Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A003506: Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.
  1 · 2, 2 · 3, 6, 3 · 4, 12, 12, 4 … Σ 1, 4, 12, 32, 80, 192, 448, 1024, …
  (−∞, 0)nonnegativerational roots
• A038207 → Veronese 2 (j=1) → (θ+1)²
  1 · 4 · 12, 4 · 32, 32 … Σ 0, 1, 4, 16, 64, 249, 940, 3460, …
  (−∞, 0]nonnegative
• A178821 → basis A013611 → basis A007318
  1 · 30, 20 · 540, 720, 240 · 7560, 15120, 10080, 2240 … Σ 1, 50, 1500, 35000, 700000, 12600000, …
  (−∞, −1]nonnegativerational roots
• A172431 → basis A013609 → x ↦ x/(x+1)
  1 · 3, 7 · 8, 36, 40 · 21, 139, 303, 217 … Σ 1, 10, 84, 680, 5456, 43680, 349504, …
  [−1, 0]nonnegative
• theta_plus_one of 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, 4, 3 · 1, 12, 30, 24, 5 … Σ 1, 8, 72, 768, 9600, 138240, 2257920, …
  (−∞, 0]nonnegative
• basis-map coefficients of A132812 → D
  2 · 3, 6 · -12, 24, 12 · -20, -40, 90, 20 … Σ 0, 2, 9, 24, 50, 90, 147, 224, …
  (−∞, ∞)
• A038333 → basis A013610 → θ+1
  361, 1596, 1323 · 6859, 45486, 75411, 37044 … Σ 3280, 164800, 7936000, 371200000, …
  (−∞, 0]nonnegativerational roots
• A056242 → reverse → D
  1 · 5, 2 · 16, 18, 3 · 44, 82, 42, 4 … Σ 0, 1, 7, 37, 172, 744, 3076, 12332, …
  (−∞, 0]nonnegative
• A159854 → basis A013611 → x ↦ x/(x+1)
  1, 10, 25 · 2, 26, 110, 150 · 4, 64, 376, 960, 900 … Σ 36, 288, 2304, 18432, 147456, 1179648, …
  [−1, 0]nonnegativerational roots
• A128100 → D → D+1
  1 · 2 · 7, 2 · 16, 6 … Σ 0, 0, 1, 2, 9, 22, 65, 162, …
  (−∞, 0]nonnegative
• A103406 → Veronese 2 → θ+1
  1 · 1 · 1, 6 · 1, 18 … Σ 1, 1, 7, 19, 52, 136, 344, 848, …
  (−∞, 0]nonnegative
• A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
  1 · 5, 1 · 25, 10, 1 · 125, 75, 15, 1 … Σ 1, 6, 36, 216, 1296, 7776, 46656, …
  (−∞, −1]nonnegativerational roots
• A091044 → θ+1
  1 · 2, 4 · 3, 20, 9 · 4, 56, 84, 16 … Σ 1, 6, 32, 160, 768, 3584, 16384, 73728, …
  (−∞, 0]nonnegative
• A086645 → D → D
  2 · 30, 6 · 140, 168, 12 · 420, 1260, 540, 20 … Σ 0, 0, 2, 36, 320, 2240, 13824, 78848, …
  (−∞, 0]nonnegative
• A122542 → D → D+1
  1 · 4, 2 · 10, 14, 3 · 18, 52, 30, 4 … Σ 0, 1, 6, 27, 104, 365, 1206, 3815, …
  (−∞, 0]nonnegative
• A119900 → x ↦ x+1
  1 · 2, 2 · 4, 7, 3 · 8, 20, 16, 4 … Σ 1, 4, 14, 48, 164, 560, 1912, 6528, …
  (−∞, −1]nonnegative
• A079621 → (θ+1)²
  1 · 4, 4 · 24, 48, 9 · 192, 576, 216, 16 … Σ 1, 8, 81, 1000, 14585, 245376, 4675993, …
  (−∞, 0]nonnegative
• A082985 → x ↦ x+1 → θ+1
  1 · 4, 6 · 11, 30, 15 · 29, 112, 105, 28 … Σ 1, 10, 56, 274, 1276, 5798, 25936, …
  (−∞, 0]nonnegative
• A257625 → x ↦ x+1
  1 · 6, 3 · 72, 72, 9 · 1296, 1944, 702, 27 … Σ 1, 9, 153, 3969, 137457, 5949369, …
  (−∞, −1]nonnegative
• A152828 → basis A007318 → D
  1 · 1 · 1 · 4, 2 … Σ 0, 0, 0, 1, 1, 1, 6, 6, …
  (−∞, 0]nonnegativerational roots
• A236076 → D+1
  1 · 2, 2 · 1, 7, 3 · 0, 6, 18, 5 … Σ 1, 4, 11, 29, 71, 169, 391, 889, …
  (−∞, 0]nonnegative
• A272867 → x ↦ x+1 → x ↦ x+1
  1 · 13, 8, 1 · 169, 208, 90, 16, 1 … Σ 1, 22, 484, 10648, 234256, 5153632, …
  (−∞, −1]nonnegative
• A139548 → D+1
  1 · 6, 8, 2 · 20, 64, 72, 32, 4 … Σ 1, 16, 192, 2048, 20480, 196608, …
  (−∞, 0]nonnegativerational roots
• A137320 → basis A013610 → basis A007318
  1 · 8, 6 · 72, 102, 36 · 720, 1452, 972, 216 … Σ 1, 14, 210, 3360, 57120, 1028160, …
  (−∞, −1]nonnegativerational roots
• A075502 → D
  21, 2 · 343, 84, 3 · 5145, 2450, 210, 4 … Σ 23, 430, 7809, 146151, 2892574, …
  (−∞, 0]nonnegative
• A145596 → D
  2 · 8, 6 · 20, 40, 12 · 40, 150, 120, 20 … Σ 0, 2, 14, 72, 330, 1430, 6006, 24752, …
  (−∞, 0]nonnegative
• A055356 → θ+1
  1 · 1 · 1, 2 · 1, 8, 6 … Σ 1, 1, 3, 15, 101, 851, 8603, 101423, …
  (−∞, 0]nonnegative
• A178820 → Veronese 2 (j=1) → D+1
  4 · 20 · 80, 20 · 280, 140 … Σ 0, 4, 20, 100, 420, 1568, 5376, 17280, …
  (−∞, 0]nonnegative
• A118884: Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0011 (n,k>=0).
  A318772 → x ↦ (1/48)x−1/48
  1 · 2 · 4 · 8 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, 1]nonnegative
• A038301 → basis A013611 → θ+1
  400, 3520, 5808 · 8000, 105600, 348480, 340736 … Σ 9728, 802816, 62914560, 4764729344, …
  (−∞, 0]nonnegativerational roots
• A028338 → basis A013609 → D+1
  1 · 4, 2 · 20, 20, 4 · 136, 184, 72, 8 … Σ 1, 6, 44, 400, 4384, 56448, 836352, …
  (−∞, 0]nonnegative
• A386789 → (θ+1)²
  1 · 0, 8 · 0, 12, 54 · 0, 16, 180, 320 … Σ 1, 8, 66, 516, 3855, 27828, 195664, …
  (−∞, 0]nonnegative
• A075497 → D+1 → x ↦ x/(x+1)
  1 · 3, 4 · 10, 28, 19 · 36, 160, 227, 104 … Σ 1, 7, 57, 527, 5425, 61463, 760809, …
  [−1, 0]nonnegative
• A391533 → basis A013609 → reverse
  4, 12, 9 · 8, 32, 42, 18 · 16, 80, 148, 120, 36 … Σ 25, 100, 400, 1600, 6400, 25600, …
  (−∞, 0]nonnegativerational roots
• A030528 → basis A013611 → D+1
  1 · 6, 4 · 19, 48, 16 · 45, 232, 288, 64 … Σ 1, 10, 83, 629, 4520, 31365, 212325, …
  (−∞, 0]nonnegative
• A225471 → basis A008277 → θ+1
  1 · 4, 2 · 32, 26, 3 · 384, 402, 81, 4 … Σ 1, 6, 61, 871, 15996, 358891, 9509641, …
  (−∞, 0]nonnegative
• A071951 → x ↦ x+1
  1 · 3, 1 · 13, 10, 1 · 81, 95, 23, 1 … Σ 1, 4, 24, 200, 2160, 29088, 474688, …
  (−∞, −1]nonnegative
• A286718 → D
  1 · 5, 2 · 39, 24, 3 · 418, 318, 66, 4 … Σ 0, 1, 7, 66, 806, 12164, 219108, …
  (−∞, 0]nonnegative
• Hadamard-factor parent of A155495 → D
  6, 2 · 15, 30, 3 · 28, 140, 84, 4 … Σ 8, 48, 256, 1280, 6144, 28672, 131072, …
  (−∞, 0]nonnegative
• A003506 → x ↦ x+1
  1 · 4, 2 · 12, 12, 3 · 32, 48, 24, 4 … Σ 0, 1, 6, 27, 108, 405, 1458, 5103, …
  (−∞, −1]nonnegativerational roots
• A391403 → x ↦ x+1 → x ↦ x+1
  1 · 1 · 5, 2 · 13, 6 … Σ 1, 1, 7, 19, 91, 331, 1441, 5797, …
  (−∞, −1]nonnegative
• A027465 → basis A013611 → (θ+1)²
  1 · 4, 16 · 16, 128, 144 · 64, 768, 1728, 1024 … Σ 1, 20, 288, 3584, 40960, 442368, …
  (−∞, 0]nonnegative
• A286718 → basis A013611 → D+1
  1 · 6, 4 · 38, 60, 16 · 344, 744, 432, 64 … Σ 1, 10, 114, 1584, 26352, 513216, …
  (−∞, 0]nonnegative
• A120910 → basis A013611 → x ↦ x/(x+1)
  3 · 9, 21 · 27, 126, 147 · 81, 567, 1323, 1029 … Σ 3, 30, 300, 3000, 30000, 300000, …
  [−1, 0]nonnegativerational roots
• A062190 → reverse → D
  1 · 14, 2 · 84, 48, 3 · 336, 432, 108, 4 … Σ 0, 1, 16, 135, 880, 5005, 26208, …
  (−∞, 0]nonnegative
• A145324 → basis A007318 → reverse
  1 · 2, 3 · 6, 17, 12 · 24, 98, 133, 60 … Σ 1, 5, 35, 315, 3465, 45045, 675675, …
  (−∞, 0]nonnegativerational roots
• A062190 → reverse → (θ+1)²
  1 · 6, 4 · 21, 56, 9 · 56, 336, 216, 16 … Σ 1, 10, 86, 624, 4015, 23738, 131950, …
  (−∞, 0]nonnegative
• A091044 → basis A013609 → x ↦ x+1
  1 · 8, 4 · 60, 56, 12 · 448, 608, 256, 32 … Σ 1, 12, 128, 1344, 14080, 147456, …
  (−∞, −1]nonnegative
• A113413 → basis A013609 → x ↦ x+1
  1 · 5, 2 · 23, 20, 4 · 107, 142, 60, 8 … Σ 1, 7, 47, 317, 2137, 14407, 97127, …
  (−∞, −1]nonnegative
• A099040 → x ↦ x−4
  1 · -8, 2 · 56, -30, 4 · -384, 320, -88, 8 … Σ 1, -6, 30, -144, 684, -3240, 15336, …
  (2, 4]