Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A091044 → ⊙ C(n+2,k+1) → θ+1
  1 · 1, 2 · 1, 10, 3 · 1, 28, 42, 4 … Σ 1, 3, 14, 75, 438, 2702, 17324, 114291, …
  (−∞, 0]nonnegative
• A103406 → (θ+1)²
  1 · 1, 8 · 1, 16, 27 · 1, 24, 81, 64 … Σ 1, 9, 44, 170, 576, 1792, 5248, 14688, …
  (−∞, 0]nonnegative
• A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by co…
  1 · 0, 2 · 0, 0, 4 · 0, 0, 0, 8 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  [−1, 0]nonnegativerational roots
• A008459 → D → D+1
  1 · 6, 2 · 27, 24, 3 · 88, 168, 60, 4 … Σ 0, 1, 8, 54, 320, 1750, 9072, 45276, …
  (−∞, 0]nonnegative
• A062264 → x ↦ x+1 → D
  5 · 42, 30 · 252, 336, 105 · 1320, 2520, 1512, 280 … Σ 0, 5, 72, 693, 5632, 41805, 293640, …
  (−∞, 0]nonnegative
• A028326 → D → D+1
  2 · 8, 4 · 18, 24, 6 · 32, 72, 48, 8 … Σ 0, 2, 12, 48, 160, 480, 1344, 3584, …
  (−∞, 0]nonnegativerational roots
• A143494 → reverse → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 10 · 1, 12, 40, 37 … Σ 1, 4, 18, 90, 494, 2946, 18926, 130066, …
  [−1, 0]nonnegative
• A011973 → basis A008277 → (θ+1)²
  1 · 1 · 2, 4 · 3, 8 … Σ 1, 1, 6, 11, 38, 87, 257, 647, …
  (−∞, 0]nonnegative
• A038321 → D+1
  275, 252, 49 · 3872, 5775, 2646, 343 … Σ 576, 12636, 268272, 5563728, 113374080, …
  (−∞, 0]nonnegativerational roots
• A038323 → basis A013609 → x ↦ x/(x+1)
  1 · 20, 38 · 400, 1520, 1444 … Σ 1, 58, 3364, 195112, 11316496, …
  [−1, 0]nonnegativerational roots
• A169803 → x ↦ x+1
  1 · 2, 1 · 3, 2 · 5, 5, 1 … Σ 1, 3, 5, 11, 21, 43, 85, 171, …
  (−∞, −1]nonnegative
• A085880 → basis A013609 → D+1
  1 · 4, 2 · 24, 32, 8 · 160, 360, 240, 40 … Σ 1, 6, 64, 800, 10752, 150528, 2162688, …
  (−∞, 0]nonnegativerational roots
• A122542 → basis A013610 → x ↦ x/(x+1)
  1 · 1, 4 · 3, 18, 24 · 7, 60, 162, 136 … Σ 1, 5, 45, 365, 3005, 24685, 202845, …
  [−1, 0]nonnegative
• A290598 → D+1
  1 · 5, 1 · 42, 16, 1 · 490, 270, 33, 1 … Σ 1, 6, 59, 794, 13473, 274966, 6544291, …
  (−∞, 0]nonnegative
• Veronese parent of A109447 → (θ+1)²
  0, 4, 9 · 1, 8, 9 · 0, 4, 27, 48, 25 … Σ 13, 18, 104, 160, 688, 1120, 4096, …
  (−∞, 0]nonnegative
• A038281 → basis A013611 → D+1
  385, 552, 144 · 5687, 13860, 9936, 1728 … Σ 1081, 31211, 863857, 23226803, …
  (−∞, 0]nonnegativerational roots
• A059364 → basis A013611 → D
  4 · 44, 64 · 488, 1504, 1152 … Σ 0, 4, 108, 3144, 106264, 4158684, …
  (−∞, 0]nonnegative
• A139548 → basis A013610 → θ+1
  1 · 8, 48, 54 · 64, 768, 2592, 3456, 1620 … Σ 1, 110, 8500, 575000, 36250000, …
  (−∞, 0]nonnegativerational roots
• A008292 → γ (scale 1/4) → (θ+1)²
  1 · 2 · 4, 8 · 8, 64 … Σ 1, 2, 12, 72, 512, 4144, 37632, 379008, …
  (−∞, 0]nonnegative
• A249139 → basis A013609 → basis A007318
  1 · 6, 2 · 11, 4 · 41, 26, 4 … Σ 1, 8, 15, 71, 176, 673, 1905, 6616, …
  (−∞, −1]nonnegative
• A034839 → (θ+1)²
  1 · 1 · 1, 4 · 1, 12 … Σ 1, 1, 5, 13, 34, 86, 212, 512, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → basis A013611 → x ↦ x/(x+1)
  1, 10, 25 · 1, 15, 75, 125 · 1, 20, 150, 500, 625 … Σ 36, 216, 1296, 7776, 46656, 279936, …
  [−1, 0]nonnegativerational roots
• delannoy → x ↦ x+1 → (θ+1)²
  1 · 2, 4 · 5, 20, 9 · 12, 72, 72, 16 … Σ 1, 6, 34, 172, 822, 3768, 16754, 72756, …
  (−∞, 0]nonnegative
• A013611 → D → (θ+1)²
  4 · 8, 128 · 12, 384, 1728 · 16, 768, 6912, 16384 … Σ 0, 4, 136, 2124, 24080, 228500, …
  (−∞, 0]nonnegative
• A013622 → basis A013610 → D+1
  1 · 23, 15 · 304, 690, 225 · 3392, 13680, 15525, 3375 … Σ 1, 38, 1219, 35972, 1009861, 27424418, …
  (−∞, 0]nonnegativerational roots
• A133214 → D+1
  9, 16, 4 · 19, 90, 60, 8 · 33, 320, 528, 192, 16 … Σ 29, 177, 1089, 6693, 41005, 250337, …
  (−∞, 0]nonnegative
• A052553 → basis A013610 → D+1
  1 · 1 · 5, 3 · 9, 6 … Σ 1, 1, 8, 15, 62, 149, 484, 1275, …
  (−∞, 0]nonnegative
• 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
• A038259 → basis A013610 → x ↦ x/(x+1)
  1 · 11, 26 · 121, 572, 676 · 1331, 9438, 22308, 17576 … Σ 1, 37, 1369, 50653, 1874161, 69343957, …
  [−1, 0]nonnegativerational roots
• A060920 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 13, 59, 88, 43 … Σ 1, 5, 31, 203, 1345, 8933, 59359, …
  [−1, 0]nonnegative
• A013620 → basis A008288 → D
  3 · 39, 18 · 333, 378, 81 · 2391, 4698, 2349, 324 … Σ 0, 3, 57, 792, 9762, 112785, 1250919, …
  (−∞, 0]nonnegative
• A038321 → (θ+1)²
  121, 616, 441 · 1331, 10164, 14553, 5488 … Σ 1178, 31536, 785376, 18627408, …
  (−∞, 0]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
• A027466 → θ+1
  1 · 7, 2 · 49, 28, 3 · 343, 294, 63, 4 … Σ 1, 9, 80, 704, 6144, 53248, 458752, …
  (−∞, 0]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
• A063007 → basis A013610 → D+1
  1 · 9, 6 · 67, 162, 54 · 459, 2016, 2430, 540 … Σ 1, 15, 283, 5445, 104541, 1995939, …
  (−∞, 0]nonnegative
• A139548 → x ↦ x+1 → basis A013609
  1 · 18, 24, 8 · 324, 864, 864, 384, 64 … Σ 1, 50, 2500, 125000, 6250000, …
  (−∞, 0]nonnegativerational roots
• A038314 → x ↦ x+1 → θ+1
  484, 1056, 432 · 10648, 34848, 28512, 6912 … Σ 1972, 80920, 3222928, 125615584, …
  (−∞, 0]nonnegativerational 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
• A055151 → D → θ+1
  1 · 3 · 6, 8 · 10, 40 … Σ 0, 0, 1, 3, 14, 50, 180, 616, …
  (−∞, 0]nonnegative
• A110441 → basis A013611 → D
  4 · 32, 32 · 176, 384, 192 · 832, 2880, 3072, 1024 … Σ 0, 4, 64, 752, 7808, 75824, 706048, …
  (−∞, 0]nonnegative
• A028326 → D → θ+1
  2 · 4, 8 · 6, 24, 18 · 8, 48, 72, 32 … Σ 0, 2, 12, 48, 160, 480, 1344, 3584, …
  (−∞, 0]nonnegativerational roots
• A106246 → reverse → D
  2 · 4, 2 · 6, 6 · 8, 12 … Σ 0, 2, 6, 12, 20, 30, 42, 56, …
  (−∞, 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]
• A038305 → x ↦ x+1 → x ↦ x+1
  256, 96, 9 · 4096, 2304, 432, 27 … Σ 361, 6859, 130321, 2476099, 47045881, …
  (−∞, −1]nonnegativerational roots
• A375853 → (θ+1)²
  2 · 8, 32 · 20, 224, 180 · 40, 864, 1944, 640 … Σ 2, 40, 424, 3488, 24832, 160768, …
  (−∞, 0]nonnegative
• A034870 → basis A013609 → θ+1
  1 · 4, 16, 12 · 16, 128, 288, 256, 80 … Σ 1, 32, 768, 16384, 327680, 6291456, …
  (−∞, 0]nonnegativerational roots
• A108299 → D+1
  1 · 0, -1 · 0, -3, -1 · 0, -5, 1, 1 … Σ 1, -1, -4, -3, 3, 8, 5, -5, …
  (−∞, ∞)