Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A019538 → D+1
  1 · 3, 2 · 7, 18, 6 · 15, 86, 108, 24 … Σ 1, 5, 31, 233, 2071, 21305, 249271, …
  (−∞, 0]nonnegative
• A038336 → basis A013609 → x ↦ x/(x+1)
  484, 1848, 1764 · 10648, 60984, 116424, 74088 … Σ 4096, 262144, 16777216, 1073741824, …
  [−1, 0]nonnegativerational roots
• A081577 → Veronese 2 → x ↦ x+1
  1 · 1 · 2, 1 · 8, 7 … Σ 1, 1, 3, 15, 49, 145, 483, 1623, …
  (−∞, −1]nonnegative
• A013610 → basis A013610 → D
  9 · 72, 162 · 432, 1944, 2187 … Σ 0, 9, 234, 4563, 79092, 1285245, …
  (−∞, 0]nonnegativerational roots
• A128813 → basis A007318 → (θ+1)²
  1 · 2, 4 · 8, 40, 27 · 56, 472, 729, 288 … Σ 1, 6, 75, 1545, 47315, 2017070, …
  (−∞, 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]
• basis-map coefficients of A026729 → basis A013609
  1 · 1, 2 · 1, 4, 4 · 1, 6, 12, 8 … Σ 1, 3, 9, 27, 81, 243, 729, 2187, …
  [−1, 0]nonnegativerational roots
• A143495 → θ+1 → reverse
  1 · 2, 3 · 3, 14, 9 · 4, 36, 74, 27 … Σ 1, 5, 26, 141, 799, 4736, 29371, …
  (−∞, 0]nonnegative
• A128099 → basis A013609 → θ+1
  1 · 1 · 3, 8 · 5, 16 … Σ 1, 1, 11, 21, 115, 293, 1203, 3493, …
  (−∞, 0]nonnegative
• A049020 → reverse
  1 · 1, 1 · 1, 3, 2 · 1, 6, 10, 5 … Σ 1, 2, 6, 22, 94, 454, 2430, 14214, …
  (−∞, 0]nonnegative
• A161198 → basis A007318 → D
  2 · 16, 8 · 142, 120, 24 · 1488, 1648, 576, 64 … Σ 0, 2, 24, 286, 3776, 56018, 930360, …
  (−∞, 0]nonnegative
• A345013 → θ+1
  1 · 4, 6 · 15, 40, 18 · 56, 210, 180, 40 … Σ 1, 10, 73, 486, 3113, 19538, 121105, …
  (−∞, 0]nonnegative
• A274496 → D → (θ+1)²
  2 · 4 · 8, 32 · 16, 64 … Σ 0, 0, 2, 4, 40, 80, 432, 864, …
  (−∞, 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
• A131431 → D+1 → basis A007318
  1 · 8, 4 · 21, 28, 7 · 40, 90, 60, 10 … Σ 1, 12, 56, 200, 624, 1792, 4864, 12672, …
  (−∞, −1]nonnegativerational roots
• A135089 → D
  5 · 10, 10 · 15, 30, 15 · 20, 60, 60, 20 … Σ 0, 5, 20, 60, 160, 400, 960, 2240, …
  (−∞, 0]nonnegativerational roots
• A038331 → basis A013610 → θ+1
  289, 1020, 675 · 4913, 26010, 34425, 13500 … Σ 1984, 78848, 3014656, 112197632, …
  (−∞, 0]nonnegativerational roots
• A036969: Triangle read by rows: T(n,k) = T(n-1,k-1) + k^2*T(n-1,k), 1 < k <= n, T(n,1) = 1.
  1 · 1, 1 · 1, 5, 1 · 1, 21, 14, 1 … Σ 1, 2, 7, 37, 264, 2433, 27913, 386906, …
  (−∞, 0]nonnegative
• A232535: Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.
  1 · 1, 2 · 1, 8, 3 · 1, 18, 25, 4 … Σ 1, 3, 12, 48, 192, 768, 3072, 12288, …
  (−∞, 0)nonnegative
• A113413 → x ↦ x+1 → D
  1 · 6, 2 · 23, 18, 3 · 76, 96, 36, 4 … Σ 0, 1, 8, 44, 212, 952, 4092, 17072, …
  (−∞, 0]nonnegative
• A062190 → D+1
  1 · 7, 6 · 15, 56, 21 · 25, 192, 252, 56 … Σ 1, 13, 92, 525, 2695, 13013, 60424, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729 → basis A013611 → (θ+1)²
  1 · 1, 16 · 1, 32, 144 · 1, 48, 432, 1024 … Σ 1, 17, 177, 1505, 11425, 80625, 540625, …
  (−∞, 0]nonnegative
• A136124 → θ+1
  1 · 2, 2 · 6, 10, 3 · 24, 52, 27, 4 … Σ 1, 4, 19, 107, 702, 5274, 44712, …
  (−∞, 0]nonnegative
• A137320 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 2, 4 · 6, 22, 20 · 24, 124, 212, 120 … Σ 1, 6, 48, 480, 5760, 80640, 1290240, …
  [−1, 0]nonnegativerational roots
• A038221 → basis A013611 → θ+1
  1 · 6, 24 · 36, 288, 432 · 216, 2592, 7776, 6912 … Σ 1, 30, 756, 17496, 384912, 8188128, …
  (−∞, 0]nonnegativerational roots
• A053117 → x ↦ x+1 → (θ+1)²
  1 · 2, 8 · 3, 32, 36 · 4, 80, 216, 128 … Σ 1, 10, 71, 428, 2345, 12062, 59311, …
  (−∞, 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
• A106246 → D+1 → θ+1
  1 · 3, 2 · 5, 12, 3 · 3, 30, 27, 4 … Σ 1, 5, 20, 64, 167, 371, 730, 1310, …
  (−∞, 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
• A008459 → x ↦ x+1 → basis A007318
  1 · 3, 1 · 13, 8, 1 · 63, 57, 15, 1 … Σ 1, 4, 22, 136, 886, 5944, 40636, …
  (−∞, −1]nonnegative
• A122542 → basis A013611 → D+1
  1 · 5, 4 · 19, 48, 16 · 59, 276, 304, 64 … Σ 1, 9, 83, 703, 5713, 45073, 348083, …
  (−∞, 0]nonnegative
• A106246 → basis A007318 → x ↦ x/(x+1)
  1 · 3, 4 · 6, 18, 13 · 10, 48, 75, 38 … Σ 1, 7, 37, 171, 729, 2943, 11421, 43011, …
  [−1, 0]nonnegative
• A134991 → x ↦ x+1
  1 · 4, 3 · 26, 40, 15 · 236, 550, 420, 105 … Σ 1, 7, 81, 1311, 27273, 693351, …
  (−2, −1)nonnegative
• A056939 → D
  1 · 4, 2 · 10, 20, 3 · 20, 100, 60, 4 … Σ 0, 1, 6, 33, 184, 1055, 6222, 37639, …
  (−∞, 0]nonnegative
• A088729 → x ↦ x+1
  23, 11, 1 · 173, 118, 21, 1 · 1602, 1389, 361, 34, 1 … Σ 35, 313, 3387, 42857, 619355, 10048073, …
  (−∞, −1]nonnegative
• A026729 → D
  1 · 1, 2 · 0, 4, 3 · 0, 2, 9, 4 … Σ 0, 1, 3, 7, 15, 30, 58, 109, …
  (−∞, 0]nonnegative
• A108426 → x ↦ x+1
  1 · 2, 1 · 10, 9, 2 · 66, 85, 36, 5 … Σ 1, 3, 21, 192, 2001, 22539, 267276, …
  (−∞, −1]nonnegative
• A085478 → Veronese 2 → x ↦ x+1
  1 · 1 · 1 · 2, 1 … Σ 1, 1, 1, 3, 11, 35, 107, 329, …
  (−∞, −1]nonnegative
• A232535 → basis A013610 → (θ+1)²
  1 · 3, 24 · 12, 168, 243 · 48, 960, 2997, 1728 … Σ 1, 27, 423, 5733, 72117, 863127, …
  (−∞, 0]nonnegative
• A161198 → basis A007318 → D+1
  1 · 5, 2 · 31, 24, 4 · 247, 262, 84, 8 … Σ 1, 7, 59, 601, 7241, 101063, 1606035, …
  (−∞, 0]nonnegative
• A127773 → (θ+1)² → reverse
  1 · 12 · 54 · 160 … Σ 1, 12, 54, 160, 375, 756, 1372, 2304, …
  (−∞, 0]nonnegativerational roots
• A108838: Triangle of Dyck paths counted by number of long interior inclines.
  2 · 3, 2 · 4, 8, 2 · 5, 20, 15, 2 … Σ 2, 5, 14, 42, 132, 429, 1430, 4862, …
  (−∞, 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
• A130123 → basis A013611
  1 · 2, 8 · 4, 32, 64 · 8, 96, 384, 512 … Σ 1, 10, 100, 1000, 10000, 100000, …
  [−1, 0]nonnegativerational roots
• A143494 → reverse
  1 · 1, 2 · 1, 5, 4 · 1, 9, 19, 8 … Σ 1, 3, 10, 37, 151, 674, 3263, 17007, …
  (−∞, 0]nonnegative
• A120909 → Veronese 2 (j=1) → D
  24 · 96 · 240, 192 · 480, 1152 … Σ 0, 0, 0, 24, 96, 432, 1632, 6024, …
  (−∞, 0]nonnegative
• A104684 → basis A013609 → D+1
  1 · 5, 2 · 29, 24, 4 · 177, 234, 84, 8 … Σ 1, 7, 57, 503, 4577, 42135, 389625, …
  (−∞, 0]nonnegative