Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A052553 → D → (θ+1)²
  1 · 2 · 3, 8 · 4, 24 … Σ 0, 0, 1, 2, 11, 28, 80, 194, …
  (−∞, 0]nonnegative
• A030528 → x ↦ x−1
  1 · 0, 1 · -1, 0, 1 · 1, -2, 0, 1 … Σ 1, 1, 0, 0, 0, 0, 0, 0, …
  (−3, 1]
• A038323 → basis A013610 → reverse
  1 · 27, 20 · 729, 1080, 400 … Σ 1, 47, 2209, 103823, 4879681, …
  (−∞, 0]nonnegativerational roots
• A128813 → basis A013610 → D+1
  1 · 5, 3 · 38, 84, 27 · 410, 1812, 2187, 486 … Σ 1, 8, 149, 4895, 249445, 18214270, …
  (−∞, 0]nonnegative
• A115951 → D+1
  1 · 2, 2 · 2, 14, 6 · 0, 24, 72, 20 … Σ 1, 4, 22, 116, 608, 3152, 16208, 82784, …
  (−∞, 0]nonnegative
• A112493 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 8 · 1, 14, 50, 52 … Σ 1, 3, 15, 117, 1269, 17667, 300375, …
  [−1, 0)nonnegative
• A131222 → θ+1
  1 · 0, 2 · 0, 6, 3 · 0, 28, 27, 4 … Σ 1, 2, 9, 59, 506, 5354, 67344, 981336, …
  (−∞, 0]nonnegative
• A272867 → D+1
  1 · 5, 6, 1 · 9, 44, 42, 12, 1 … Σ 1, 12, 108, 864, 6480, 46656, 326592, …
  (−∞, 0]nonnegative
• A249248 → reverse → basis A007318
  1 · 4, 1 · 11, 9, 1 · 38, 49, 15, 1 … Σ 1, 5, 21, 103, 519, 2793, 15645, 91623, …
  (−∞, −1]nonnegative
• A062190 → θ+1
  1 · 1, 12 · 1, 28, 63 · 1, 48, 252, 224 … Σ 1, 13, 92, 525, 2695, 13013, 60424, …
  (−∞, 0]nonnegative
• A080248 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 16, 39, 25 … Σ 1, 3, 13, 81, 669, 6955, 88505, …
  [−1, 0]nonnegative
• A013622 → basis A013609 → θ+1
  1 · 8, 20 · 64, 320, 300 · 512, 3840, 7200, 4000 … Σ 1, 28, 684, 15552, 338256, 7138368, …
  (−∞, 0]nonnegativerational roots
• A038221 → basis A013609 → D+1
  1 · 12, 6 · 108, 144, 36 · 864, 1944, 1296, 216 … Σ 1, 18, 288, 4320, 62208, 870912, …
  (−∞, 0]nonnegativerational roots
• A055151 → basis A007318 → x ↦ x+1
  1 · 1 · 3, 1 · 7, 3 … Σ 1, 1, 4, 10, 37, 121, 451, 1639, …
  (−∞, −1]nonnegative
• A116666 → Veronese 2 (j=1) → D+1
  3 · 6 · 16, 7 · 40, 28 … Σ 0, 3, 6, 23, 68, 188, 496, 1264, …
  (−∞, 0]nonnegative
• A173018 → θ+1
  1 · 1 · 1, 2 · 1, 8, 3 … Σ 1, 1, 3, 12, 60, 360, 2520, 20160, …
  (−∞, 0]nonnegative
• A056242: Triangle read by rows: T(n,k) = number of k-part order-consecutive partition of {1,2,...,n} (1 <= k <= n).
  1 · 1, 2 · 1, 5, 4 · 1, 9, 16, 8 … Σ 1, 3, 10, 34, 116, 396, 1352, 4616, …
  [−1, 0]nonnegative
• A008299 → x ↦ x+1
  1 · 1 · 4, 3 · 11, 10 … Σ 1, 1, 7, 21, 111, 533, 3039, 18245, …
  (−∞, −1]nonnegative
• type_b_eulerian → γ (scale 1/4) → D → x ↦ x+1
  1 · 5 · 28, 10 · 180, 122 … Σ 0, 0, 1, 5, 38, 302, 2827, 29607, …
  (−∞, −1]nonnegative
• A046716 → basis A013609 → x ↦ x+1
  1 · 4, 2 · 19, 18, 4 · 118, 162, 68, 8 … Σ 1, 6, 41, 356, 4401, 78826, 1893481, …
  (−∞, −1]nonnegative
• A039756 → reverse → D
  1 · 4, 2 · 13, 18, 3 · 40, 116, 48, 4 … Σ 0, 1, 6, 34, 208, 1396, 10232, 81320, …
  (−∞, 0]nonnegative
• A207607 → basis A007318 → x ↦ x+1
  1 · 5, 2 · 19, 13, 2 · 71, 69, 21, 2 … Σ 1, 7, 34, 163, 781, 3742, 17929, 85903, …
  (−6, −2)nonnegative
• A039756: Triangle of B-analogs of Stirling numbers of 2nd kind.
  1 · 1, 1 · 1, 4, 1 · 1, 9, 13, 1 … Σ 1, 2, 6, 24, 116, 648, 4088, 28640, …
  (−∞, 0)nonnegative
• A352880 → reverse → (θ+1)²
  8 · 18, 48 · 32, 256, 144 · 50, 800, 1350, 320 … Σ 8, 66, 432, 2520, 13680, 70686, 352352, …
  (−∞, 0]nonnegative
• A063007 → basis A013610 → (θ+1)²
  1 · 3, 24 · 13, 216, 486 · 63, 1584, 7290, 8640 … Σ 1, 27, 715, 17577, 409641, 9186939, …
  (−∞, 0]nonnegative
• A172431 → Veronese 2 → θ+1
  1 · 1 · 1, 6 · 1, 20 … Σ 1, 1, 7, 21, 58, 178, 517, 1483, …
  (−∞, 0]nonnegative
• A105070: T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.
  1 · 2 · 3, 2 · 4, 8 … Σ 1, 2, 5, 12, 29, 70, 169, 408, …
  (−∞, 0)nonnegative
• A144438: Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 5, 1 · 1, 14, 14, 1 … Σ 1, 2, 7, 30, 157, 972, 6961, 56660, …
  (−∞, 0)nonnegative
• A086645 → basis A013610 → x ↦ x+1
  1 · 5, 3 · 41, 42, 9 · 365, 549, 243, 27 … Σ 1, 8, 92, 1184, 15632, 207488, 2757056, …
  (−∞, −1]nonnegative
• A038259 → basis A007318 → (θ+1)²
  1 · 11, 20 · 121, 440, 225 · 1331, 7260, 7425, 2000 … Σ 1, 31, 786, 18016, 388096, 8011776, …
  (−∞, 0]nonnegative
• A120910 → D → x ↦ x+1
  3 · 18, 6 · 81, 54, 9 · 324, 324, 108, 12 … Σ 0, 3, 24, 144, 768, 3840, 18432, 86016, …
  (−∞, −1]nonnegativerational roots
• A091042 → basis A007318
  1 · 4, 3 · 16, 20, 5 · 64, 112, 56, 7 … Σ 1, 7, 41, 239, 1393, 8119, 47321, …
  (−∞, −1]nonnegative
• A204420 → basis A013609 → x ↦ x+1
  1 · 3, 2 · 45, 48, 12 · 1575, 2130, 900, 120 … Σ 1, 5, 105, 4725, 363825, 42567525, …
  (−∞, −1]nonnegativerational roots
• A008955 → basis A013610 → D+1
  1 · 5, 3 · 49, 111, 36 · 760, 3486, 4329, 972 … Σ 1, 8, 196, 9547, 771515, 93254890, …
  (−∞, 0]nonnegative
• Veronese parent of A109447 → θ+1
  0, 2, 3 · 1, 4, 3 · 0, 2, 9, 12, 5 · 1, 8, 18, 16, 5 … Σ 5, 8, 28, 48, 144, 256, 704, 1280, …
  (−∞, 0]nonnegativerational roots
• A026729 → basis A013611 → D+1
  1 · 5, 4 · 14, 44, 16 · 31, 188, 272, 64 … Σ 1, 9, 74, 555, 3965, 27400, 184925, …
  (−∞, 0]nonnegative
• A081581 → D
  1 · 8, 2 · 15, 30, 3 · 22, 156, 66, 4 … Σ 0, 1, 10, 48, 248, 1100, 4872, 20608, …
  (−∞, 0]nonnegative
• A159854 → basis A013609 → basis A007318
  9, 12, 4 · 36, 66, 40, 8 · 144, 336, 292, 112, 16 … Σ 25, 150, 900, 5400, 32400, 194400, …
  (−∞, −1]nonnegativerational roots
• A102547 → reverse → D
  1 · 0, 2 · 0, 0, 3 · 0, 0, 3, 4 … Σ 1, 2, 3, 7, 13, 21, 36, 61, …
  (−∞, 0]nonnegative
• A081577 → x ↦ x+1 → D+1
  1 · 3, 1 · 12, 8, 1 · 40, 44, 13, 1 … Σ 1, 4, 21, 98, 455, 2064, 9241, 40894, …
  (−∞, 0]nonnegative
• A053117 → x ↦ x+1 → D+1
  1 · 4, 2 · 11, 16, 4 · 24, 68, 48, 8 … Σ 1, 6, 31, 148, 673, 2962, 12735, 53800, …
  (−∞, 0]nonnegative
• A059364 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 6 · 15, 63, 66 · 105, 681, 1470, 1056 … Σ 1, 9, 144, 3312, 99360, 3676320, …
  [−1, 0]nonnegativerational roots
• A178822 → Veronese 2 → D
  21 · 168 · 756, 252 · 2520, 2520 … Σ 0, 0, 21, 168, 1008, 5040, 22176, …
  (−∞, 0]nonnegative
• A008277 → θ+2 (shift +1) → x ↦ x/(x+1)
  2 · 2, 5 · 2, 13, 15 · 2, 27, 72, 52 … Σ 2, 7, 30, 153, 898, 5919, 43102, …
  [−1, 0]nonnegative
• A046716 → x ↦ x+1 → x ↦ x+1
  1 · 3, 1 · 11, 7, 1 · 53, 50, 14, 1 … Σ 1, 4, 19, 118, 1021, 12088, 183727, …
  (−∞, −1]nonnegative
• A137320 → basis A013610 → reverse
  1 · 6, 2 · 36, 30, 6 · 216, 324, 156, 24 … Σ 1, 8, 72, 720, 7920, 95040, 1235520, …
  (−∞, 0]nonnegativerational roots
• A059364 → basis A013611 → (θ+1)²
  1 · 3, 16 · 15, 176, 288 · 105, 1952, 6768, 6144 … Σ 1, 19, 479, 14969, 557825, 24136635, …
  (−∞, 0]nonnegative
• narayana → D
  1 · 3, 2 · 6, 12, 3 · 10, 40, 30, 4 … Σ 0, 1, 5, 21, 84, 330, 1287, 5005, …
  (−∞, 0]nonnegative
• A075497 → D
  1 · 6, 2 · 28, 24, 3 · 120, 200, 60, 4 … Σ 0, 1, 8, 55, 384, 2841, 22504, 190863, …
  (−∞, 0]nonnegative
• A099093 → x ↦ x+1 → D
  3 · 21, 18 · 117, 198, 81 · 585, 1476, 1215, 324 … Σ 0, 3, 39, 396, 3600, 30780, 253044, …
  (−∞, 0]nonnegative