Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A352880 → reverse → D+1
  8 · 30, 12 · 96, 96, 16 · 250, 500, 210, 20 … Σ 8, 42, 208, 980, 4464, 19866, 86944, …
  (−∞, 0]nonnegative
• A176230 → x ↦ x/(x+1)
  1 · 1, 2 · 3, 12, 10 · 15, 90, 150, 76 … Σ 1, 3, 25, 331, 5937, 133651, 3609673, …
  [−1, 0]nonnegative
• A391403 → D+1
  1 · 1 · 3, 2 · 7, 6 … Σ 1, 1, 5, 13, 43, 131, 411, 1275, …
  (−∞, 0]nonnegative
• A136572 → Veronese 2 (j=1) → D
  6 · 0, 240 · 0, 0, 15120 · 0, 0, 0, 1451520 … Σ 0, 0, 0, 6, 0, 240, 0, 15120, …
  (−∞, 0]nonnegativerational roots
• A103327 → basis A013609 → D
  2 · 24, 8 · 160, 192, 24 · 896, 1920, 960, 64 … Σ 0, 2, 32, 376, 3840, 36384, 329216, …
  (−∞, 0]nonnegative
• A257622: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 4.
  1 · 4, 4 · 16, 56, 16 · 64, 552, 552, 64 … Σ 1, 8, 88, 1232, 20944, 418880, 9634240, …
  (−∞, 0)nonnegative
• A038226 → (θ+1)²
  9, 192, 576 · 27, 864, 5184, 8192 … Σ 777, 14267, 235345, 3621651, 53073625, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → Veronese 2 → θ+1
  1 · -1 · 1, 2 · -1, -6 … Σ 1, -1, 3, -7, 16, -36, 80, -176, …
  (−∞, 0]
• A229032 → x ↦ x+1 → D+1
  1 · 2 · 11, 4 · 36, 16 … Σ 1, 2, 15, 52, 253, 1014, 4395, 18088, …
  (−∞, 0]nonnegative
• A104029 → (θ+1)²
  1 · 2, 4 · 3, 20, 9 · 4, 52, 81, 16 … Σ 1, 6, 32, 153, 673, 2786, 11028, 42193, …
  (−∞, 0]nonnegative
• A257180 → x ↦ x+1
  42, 42, 9 · 336, 504, 222, 27 … Σ 93, 1089, 14277, 207369, 3307053, …
  (−∞, −1]nonnegative
• A123125: Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.
  A173018 → reverse (shift +1)
  1 · 0, 1 · 0, 1, 1 · 0, 1, 4, 1 … Σ 1, 1, 2, 6, 24, 120, 720, 5040, …
  (−∞, 0]nonnegative
• A038207 → Veronese 2 (j=1) → D+1
  1 · 4 · 13, 1 · 40, 8 … Σ 0, 1, 4, 14, 48, 163, 548, 1824, …
  (−∞, 0]nonnegative
• A133087 → reverse → D
  2 · 2, 2 · 4, 10, 6 · 4, 12, 12, 4 … Σ 0, 2, 4, 20, 32, 128, 192, 704, …
  (−∞, 0]nonnegativerational roots
• A062264 → reverse → x ↦ x/(x+1)
  1 · 5, 6 · 15, 42, 28 · 35, 168, 252, 120 … Σ 1, 11, 85, 575, 3649, 22363, 134245, …
  [−1, 0]nonnegative
• A056242 → x ↦ x+1 → θ+1
  1 · 3, 4 · 10, 26, 12 · 34, 130, 120, 32 … Σ 1, 7, 48, 316, 2019, 12621, 77598, …
  (−∞, 0]nonnegative
• A132393 → basis A013610 → D+1
  1 · 4, 3 · 11, 27, 9 · 39, 141, 135, 27 … Σ 1, 7, 47, 342, 2754, 24552, 241128, …
  (−∞, 0]nonnegative
• basis-map coefficients of A371080, shifted → basis A013611 → (θ+1)²
  1 · 1, 16 · 4, 80, 144 · 28, 624, 1728, 1024 … Σ 1, 17, 228, 3404, 58792, 1169448, …
  (−∞, 0]nonnegative
• A038291 → basis A013611 → reverse
  1 · 4, 10 · 16, 80, 100 · 64, 480, 1200, 1000 … Σ 1, 14, 196, 2744, 38416, 537824, …
  (−∞, 0]nonnegativerational roots
• A094638 → basis A013610 → basis A007318
  1 · 5, 3 · 45, 57, 18 · 585, 1146, 747, 162 … Σ 1, 8, 120, 2640, 76560, 2756160, …
  (−∞, −1]nonnegativerational roots
• A038226 → θ+1
  9, 96, 192 · 27, 432, 1728, 2048 … Σ 297, 4235, 57233, 746691, 9502009, …
  (−∞, 0]nonnegativerational roots
• A008277 → D → θ+1
  1 · 3, 4 · 7, 24, 9 · 15, 100, 90, 16 … Σ 0, 1, 7, 40, 221, 1241, 7218, 43807, …
  (−∞, 0]nonnegative
• A102662 → reverse → x ↦ x+1
  1 · 4, 1 · 9, 7, 1 · 22, 28, 10, 1 … Σ 1, 5, 17, 61, 217, 773, 2753, 9805, …
  (−∞, −1]nonnegative
• A081578 → (θ+1)²
  1 · 1, 4 · 1, 20, 9 · 1, 36, 81, 16 … Σ 1, 5, 30, 134, 583, 2355, 9196, 34748, …
  (−∞, 0]nonnegative
• A036969 → x ↦ x+1
  1 · 2, 1 · 7, 7, 1 · 37, 52, 17, 1 … Σ 1, 3, 15, 107, 1015, 12243, 181311, …
  (−∞, −1]nonnegative
• A008277 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 14, 18, 4 … Σ 0, 1, 3, 10, 37, 151, 674, 3263, …
  (−∞, 0]nonnegative
• A099093 → x ↦ x−10
  1 · -30, 3 · 870, -177, 9 · -25200, 7740, -792, 27 … Σ 1, -27, 702, -18225, 473121, -12282192, …
  (26/3, 10]
• A200536 → basis A013610 → reverse
  1 · 18, 21, 6 · 324, 756, 657, 252, 36 … Σ 1, 45, 2025, 91125, 4100625, 184528125, …
  (−∞, 0]nonnegativerational roots
• A116414 → basis A007318 → θ+1
  1 · 5, 2 · 22, 20, 3 · 95, 138, 45, 4 … Σ 1, 7, 45, 282, 1737, 10557, 63477, …
  (−∞, 0]nonnegative
• A038279 → (θ+1)²
  1 · 8, 4 · 64, 64, 9 · 512, 768, 216, 16 … Σ 1, 12, 137, 1512, 16281, 172044, …
  (−∞, 0]nonnegative
• A038291 → basis A013610 → x ↦ x/(x+1)
  1 · 10, 13 · 100, 260, 169 · 1000, 3900, 5070, 2197 … Σ 1, 23, 529, 12167, 279841, 6436343, …
  [−1, 0]nonnegativerational roots
• basis-map coefficients of A030528, shifted → (θ+1)² → θ+1
  1 · 0, -8 · 0, 0, 27 · 0, 0, 0, -64 … Σ 1, -8, 27, -64, 125, -216, 343, -512, …
  (−∞, 0]rational roots
• A133081 → reverse (shift +1) → D+1
  1 · 1, 3, 1 · 0, 0, 0, 4, 1 · 0, 0, 0, 0, 5, 7, 1 … Σ 1, 5, 5, 13, 9, 21, 13, 29, …
  (−∞, 0]nonnegative
• A136124 → basis A013609 → D+1
  1 · 5, 2 · 26, 22, 4 · 154, 190, 72, 8 … Σ 1, 7, 52, 424, 3812, 37668, 406920, …
  (−∞, 0]nonnegative
• A038323 → basis A013610 → θ+1
  1 · 20, 54 · 400, 2160, 2187 … Σ 1, 74, 4747, 282752, 16092565, …
  (−∞, 0]nonnegativerational roots
• A081577 → basis A013609 → x ↦ x/(x+1)
  1 · 2, 4 · 6, 24, 22 · 16, 96, 184, 112 … Σ 1, 6, 52, 408, 3280, 26208, 209728, …
  [−1, 0]nonnegative
• shift_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 between …
  2, 1 · 8, 12, 6, 1 · 48, 120, 116, 54, 12, 1 … Σ 3, 27, 351, 6075, 131463, 3413907, …
  (−∞, −1]nonnegative
• A081579 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 3 · 8, 24, 17 · 24, 108, 158, 75 … Σ 1, 5, 49, 365, 3001, 23765, 190849, …
  [−1, 0]nonnegative
• A121314 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 1, 5, 5 … Σ 1, 1, 3, 11, 41, 153, 571, 2131, …
  [−1, 0]nonnegative
• A178820 → basis A013610 → D+1
  1 · 20, 12 · 160, 300, 90 · 880, 2880, 2700, 540 … Σ 1, 32, 550, 7000, 74375, 700000, …
  (−∞, 0]nonnegativerational roots
• A207607 → basis A013611 → D
  8 · 36, 64 · 132, 480, 384 · 444, 2432, 4032, 2048 … Σ 0, 8, 100, 996, 8956, 75980, 620808, …
  (−∞, 0]nonnegative
• A372868 → (θ+1)²
  1 · 2 · 4, 4 · 8, 24 … Σ 1, 2, 8, 32, 121, 442, 1580, 5560, …
  (−∞, 0]nonnegative
• A137320 → basis A013610 → D
  6 · 30, 72 · 156, 648, 648 · 924, 5112, 9072, 5184 … Σ 0, 6, 102, 1452, 20292, 291024, …
  (−∞, 0]nonnegative
• A078812 → basis A013610 → D+1
  1 · 6, 3 · 26, 36, 9 · 96, 237, 162, 27 … Σ 1, 9, 71, 522, 3673, 25083, 167615, …
  (−∞, 0]nonnegative
• A131222 → reverse
  1 · 1 · 1, 3 · 1, 9, 14 … Σ 1, 1, 4, 24, 192, 1920, 23040, 322560, …
  (−∞, 0]nonnegative
• A038303 → basis A007318 → θ+1
  1 · 11, 2 · 121, 44, 3 · 1331, 726, 99, 4 … Σ 1, 13, 168, 2160, 27648, 352512, …
  (−∞, 0]nonnegativerational roots
• A225471 → basis A008277 → D+1
  1 · 5, 1 · 45, 15, 1 · 585, 255, 30, 1 … Σ 1, 6, 61, 871, 15996, 358891, 9509641, …
  (−∞, 0]nonnegative
• A127773 → (θ+1)²
  1 · 0, 12 · 0, 0, 54 · 0, 0, 0, 160 … Σ 1, 12, 54, 160, 375, 756, 1372, 2304, …
  [−1, 0]nonnegativerational roots
• A111125 → reverse → D
  3 · 5, 10 · 7, 28, 21 · 9, 54, 90, 36 … Σ 0, 3, 15, 56, 189, 605, 1872, 5655, …
  (−∞, 0]nonnegative
• A261711: Triangle read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string 01, where n>=0 and k>=0.
  1 · 4 · 15, 1 · 56, 8 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  (−∞, 0)nonnegative