Proof methods / preserver_image

Image under a real-rootedness preserver

1,387 sequences

• A081578: Pascal-(1,3,1) array.
  1 · 1, 1 · 1, 5, 1 · 1, 9, 9, 1 … Σ 1, 2, 7, 20, 61, 182, 547, 1640, …
  (−∞, 0)nonnegative
• A178821 → (θ+1)²
  1 · 5, 20 · 15, 120, 135 · 35, 420, 945, 560 … Σ 1, 25, 270, 1960, 11200, 54432, 235200, …
  (−∞, 0]nonnegative
• A019538: Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
  1 · 1, 2 · 1, 6, 6 · 1, 14, 36, 24 … Σ 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  (−1, 0)nonnegative
• A249248 → D+1
  1 · 4, 3 · 8, 13, 3 · 13, 56, 31, 3 … Σ 1, 7, 24, 103, 419, 1815, 7970, 36277, …
  (−∞, 0]nonnegative
• A094816 → D+1
  1 · 2, 1 · 4, 5, 1 · 9, 20, 9, 1 … Σ 1, 3, 10, 39, 181, 995, 6386, 47075, …
  (−∞, 0]nonnegative
• A039756 → reverse
  1 · 1, 1 · 1, 4, 1 · 1, 13, 9, 1 … Σ 1, 2, 6, 24, 116, 648, 4088, 28640, …
  (−∞, 0]nonnegative
• A008277 → D
  1 · 3, 2 · 7, 12, 3 · 15, 50, 30, 4 … Σ 0, 1, 5, 22, 99, 471, 2386, 12867, …
  (−∞, 0]nonnegative
• A108838 → (θ+1)²
  2 · 3, 8 · 4, 32, 18 · 5, 80, 135, 32 … Σ 2, 11, 54, 252, 1140, 5049, 22022, …
  (−∞, 0]nonnegative
• A113953 → x ↦ x−7
  1 · -7, 1 · 35, -12, 1 · -147, 91, -17, 1 … Σ 1, -6, 24, -72, 144, 0, -1728, 10368, …
  (−1, 7]
• A178821 → θ+1
  1 · 5, 10 · 15, 60, 45 · 35, 210, 315, 140 … Σ 1, 15, 120, 700, 3360, 14112, 53760, …
  (−∞, 0]nonnegativerational roots
• A003506 → γ
  1 · 2 · 3 · 4 … Σ 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 0]nonnegativerational roots
• A103327 → (θ+1)²
  1 · 3, 4 · 5, 40, 9 · 7, 140, 189, 16 … Σ 1, 7, 54, 352, 2080, 11520, 60928, …
  (−∞, 0]nonnegative
• A062190 → D
  6 · 14, 42 · 24, 168, 168 · 36, 432, 1008, 504 … Σ 0, 6, 56, 360, 1980, 10010, 48048, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/4) (shift +1) → D
  2 · 16 · 88, 32 · 416, 544 … Σ 0, 0, 2, 16, 120, 960, 8400, 80640, …
  (−∞, 0]nonnegative
• A165253 → θ+1
  1 · 1 · 1, 2 · 1, 6, 3 … Σ 1, 1, 3, 10, 32, 99, 299, 887, …
  (−∞, 0]nonnegative
• A103327 → x ↦ x/(x+1)
  1 · 3, 4 · 5, 20, 16 · 7, 56, 112, 64 … Σ 1, 7, 41, 239, 1393, 8119, 47321, …
  [−1, 0]nonnegative
• A026729 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 2 · 0, 0, 2, 3 … Σ 1, 1, 3, 5, 11, 21, 43, 85, …
  [−1, 0]nonnegative
• A085478 → D
  1 · 3, 2 · 6, 10, 3 · 10, 30, 21, 4 … Σ 0, 0, 1, 5, 19, 65, 210, 654, …
  (−∞, 0]nonnegative
• basis-map coefficients of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
  1 · -3, 3 · 6, -12, 6 · -10, 30, -30, 10 … Σ 1, 0, 0, 0, 0, 0, 0, 0, …
  (−∞, 1][0, ∞)rational roots
• A114656 → (θ+1)²
  1 · 2, 4 · 4, 24, 9 · 8, 96, 108, 16 … Σ 1, 6, 37, 228, 1401, 8578, 52333, …
  (−∞, 0]nonnegative
• A100861 → D+1
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 22, 66, 226, 778, …
  (−∞, 0]nonnegative
• A099040 → x ↦ x−10
  1 · -20, 2 · 380, -78, 4 · -7200, 2240, -232, 8 … Σ 1, -18, 306, -5184, 87804, -1487160, …
  (8, 10]
• A145596 → (θ+1)²
  1 · 2, 8 · 3, 32, 27 · 4, 80, 180, 64 … Σ 1, 10, 62, 328, 1605, 7502, 34034, …
  (−∞, 0]nonnegative
• A344565: Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
  3 · 15, 15 · 45, 90, 45 · 105, 315, 315, 105 … Σ 3, 30, 180, 840, 3360, 12096, 40320, …
  (−∞, 0)nonnegativerational roots
• basis-map coefficients of A112227 → x ↦ x+1
  1, 2, 1 · 1, 3, 3, 1 · 1, 4, 6, 4, 1 … Σ 4, 8, 16, 32, 64, 128, 256, 512, …
  (−∞, −1]nonnegativerational roots
• A207606 → basis A007318
  13, 11, 2 · 34, 44, 17, 2 · 89, 155, 93, 23, 2 … Σ 26, 97, 362, 1351, 5042, 18817, 70226, …
  (−5, −1)nonnegative
• A062145 → (θ+1)²
  1 · 1, 16 · 1, 40, 90 · 1, 72, 405, 320 … Σ 1, 17, 131, 798, 4362, 22407, 110539, …
  (−∞, 0]nonnegative
• basis-map coefficients of A030528, shifted → D
  -1 · 0, 2 · 0, 0, -3 · 0, 0, 0, 4 … Σ 0, -1, 2, -3, 4, -5, 6, -7, …
  (−∞, 0]rational roots
• basis-map coefficients of A030528, shifted → D+1
  1 · -1, -1 · 0, 2, 1 · 0, 0, -3, -1 … Σ 1, -2, 3, -4, 5, -6, 7, -8, …
  (−∞, 0]rational roots
• A114656 → D
  1 · 6, 2 · 24, 24, 3 · 80, 160, 60, 4 … Σ 0, 1, 8, 51, 304, 1765, 10128, 57799, …
  (−∞, 0]nonnegative
• A232535 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 10, 12 · 1, 21, 64, 48 … Σ 1, 4, 23, 134, 781, 4552, 26531, …
  [−1, 0]nonnegative
• A102662 → D+1
  1 · 4, 3 · 6, 11, 3 · 8, 29, 20, 3 … Σ 1, 7, 20, 60, 171, 477, 1306, 3526, …
  (−∞, 0]nonnegative
• A075497 → D+1
  1 · 3, 1 · 10, 8, 1 · 36, 52, 15, 1 … Σ 1, 4, 19, 104, 641, 4380, 32803, …
  (−∞, 0]nonnegative
• delannoy
  1 · 1, 1 · 1, 3, 1 · 1, 5, 5, 1 … Σ 1, 2, 5, 12, 29, 70, 169, 408, …
  [−3−2√2, −3+2√2]nonnegative
• A039991 → x ↦ x+1
  1 · 1 · 1, -2, -1 · 1, -6, -3 … Σ 1, 1, -2, -8, -8, 16, 64, 64, …
  (−∞, ∞)
• A099040 → x ↦ x−8
  1 · -16, 2 · 240, -62, 4 · -3584, 1408, -184, 8 … Σ 1, -14, 182, -2352, 30380, -392392, …
  (6, 8]
• A116414 → D
  1 · 8, 2 · 42, 24, 3 · 184, 174, 48, 4 … Σ 0, 1, 10, 69, 410, 2252, 11790, 59767, …
  (−∞, 0]nonnegative
• A008459 → (θ+1)²
  1 · 1, 4 · 1, 16, 9 · 1, 36, 81, 16 … Σ 1, 5, 26, 134, 670, 3262, 15540, 72732, …
  (−∞, 0]nonnegative
• A108299 → x ↦ x+1
  1 · 0, -1 · -1, -3, -1 · -1, -2, 1, 1 … Σ 1, -1, -5, -1, 19, 23, -53, -145, …
  (−∞, ∞)
• A008278 → D
  1 · 3, 2 · 6, 14, 3 · 10, 50, 45, 4 … Σ 0, 1, 5, 23, 109, 544, 2876, 16113, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → D
  0, 2 · 0, 0, 3 · 0, 0, 0, 4 · 0, 0, 0, 0, 5 … Σ 2, 3, 4, 5, 6, 7, 8, 9, …
  (−∞, 0]nonnegativerational roots
• A103327 → D+1
  1 · 4, 1 · 15, 12, 1 · 42, 77, 24, 1 … Σ 1, 5, 28, 144, 704, 3328, 15360, 69632, …
  (−∞, 0]nonnegative
• A391403 → D+1
  1 · 1 · 3, 2 · 7, 6 … Σ 1, 1, 5, 13, 43, 131, 411, 1275, …
  (−∞, 0]nonnegative
• A123125: Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.
  A173018 → reverse (shift +1)
  1 · 0, 1 · 0, 1, 1 · 0, 1, 4, 1 … Σ 1, 1, 2, 6, 24, 120, 720, 5040, …
  (−∞, 0]nonnegative
• A081578 → (θ+1)²
  1 · 1, 4 · 1, 20, 9 · 1, 36, 81, 16 … Σ 1, 5, 30, 134, 583, 2355, 9196, 34748, …
  (−∞, 0]nonnegative
• A008277 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 14, 18, 4 … Σ 0, 1, 3, 10, 37, 151, 674, 3263, …
  (−∞, 0]nonnegative
• A099093 → x ↦ x−10
  1 · -30, 3 · 870, -177, 9 · -25200, 7740, -792, 27 … Σ 1, -27, 702, -18225, 473121, -12282192, …
  (26/3, 10]
• A127773 → (θ+1)²
  1 · 0, 12 · 0, 0, 54 · 0, 0, 0, 160 … Σ 1, 12, 54, 160, 375, 756, 1372, 2304, …
  (−∞, 0]nonnegativerational roots
• A261711: Triangle read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string 01, where n>=0 and k>=0.
  1 · 4 · 15, 1 · 56, 8 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  (−∞, 0)nonnegative
• A128099: Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= …
  1 · 1 · 1, 2 · 1, 4 … Σ 1, 1, 3, 5, 11, 21, 43, 85, …
  (−∞, 0)nonnegative