Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A109692 → basis A013610 → D+1
  1 · 5, 3 · 38, 84, 27 · 348, 1524, 1827, 405 … Σ 1, 8, 149, 4104, 147681, 6532992, …
  (−∞, 0]nonnegative
• A030528 → (θ+1)²
  1 · 1, 4 · 0, 8, 9 · 0, 4, 27, 16 … Σ 1, 5, 17, 47, 116, 266, 579, 1213, …
  (−∞, 0]nonnegative
• A352880 → reverse → x ↦ x/(x+1)
  8 · 18, 30 · 32, 128, 112 · 50, 350, 700, 420 … Σ 8, 48, 272, 1520, 8472, 47264, 264224, …
  [−1, 0]nonnegative
• A084534 → reverse
  1 · 2, 1 · 2, 4, 1 · 2, 9, 6, 1 … Σ 1, 3, 7, 18, 47, 123, 322, 843, …
  (−∞, 0]nonnegative
• A094262 → D
  2, 2 · 6, 24, 30, 12 · 14, 122, 372, 524, 350, 90 … Σ 0, 4, 72, 1472, 36480, 1078528, …
  (−∞, 0]nonnegative
• A152198 → x ↦ x+1 → (θ+1)²
  1 · 1 · 2, 4 · 2, 4 … Σ 1, 1, 6, 6, 29, 29, 126, 126, …
  (−∞, 0]nonnegative
• A086645 → basis A013609 → x ↦ x/(x+1)
  1 · 2, 4 · 8, 32, 28 · 32, 192, 360, 208 … Σ 1, 6, 68, 792, 9232, 107616, 1254464, …
  [−1, 0]nonnegative
• A038338 → basis A013610 → D
  1728, 2592 · 62208, 186624, 139968 … Σ 4320, 388800, 31104000, 2332800000, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A026729 → basis A013610 → (θ+1)²
  1 · 1, 12 · 1, 24, 81 · 1, 36, 243, 432 … Σ 1, 13, 106, 712, 4288, 24064, 128512, …
  (−∞, 0]nonnegative
• A304236 → basis A013610 → D+1
  1 · 1 · 13, 9 · 25, 18 … Σ 1, 1, 22, 43, 424, 1165, 7882, 26191, …
  (−∞, 0]nonnegative
• A081577 → basis A013610 → D
  3 · 18, 18 · 72, 180, 81 · 264, 1044, 1134, 324 … Σ 0, 3, 36, 333, 2766, 21591, 162072, …
  (−∞, 0]nonnegative
• A111006 → θ+1
  1 · 0, 2 · 0, 2, 6 · 0, 0, 6, 12 … Σ 1, 2, 8, 18, 48, 110, 260, 586, …
  (−∞, 0]nonnegative
• A103327 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 4, 5 · 16, 44, 29 · 64, 272, 376, 169 … Σ 1, 9, 89, 881, 8721, 86329, 854569, …
  [−1, 0]nonnegative
• A103327 → basis A013610 → D
  3 · 36, 18 · 240, 432, 81 · 1344, 4320, 3240, 324 … Σ 0, 3, 54, 753, 9228, 105183, 1145922, …
  (−∞, 0]nonnegative
• A123519 → reverse → x ↦ x+1
  11, 8, 1 · 41, 47, 15, 1 · 153, 240, 126, 24, 1 … Σ 20, 104, 544, 2848, 14912, 78080, …
  (−∞, −1]nonnegative
• A013612 → x ↦ x+1 → reverse
  25, 60, 36 · 125, 450, 540, 216 … Σ 121, 1331, 14641, 161051, 1771561, …
  (−∞, 0]nonnegativerational roots
• A039756 → D
  1 · 4, 2 · 9, 26, 3 · 16, 116, 120, 4 … Σ 0, 1, 6, 38, 256, 1844, 14296, 119160, …
  (−∞, 0]nonnegative
• A085881 → basis A007318 → D+1
  1 · 3, 1 · 24, 18, 3 · 300, 360, 135, 15 … Σ 1, 4, 45, 810, 19845, 612360, 22733865, …
  (−∞, 0]nonnegativerational roots
• A391045 → reverse
  1 · 1, 1 · 1, 4, 1 · 1, 10, 9, 1 … Σ 1, 2, 6, 21, 80, 322, 1347, 5798, …
  (−∞, 0]nonnegative
• A281000 → reverse
  1 · 1, 3 · 1, 10, 10 · 1, 21, 70, 35 … Σ 1, 4, 21, 127, 835, 5798, 41835, …
  (−∞, 0]nonnegative
• A038326 → x ↦ x+1 → (θ+1)²
  1 · 23, 48 · 529, 2208, 1296 … Σ 1, 71, 4033, 205415, 9791425, …
  (−∞, 0]nonnegative
• A142460 → θ+1
  1 · 1, 2 · 1, 24, 3 · 1, 166, 249, 4 … Σ 1, 3, 28, 420, 8568, 219912, 6785856, …
  (−∞, 0]nonnegative
• A299989 → (θ+1)²
  0, 4 · 0, 12, 36, 16 · 0, 36, 216, 352, 200, 36 … Σ 4, 64, 840, 9920, 109568, 1155072, …
  (−∞, 0]nonnegative
• A126216 → D → D+1
  1 · 7, 2 · 39, 24, 3 · 196, 196, 54, 4 … Σ 0, 1, 9, 66, 450, 2955, 18963, 119812, …
  (−∞, 0]nonnegative
• A156289 → D+1
  1 · 4, 3 · 16, 45, 15 · 64, 483, 525, 105 … Σ 1, 7, 76, 1177, 24451, 650992, …
  (−∞, 0]nonnegative
• A249250 → reverse → x ↦ x+1
  1 · 4, 1 · 10, 8, 1 · 28, 36, 12, 1 … Σ 1, 5, 19, 77, 307, 1229, 4915, 19661, …
  (−∞, −1]nonnegative
• A285061 → x ↦ x+1
  1 · 5, 4 · 41, 56, 16 · 429, 796, 432, 64 … Σ 1, 9, 113, 1721, 30305, 600297, …
  (−∞, −1]nonnegative
• A090582: T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.
  1 · 2, 1 · 6, 6, 1 · 24, 36, 14, 1 … Σ 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  (−∞, −1]nonnegative
• A178821 → basis A007318 → θ+1
  1 · 10, 10 · 60, 120, 45 · 280, 840, 630, 140 … Σ 1, 20, 225, 1890, 13230, 81648, 459270, …
  (−∞, 0]nonnegativerational roots
• A269945 → D
  1 · 1, 2 · 1, 10, 3 · 1, 42, 42, 4 … Σ 0, 1, 3, 14, 89, 737, 7696, 98093, …
  (−∞, 0]nonnegative
• A027466 → basis A013611 → D
  4 · 64, 32 · 768, 768, 192 · 8192, 12288, 6144, 1024 … Σ 0, 4, 96, 1728, 27648, 414720, 5971968, …
  (−∞, 0]nonnegativerational roots
• A130534 → Veronese 2 (j=1) → D
  1 · 10 · 85, 2 · 735, 42 … Σ 0, 0, 0, 1, 10, 87, 777, 7416, …
  (−∞, 0]nonnegative
• A094305 → x ↦ x/(x+1)
  1 · 3, 6 · 6, 24, 24 · 10, 60, 120, 80 … Σ 1, 9, 54, 270, 1215, 5103, 20412, …
  [−1, 0]nonnegativerational roots
• A104684 → D → θ+1
  1 · 6, 4 · 30, 48, 9 · 140, 360, 180, 16 … Σ 0, 1, 10, 87, 696, 5265, 38298, 270655, …
  (−∞, 0]nonnegative
• A055151 → basis A007318 → D+1
  1 · 1 · 3, 1 · 7, 3 … Σ 1, 1, 4, 10, 35, 111, 386, 1324, …
  (−∞, 0]nonnegative
• A094816 → x ↦ x+1 → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 9, 23, 16 … Σ 1, 3, 11, 49, 261, 1631, 11743, 95901, …
  (−∞, 0]nonnegative
• A143491 → x ↦ x/(x+1)
  6, 17, 12 · 24, 98, 133, 60 … Σ 35, 315, 3465, 45045, 675675, 11486475, …
  [−1, 0]nonnegativerational roots
• A124038 → D+1
  1 · -1, 1 · -3, 0, 1 · 0, -6, 1, 1 … Σ 1, 0, -2, -4, -3, 2, 7, 6, …
  (−∞, ∞)
• A108838 → x ↦ x+1
  2 · 5, 2 · 14, 12, 2 · 42, 56, 21, 2 … Σ 2, 7, 28, 121, 550, 2591, 12536, 61921, …
  (−∞, −1]nonnegative
• A038301 → x ↦ x+1 → x ↦ x+1
  961, 682, 121 · 29791, 31713, 11253, 1331 … Σ 1764, 74088, 3111696, 130691232, …
  (−∞, −1]nonnegativerational roots
• A132774 → Veronese 2 (j=1) → x ↦ x+1
  3 · 4 · 7, 7 · 8, 8 … Σ 0, 3, 4, 14, 16, 44, 48, 120, …
  (−∞, −1]nonnegativerational roots
• A159854 → θ+1
  0, 0, 3 · 0, 0, 3, 4 · 0, 0, 3, 8, 5 … Σ 3, 7, 16, 36, 80, 176, 384, 832, …
  (−∞, 0]nonnegativerational roots
• A110441 → basis A007318 → reverse
  1 · 1, 4 · 1, 8, 14 · 1, 12, 44, 48 … Σ 1, 5, 23, 105, 479, 2185, 9967, 45465, …
  (−1, 0]nonnegative
• A086645 → basis A007318 → θ+1
  1 · 2, 2 · 8, 16, 3 · 32, 96, 54, 4 … Σ 1, 4, 27, 186, 1253, 8288, 54047, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → basis A008277 → x ↦ x/(x+1)
  1, 5, 5 · 1, 10, 23, 15 · 1, 19, 76, 109, 52 … Σ 11, 49, 257, 1539, 10299, 75905, …
  [−1, 0]nonnegative
• A110441 → basis A013609 → reverse
  1 · 2, 4 · 4, 16, 14 · 8, 48, 88, 48 … Σ 1, 6, 34, 192, 1084, 6120, 34552, …
  (−∞, 0]nonnegative
• basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
  1 · 0, 2 · -3, 3, 3 · 0, -12, 12, 4 … Σ 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 1]
• A142461: Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = …
  1 · 1, 1 · 1, 14, 1 · 1, 111, 111, 1 … Σ 1, 2, 16, 224, 4480, 116480, 3727360, …
  (−∞, 0)nonnegative
• A207824 → basis A013610 → D+1
  1 · 9, 3 · 71, 54, 9 · 522, 642, 243, 27 … Σ 1, 12, 134, 1434, 14905, 151668, …
  (−∞, 0]nonnegative
• A144436 → D+1
  1 · 2, 1 · 9, 10, 1 · 24, 69, 26, 1 … Σ 1, 3, 20, 120, 840, 6552, 56896, …
  (−∞, 0]nonnegative