Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A089231 → (θ+1)²
  1 · 1, 8 · 1, 24, 54 · 1, 48, 324, 384 … Σ 1, 9, 79, 757, 8001, 92941, 1179319, …
  (−∞, 0]nonnegative
• A038208 → Veronese 2 → basis A007318
  1 · 2 · 8, 4 · 32, 24 … Σ 1, 2, 12, 56, 272, 1312, 6336, 30592, …
  (−∞, −1]nonnegative
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
  4, 4, 1 · 24, 36, 18, 3 · 96, 192, 144, 48, 6 … Σ 0, 0, 9, 81, 486, 2430, 10935, 45927, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A112227 → basis A013611 → D+1
  9, 40, 16 · 13, 108, 240, 64 … Σ 65, 425, 2625, 15625, 90625, 515625, …
  (−∞, 0]nonnegativerational roots
• A038277 → x ↦ x/(x+1) → x ↦ x/(x+1)
  1 · 7, 25 · 49, 350, 625 · 343, 3675, 13125, 15625 … Σ 1, 32, 1024, 32768, 1048576, 33554432, …
  [−1, 0]nonnegativerational roots
• A038221 → basis A013611 → D
  12 · 144, 288 · 1296, 5184, 5184 … Σ 0, 12, 432, 11664, 279936, 6298560, …
  (−∞, 0]nonnegativerational roots
• A059364 → basis A013609 → (θ+1)²
  1 · 3, 8 · 15, 88, 72 · 105, 976, 1692, 768 … Σ 1, 11, 175, 3541, 86461, 2467391, …
  (−∞, 0]nonnegative
• A196347 → basis A013610 → basis A007318
  1 · 5, 3 · 50, 60, 18 · 750, 1350, 810, 162 … Σ 1, 8, 128, 3072, 98304, 3932160, …
  (−∞, −1]nonnegativerational roots
• A026374 → θ+1 → D
  2 · 6, 6 · 8, 24, 12 · 12, 66, 72, 20 … Σ 0, 2, 12, 44, 170, 490, 1650, 4300, …
  (−∞, 0]nonnegative
• A134082: Triangle read by rows, (n-1) zeros followed by (2n, 1).
  1 · 2, 1 · 0, 4, 1 · 0, 0, 6, 1 … Σ 1, 3, 5, 7, 9, 11, 13, 15, …
  (−∞, 0]nonnegativerational roots
• A038324 → basis A007318
  1 · 21, 10 · 441, 420, 100 · 9261, 13230, 6300, 1000 … Σ 1, 31, 961, 29791, 923521, 28629151, …
  (−∞, −1]nonnegativerational roots
• A033282: Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
  1 · 1, 2 · 1, 5, 5 · 1, 9, 21, 14 … Σ 1, 3, 11, 45, 197, 903, 4279, 20793, …
  (−1, 0)nonnegative
• A038336 → basis A013609 → D+1
  1364, 1680, 400 · 39688, 81840, 50400, 8000 … Σ 3444, 179928, 9038736, 441860832, …
  (−∞, 0]nonnegativerational roots
• A128100 → D → θ+1
  1 · 2 · 5, 4 · 10, 12 … Σ 0, 0, 1, 2, 9, 22, 65, 162, …
  (−∞, 0]nonnegative
• A089627 → θ+1
  1 · 1 · 1, 4 · 1, 12 … Σ 1, 1, 5, 13, 43, 131, 411, 1275, …
  (−∞, 0]nonnegative
• A178821 → basis A007318 → reverse
  1 · 5, 10 · 15, 60, 60 · 35, 210, 420, 280 … Σ 1, 15, 135, 945, 5670, 30618, 153090, …
  (−∞, 0]nonnegativerational roots
• A156992 → basis A013611 → x ↦ x+1
  1 · 12, 8 · 216, 288, 96 · 5184, 10368, 6912, 1536 … Σ 1, 20, 600, 24000, 1200000, 72000000, …
  (−∞, −1]nonnegativerational roots
• A249252 → basis A007318 → θ+1
  1 · 5, 8 · 17, 48, 24 · 61, 248, 240, 64 … Σ 1, 13, 89, 613, 4081, 26621, 170889, …
  (−∞, 0]nonnegative
• A178821 → Veronese 2 (j=1) → D+1
  5 · 30 · 140, 35 · 560, 280 … Σ 0, 5, 30, 175, 840, 3528, 13440, 47520, …
  (−∞, 0]nonnegative
• A306364 → D
  3 · 45 · 540, 90 · 6300, 3150 … Σ 0, 0, 3, 45, 630, 9450, 155925, …
  (−∞, 0]nonnegative
• narayana → D → θ+1
  1 · 3, 4 · 6, 24, 9 · 10, 80, 90, 16 … Σ 0, 1, 7, 39, 196, 930, 4257, 19019, …
  (−∞, 0]nonnegative
• A132774 → Veronese 2 → D+1
  1 · 2 · 5, 5 · 6, 6 … Σ 1, 2, 10, 12, 27, 30, 52, 56, …
  (−∞, 0]nonnegativerational roots
• A106246 → basis A007318 → basis A007318
  1 · 4, 1 · 13, 8, 1 · 38, 39, 12, 1 … Σ 1, 5, 22, 90, 351, 1323, 4860, 17496, …
  (−∞, −1]nonnegative
• A166343 → D
  12, 2 · 27, 54, 3 · 58, 324, 174, 4 … Σ 14, 84, 560, 4200, 35280, 329280, …
  (−∞, 0]nonnegative
• A290316 → D
  6 · 48, 180 · 234, 4428, 6480 … Σ 0, 6, 228, 11142, 684744, 50952270, …
  (−∞, 0]nonnegative
• A130534 → basis A008277 → x ↦ x+1
  1 · 3, 1 · 13, 8, 1 · 73, 63, 15, 1 … Σ 1, 4, 22, 152, 1256, 12032, 130768, …
  (−∞, −1]nonnegative
• A038331 → x ↦ x+1 → x ↦ x+1
  484, 220, 25 · 10648, 7260, 1650, 125 … Σ 729, 19683, 531441, 14348907, …
  (−∞, −1]nonnegativerational roots
• A081579 → x ↦ x+1 → reverse
  1 · 1, 2 · 1, 8, 8 · 1, 14, 36, 24 … Σ 1, 3, 17, 75, 361, 1683, 7937, 37275, …
  (−∞, 0]nonnegative
• A054142: Triangular array binomial(2*n-k, k), k=0..n, n >= 0.
  1 · 1, 1 · 1, 3, 1 · 1, 5, 6, 1 … Σ 1, 2, 5, 13, 34, 89, 233, 610, …
  (−∞, 0)nonnegative
• A143495 → D+1 → D+1
  1 · 5, 1 · 25, 11, 1 · 125, 91, 18, 1 … Σ 1, 6, 37, 235, 1540, 10427, 73013, …
  (−∞, 0]nonnegative
• A154233 → D+1
  1 · 2, 1 · 8, 9, 1 · 20, 57, 22, 1 … Σ 1, 3, 18, 100, 753, 5817, 54460, …
  (−∞, 0]nonnegative
• A100861 → basis A013609 → (θ+1)²
  1 · 1 · 2, 8 · 4, 24 … Σ 1, 1, 10, 28, 214, 886, 6436, 33160, …
  (−∞, 0]nonnegative
• A132393 → D+1 → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 2, 32, 54, 16 … Σ 1, 5, 22, 104, 564, 3526, 25178, …
  (−∞, 0]nonnegative
• A028246: Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.
  1 · 1, 1 · 1, 3, 2 · 1, 7, 12, 6 … Σ 1, 2, 6, 26, 150, 1082, 9366, 94586, …
  [−1, 0)nonnegative
• A290319 → θ+1
  1 · 1, 2 · 5, 12, 3 · 45, 118, 45, 4 … Σ 1, 3, 20, 212, 3088, 57264, 1290048, …
  (−∞, 0]nonnegative
• A028338 → basis A013611 → θ+1
  1 · 2, 8 · 8, 48, 48 · 48, 352, 576, 256 … Σ 1, 10, 104, 1232, 16704, 256896, …
  (−∞, 0]nonnegative
• A075501 → x ↦ x+1
  55, 20, 1 · 505, 327, 39, 1 · 5497, 5224, 1086, 64, 1 … Σ 76, 872, 11872, 186080, 3276352, …
  (−∞, −1]nonnegative
• A111125 → D → θ+1
  1 · 5, 4 · 14, 28, 9 · 30, 108, 81, 16 … Σ 0, 1, 9, 51, 235, 960, 3624, 12941, …
  (−∞, 0]nonnegative
• A306364 → D+1
  1 · 3 · 15, 3 · 105, 45 … Σ 1, 3, 18, 150, 1575, 19845, 291060, …
  (−∞, 0]nonnegative
• A008277 → θ+1 → reverse → θ+1
  1 · 2, 2 · 3, 12, 3 · 4, 36, 42, 4 … Σ 1, 4, 18, 86, 435, 2332, 13237, 79382, …
  (−∞, 0]nonnegative
• A106246 → D+1
  1 · 3, 1 · 5, 6, 1 · 3, 15, 9, 1 … Σ 1, 4, 12, 28, 55, 96, 154, 232, …
  (−∞, 0]nonnegative
• A099040 → basis A013611 → x ↦ x/(x+1)
  1 · 2, 10 · 6, 52, 110 · 16, 208, 880, 1200 … Σ 1, 12, 168, 2304, 31680, 435456, …
  [−1, 0]nonnegative
• A272867 → basis A013610 → x ↦ x+1
  1 · 33, 36, 9 · 1089, 2376, 1890, 648, 81 … Σ 1, 78, 6084, 474552, 37015056, …
  (−∞, −1]nonnegative
• A102547 → reverse
  0, 1 · 0, 0, 1 · 0, 0, 0, 1 · 0, 0, 0, 1, 1 … Σ 1, 1, 1, 2, 3, 4, 6, 9, …
  (−∞, 0]nonnegative
• A088996 → basis A007318
  1 · 1, 1 · 3, 5, 2 · 15, 34, 25, 6 … Σ 1, 2, 10, 80, 880, 12320, 209440, …
  (−∞, −1]nonnegativerational roots
• A026374 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 7, 15, 10 … Σ 1, 3, 11, 33, 121, 363, 1331, 3993, …
  [−1, 0]nonnegative
• A142461 → (θ+1)²
  1 · 1, 4 · 1, 56, 9 · 1, 444, 999, 16 … Σ 1, 5, 66, 1460, 41920, 1478400, …
  (−∞, 0]nonnegative
• A225467 → (θ+1)²
  1 · 3, 16 · 9, 160, 144 · 27, 1264, 3024, 1024 … Σ 1, 19, 313, 5339, 97265, 1900323, …
  (−∞, 0]nonnegative
• A081578 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 7, 7 · 1, 12, 30, 20 … Σ 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  [−1, 0]nonnegative
• A132440 → D+1 → (1/n)D²+(2/n)D+1/n → x ↦ x/(x+1)
  1 · 3, 4 · 6, 18, 13 · 6, 36, 63, 34 … Σ 1, 7, 37, 139, 361, 751, 1357, 2227, …
  [−1, 0]nonnegative