis A013609 → basis A007318
  1 · 4, 2 · 28, 22, 4 · 280, 276, 84, 8 … Σ 1, 6, 54, 648, 9720, 174960, 3674160, …
  (−∞, −1]nonnegativerational roots
• A064861 → Veronese 2 → x ↦ x+1
  3, 2 · 9, 8 · 18, 21, 4 · 54, 81, 28 … Σ 5, 17, 43, 163, 395, 1511, 3649, 13969, …
  (−∞, −1]nonnegative
• A038328 → basis A013611 → D+1
  420, 352, 64 · 7448, 10080, 4224, 512 … Σ 836, 22264, 574992, 14523872, …
  (−∞, 0]nonnegativerational roots
• A290596 → reverse
  1 · 1, 2 · 1, 10, 10 · 1, 24, 120, 80 … Σ 1, 3, 21, 225, 3213, 57051, 1207953, …
  (−∞, 0]nonnegative
• A038221 → basis A013611 → x ↦ x+1
  1 · 18, 12 · 324, 432, 144 · 5832, 11664, 7776, 1728 … Σ 1, 30, 900, 27000, 810000, 24300000, …
  (−∞, −1]nonnegativerational roots
• A131431 → θ+1 → reverse
  1 · 8 · 21 · 40 … Σ 1, 8, 21, 40, 65, 96, 133, 176, …
  (−∞, 0]nonnegativerational roots
• A059473 → x ↦ x+1 → x ↦ x+1
  1 · 6, 2 · 40, 26, 4 · 264, 256, 80, 8 … Σ 1, 8, 70, 608, 5284, 45920, 399064, …
  (−∞, −1]nonnegative
• A038324 → basis A013610
  1 · 21, 30 · 441, 1260, 900 … Σ 1, 51, 2601, 132651, 6765201, …
  [−1, 0]nonnegativerational roots
• A145324 → basis A013609 → reverse
  1 · 4, 3 · 24, 34, 12 · 192, 392, 266, 60 … Σ 1, 7, 70, 910, 14560, 276640, 6086080, …
  (−∞, 0]nonnegativerational roots
• A028326 → basis A013609 → D+1
  2 · 8, 4 · 24, 32, 8 · 64, 144, 96, 16 … Σ 2, 12, 64, 320, 1536, 7168, 32768, …
  (−∞, 0]nonnegativerational roots
• 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
• mobius_x_over_x_plus_one of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is …
  1, 2 · 1, 6, 12, 8 · 1, 12, 54, 116, 120, 48 … Σ 3, 27, 351, 6075, 131463, 3413907, …
  [−1, 0]nonnegative
• A110608 → x ↦ x/(x+1)
  6, 20, 15 · 20, 105, 168, 84 … Σ 41, 377, 3649, 36365, 369305, 3800305, …
  [−1, 0]nonnegative
• A038226 → basis A013611 → reverse
  1024, 704, 121 · 32768, 33792, 11616, 1331 … Σ 1849, 79507, 3418801, 147008443, …
  (−∞, 0]nonnegativerational roots
• A145324 → basis A013609 → basis A007318
  1 · 7, 4 · 70, 82, 24 · 910, 1626, 968, 192 … Σ 1, 11, 176, 3696, 96096, 2978976, …
  (−∞, −1]nonnegativerational roots
• A099040 → θ+1
  1 · 0, 4 · 0, 4, 12 · 0, 0, 24, 32 … Σ 1, 4, 16, 56, 188, 608, 1920, 5952, …
  (−∞, 0]nonnegative
• A101842: Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k and k+1.
  1, 1 · 1, 3, 3, 1 · 1, 7, 16, 16, 7, 1 … Σ 2, 8, 48, 384, 3840, 46080, 645120, …
  (−∞, 0]nonnegative
• A188440 → D
  2 · 2 · 4, 8 · 4, 8 … Σ 0, 0, 2, 2, 12, 12, 54, 54, …
  (−∞, 0]nonnegativerational roots
• type_b_eulerian → γ → D
  4 · 20 · 72, 160 · 232, 1952 … Σ 0, 0, 4, 20, 232, 2184, 27756, 367644, …
  (−∞, 0]nonnegative
• A091042 → basis A013609 → D
  6 · 40, 40 · 224, 448, 168 · 1152, 3456, 2880, 576 … Σ 0, 6, 80, 840, 8064, 73568, 648960, …
  (−∞, 0]nonnegative
• A069466 → γ → D
  12 · 120 · 840, 840 · 5040, 15120 … Σ 0, 0, 12, 120, 1680, 20160, 249480, …
  (−∞, 0]nonnegative