Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A257624: 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 + 5.
  1 · 5, 5 · 25, 80, 25 · 125, 915, 915, 125 … Σ 1, 10, 130, 2080, 39520, 869440, …
  (−∞, 0)nonnegative
• A257608 → (θ+1)²
  1 · 1, 4 · 1, 80, 9 · 1, 876, 1971, 16 … Σ 1, 5, 90, 2864, 119284, 6149880, …
  (−∞, 0]nonnegative
• A048993 → D+1
  1 · 1, 1 · 1, 3, 1 · 1, 7, 6, 1 … Σ 1, 2, 5, 15, 52, 203, 877, 4140, …
  (−∞, 0]nonnegative
• A272866 → D → D+1
  5, 2 · 28, 58, 30, 4 · 69, 330, 495, 300, 75, 6 … Σ 0, 7, 120, 1275, 11000, 84375, 600000, …
  (−∞, 0]nonnegative
• A081578 → basis A013610 → θ+1
  1 · 2, 6 · 7, 42, 27 · 20, 180, 324, 108 … Σ 1, 8, 76, 632, 5140, 40496, 313036, …
  (−∞, 0]nonnegative
• A166343 → D+1
  13, 14, 1 · 28, 81, 30, 1 · 59, 382, 336, 62, 1 … Σ 28, 140, 840, 5880, 47040, 423360, …
  (−∞, 0]nonnegative
• A081580 → γ → (θ+1)²
  1 · 1 · 1, 20 · 1, 40 … Σ 1, 1, 21, 41, 286, 756, 3451, 10371, …
  (−∞, 0]nonnegative
• A128100 → D → D
  2 · 6 · 18, 6 · 44, 24 … Σ 0, 0, 0, 0, 2, 6, 24, 68, …
  (−∞, 0]nonnegative
• A286724 → x ↦ x+1
  1 · 3, 1 · 17, 10, 1 · 139, 111, 21, 1 … Σ 1, 4, 28, 272, 3344, 49472, 852928, …
  (−∞, −1]nonnegative
• A093375 → Veronese 2 (j=1) → θ+1
  1 · 4 · 9, 2 · 16, 16 … Σ 0, 1, 4, 11, 32, 88, 232, 592, …
  (−∞, 0]nonnegative
• A028338 → basis A013610 → D+1
  1 · 5, 3 · 26, 36, 9 · 180, 348, 189, 27 … Σ 1, 8, 71, 744, 9129, 129072, 2071215, …
  (−∞, 0]nonnegative
• A207823 → reverse → D+1
  1 · 5, 4 · 9, 38, 15 · 13, 104, 214, 56 … Σ 1, 9, 62, 387, 2285, 13012, 72233, …
  (−∞, 0]nonnegative
• A132393 → θ+1 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 2, 5 · 0, 4, 17, 17 … Σ 1, 2, 7, 38, 281, 2634, 29919, 399342, …
  [−1, 0]nonnegative
• A049029 → (θ+1)²
  1 · 5, 4 · 45, 60, 9 · 585, 1020, 270, 16 … Σ 1, 9, 114, 1891, 38895, 955686, …
  (−∞, 0]nonnegative
• narayana → θ+1 → reverse
  1 · 2, 1 · 3, 6, 1 · 4, 18, 12, 1 … Σ 1, 3, 10, 35, 126, 462, 1716, 6435, …
  (−∞, 0]nonnegative
• A038303 → x ↦ x/(x+1)
  1 · 10, 11 · 100, 220, 121 · 1000, 3300, 3630, 1331 … Σ 1, 21, 441, 9261, 194481, 4084101, …
  [−1, 0]nonnegativerational roots
• A055151 → D → x ↦ x+1
  1 · 3 · 10, 4 · 30, 20 … Σ 0, 0, 1, 3, 14, 50, 195, 721, …
  (−∞, −1]nonnegative
• A008278 → x ↦ x+1 → x ↦ x+1
  1 · 3, 1 · 11, 7, 1 · 49, 46, 13, 1 … Σ 1, 4, 19, 109, 742, 5815, 51193, …
  (−∞, −1]nonnegative
• A274498 → D
  6 · 18 · 36, 72 · 108, 216 … Σ 0, 0, 6, 18, 108, 324, 1458, 4374, …
  (−∞, 0]nonnegativerational roots
• A088996 → (θ+1)²
  1 · 0, 4 · 0, 4, 18 · 0, 8, 63, 96 … Σ 1, 4, 22, 167, 1621, 19144, 266078, …
  (−∞, 0]nonnegative
• A034870 → x ↦ x+1 → D+1
  1 · 8, 6, 1 · 48, 80, 48, 12, 1 … Σ 1, 15, 189, 2187, 24057, 255879, …
  (−∞, 0]nonnegativerational roots
• A008292 → γ (scale 1/4) (shift +1) → x ↦ x+1 → x ↦ x+1
  1 · 2 · 8, 2 · 40, 16 … Σ 1, 2, 10, 56, 424, 3728, 38800, 458624, …
  (−∞, −1]nonnegative
• A027466 → basis A013610 → D+1
  1 · 11, 3 · 112, 66, 9 · 1088, 1008, 297, 27 … Σ 1, 14, 187, 2420, 30613, 380666, …
  (−∞, 0]nonnegativerational roots
• A086645 → basis A007318 → basis A007318
  1 · 3, 1 · 17, 10, 1 · 99, 87, 21, 1 … Σ 1, 4, 28, 208, 1552, 11584, 86464, …
  (−∞, −1]nonnegative
• A122542 → basis A013609 → x ↦ x/(x+1)
  1 · 1, 3 · 3, 14, 15 · 7, 47, 101, 69 … Σ 1, 4, 32, 224, 1600, 11392, 81152, …
  [−1, 0]nonnegative
• A048176: Triangle read by rows: T(n, m) = S1(n, m)*10^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).
  1 · -10, 1 · 200, -30, 1 · -6000, 1100, -60, 1 … Σ 1, -9, 171, -4959, 193401, -9476649, …
  [5, ∞)rational roots
• A038259 → basis A013611 → D
  20 · 440, 800 · 7260, 26400, 24000 … Σ 0, 20, 1240, 57660, 2383280, 92352100, …
  (−∞, 0]nonnegativerational roots
• A347823 → x ↦ x/(x+1)
  1 · 2, 5 · 3, 14, 16 · 4, 27, 60, 44 … Σ 1, 7, 33, 135, 513, 1863, 6561, 22599, …
  [−1, 0]nonnegativerational roots
• A132393 → basis A013611 → (θ+1)²
  1 · 1, 16 · 2, 48, 144 · 6, 176, 864, 1024 … Σ 1, 17, 194, 2070, 22504, 256808, …
  (−∞, 0]nonnegative
• basis-map coefficients of A030528, shifted → D → D
  2 · 0, -6 · 0, 0, 12 · 0, 0, 0, -20 … Σ 0, 0, 2, -6, 12, -20, 30, -42, …
  (−∞, 0]rational roots
• A133080 → basis A013609 → θ+1
  1 · 2, 4 · 1, 8, 12 · 2, 20, 48, 32 … Σ 1, 6, 21, 102, 297, 1350, 3645, 16038, …
  (−∞, 0]nonnegative
• A088996 → basis A007318 → D+1
  1 · 2, 1 · 8, 9, 2 · 49, 84, 43, 6 … Σ 1, 3, 19, 182, 2322, 36908, 701356, …
  (−∞, 0]nonnegative
• A060187 → (θ+1)²
  1 · 1, 4 · 1, 24, 9 · 1, 92, 207, 16 … Σ 1, 5, 34, 316, 3616, 48960, 764160, …
  (−∞, 0]nonnegative
• A178822 → Veronese 2 → x ↦ x+1
  1 · 6 · 42, 21 · 224, 168 … Σ 1, 6, 63, 392, 2142, 10332, 45738, …
  (−∞, −1]nonnegative
• A038243 → θ+1
  1 · 5, 2 · 25, 20, 3 · 125, 150, 45, 4 … Σ 1, 7, 48, 324, 2160, 14256, 93312, …
  (−∞, 0]nonnegativerational roots
• A103327 → x ↦ x+1 → θ+1
  1 · 4, 2 · 16, 24, 3 · 64, 160, 72, 4 … Σ 1, 6, 43, 300, 2037, 13554, 88799, …
  (−∞, 0]nonnegative
• A391533: The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
  4, 4, 1 · 4, 8, 5, 1 · 4, 12, 13, 6, 1 … Σ 9, 18, 36, 72, 144, 288, 576, 1152, …
  (−∞, 0)nonnegativerational roots
• A391045 → θ+1 → D
  2 · 8, 6 · 18, 60, 12 · 32, 252, 240, 20 … Σ 0, 2, 14, 90, 544, 3180, 18204, 102718, …
  (−∞, 0]nonnegative
• A052553 → D → D
  2 · 6 · 12, 6 · 20, 24 … Σ 0, 0, 0, 0, 2, 6, 18, 44, …
  (−∞, 0]nonnegative
• A178822 → basis A013611 → D
  24 · 336, 672 · 2688, 10752, 10752 … Σ 0, 24, 1008, 24192, 435456, 6531840, …
  (−∞, 0]nonnegativerational roots
• A038320 → basis A013610 → θ+1
  289, 1224, 972 · 4913, 31212, 49572, 23328 … Σ 2485, 109025, 4587625, 187578125, …
  (−∞, 0]nonnegativerational roots
• A132774: A natural number operator.
  1 · 2, 3 · 0, 4, 5 · 0, 0, 6, 7 … Σ 1, 5, 9, 13, 17, 21, 25, 29, …
  [−1, 0]nonnegativerational roots
• A178822 → x ↦ x+1 → x ↦ x+1
  1 · 18, 6 · 189, 126, 21 · 1512, 1512, 504, 56 … Σ 1, 24, 336, 3584, 32256, 258048, …
  (−∞, −1]nonnegativerational roots
• A144443 → D
  1 · 20, 2 · 159, 318, 3 · 1138, 8508, 3414, 4 … Σ 0, 1, 22, 480, 13064, 429380, 16605768, …
  (−∞, 0]nonnegative
• A367270 → x ↦ x+1 → (θ+1)²
  1 · 1 · 3, 8 · 10, 48, 27 … Σ 1, 1, 11, 85, 609, 4221, 28643, 191353, …
  (−∞, 0]nonnegative
• A034870 → x ↦ x+1 → (θ+1)²
  1 · 4, 16, 9 · 16, 128, 216, 128, 25 … Σ 1, 29, 513, 7533, 99873, 1240029, …
  (−∞, 0]nonnegative
• A026374 → D+1 → D
  1 · 5, 2 · 12, 14, 3 · 28, 58, 30, 4 … Σ 0, 1, 7, 29, 120, 365, 1275, 3425, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A013610 → D
  6, 18 · 12, 108, 324, 324 … Σ 0, 24, 768, 18432, 393216, 7864320, …
  (−∞, 0]nonnegativerational roots
• A116414 → basis A007318 → D
  1 · 10, 2 · 69, 30, 3 · 410, 282, 60, 4 … Σ 0, 1, 12, 102, 756, 5211, 34344, …
  (−∞, 0]nonnegative
• A103406 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 6, 8 · 1, 9, 24, 20 … Σ 1, 4, 15, 54, 189, 648, 2187, 7290, …
  [−1, 0]nonnegativerational roots