Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A035607 → basis A013609 → x ↦ x+1
  1 · 1 · 7, 4 · 31, 32, 8 … Σ 1, 1, 11, 71, 481, 3241, 21851, 147311, …
  (−∞, −1]nonnegative
• A038319 → basis A013611 → D+1
  896, 1440, 400 · 19456, 53760, 43200, 8000 … Σ 2736, 124416, 5412096, 228427776, …
  (−∞, 0]nonnegativerational roots
• A033282 → (θ+1)²
  1 · 1, 8 · 1, 20, 45 · 1, 36, 189, 224 … Σ 1, 9, 66, 450, 2955, 18963, 119812, …
  (−∞, 0]nonnegative
• A114687 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 8, 11 · 1, 15, 51, 45 … Σ 1, 4, 20, 112, 672, 4224, 27456, …
  [−1, 0]nonnegative
• A257624 → x ↦ x/(x+1)
  1 · 5, 10 · 25, 130, 130 · 125, 1290, 3120, 2080 … Σ 1, 15, 285, 6615, 181845, 5781375, …
  [−1, 0]nonnegative
• A136572 → Veronese 2 → θ+1
  1 · 0, 4 · 0, 0, 72 · 0, 0, 0, 2880 … Σ 1, 0, 4, 0, 72, 0, 2880, 0, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A026729 → (θ+1)² → (θ+1)²
  1 · 0, 16 · 0, 0, 81 · 0, 0, 0, 256 … Σ 1, 16, 81, 256, 625, 1296, 2401, 4096, …
  (−∞, 0]nonnegativerational roots
• A103327 → D → θ+1
  1 · 10, 4 · 35, 84, 9 · 84, 504, 324, 16 … Σ 0, 1, 14, 128, 928, 5888, 34304, …
  (−∞, 0]nonnegative
• A099097 → basis A007318 → θ+1
  1 · 3, 6 · 10, 38, 27 · 33, 186, 261, 108 … Σ 1, 9, 75, 588, 4436, 32580, 234596, …
  (−∞, 0]nonnegative
• A091044 → basis A013609 → D
  4 · 32, 24 · 192, 320, 96 · 1024, 2688, 1920, 320 … Σ 0, 4, 56, 608, 5952, 54976, 489088, …
  (−∞, 0]nonnegative
• delannoy → basis A013609 → reverse
  1 · 2, 2 · 4, 10, 5 · 8, 32, 36, 12 … Σ 1, 4, 19, 88, 409, 1900, 8827, 41008, …
  (−∞, 0]nonnegative
• basis-map coefficients of A030528, shifted → basis A013610
  1 · -1, -3 · 1, 6, 9 · -1, -9, -27, -27 … Σ 1, -4, 16, -64, 256, -1024, 4096, …
  [−1, 0]rational roots
• A257623 → D+1
  1 · 6, 3 · 57, 66, 9 · 528, 1503, 582, 27 … Σ 1, 9, 132, 2640, 66528, 2018016, …
  (−∞, 0]nonnegative
• A257613 → x ↦ x/(x+1)
  1 · 4, 8 · 16, 80, 80 · 64, 608, 1440, 960 … Σ 1, 12, 176, 3072, 62336, 1443072, …
  [−1, 0]nonnegative
• A130123 → basis A013609 → θ+1
  1 · 2, 8 · 4, 32, 48 · 8, 96, 288, 256 … Σ 1, 10, 84, 648, 4752, 33696, 233280, …
  (−∞, 0]nonnegativerational roots
• A038271 → x ↦ x/(x+1)
  1 · 7, 12 · 49, 168, 144 · 343, 1764, 3024, 1728 … Σ 1, 19, 361, 6859, 130321, 2476099, …
  [−1, 0]nonnegativerational roots
• A035607 → basis A007318 → D+1
  1 · 1 · 5, 2 · 15, 12, 2 … Σ 1, 1, 7, 29, 123, 505, 2039, 8117, …
  (−∞, 0]nonnegative
• A038271 → basis A013609 → θ+1
  1 · 12, 20 · 144, 480, 300 · 1728, 8640, 10800, 4000 … Σ 1, 32, 924, 25168, 660176, 16866432, …
  (−∞, 0]nonnegativerational roots
• A106800 → D+1
  1 · 1 · 2, 1 · 4, 5, 1 … Σ 1, 1, 3, 10, 38, 161, 747, 3753, …
  (−∞, 0]nonnegative
• A249250 → x ↦ x+1 → x ↦ x+1
  1 · 7, 3 · 25, 18, 3 · 103, 105, 33, 3 … Σ 1, 10, 46, 244, 1252, 6472, 33400, …
  (−∞, −1]nonnegative
• A159854 → basis A013610 → x ↦ x/(x+1)
  1, 8, 16 · 2, 21, 72, 80 · 4, 52, 249, 520, 400 … Σ 25, 175, 1225, 8575, 60025, 420175, …
  [−1, 0]nonnegativerational roots
• A257623 → D
  3 · 48, 18 · 501, 1002, 81 · 4494, 26052, 13482, 324 … Σ 0, 3, 66, 1584, 44352, 1441440, …
  (−∞, 0]nonnegative
• A038303 → basis A013610 → (θ+1)²
  1 · 11, 12 · 121, 264, 81 · 1331, 4356, 2673, 432 … Σ 1, 23, 466, 8792, 158368, 2760464, …
  (−∞, 0]nonnegative
• A207824 → basis A013609 → x ↦ x/(x+1)
  1 · 6, 8 · 35, 94, 63 · 204, 824, 1108, 496 … Σ 1, 14, 192, 2632, 36080, 494592, …
  [−1, −2/3)nonnegative
• A078812 → basis A013609 → D
  2 · 12, 8 · 50, 72, 24 · 180, 408, 288, 64 … Σ 0, 2, 20, 146, 940, 5656, 32620, …
  (−∞, 0]nonnegative
• A160562 → θ+1
  1 · 1, 2 · 1, 20, 3 · 1, 182, 105, 4 … Σ 1, 3, 24, 292, 4880, 110832, 3217664, …
  (−∞, 0]nonnegative
• A003506 → (θ+1)² → D+1
  1 · 10, 8 · 27, 78, 27 · 52, 264, 300, 64 … Σ 0, 1, 18, 132, 680, 2880, 10752, 36736, …
  (−∞, 0]nonnegative
• A038326 → x ↦ x+1 → θ+1
  1 · 23, 24 · 529, 1104, 432 … Σ 1, 47, 2065, 86975, 3558625, 142559375, …
  (−∞, 0]nonnegativerational roots
• A178822: Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
  1 · 6, 6 · 21, 42, 21 · 56, 168, 168, 56 … Σ 1, 12, 84, 448, 2016, 8064, 29568, …
  (−∞, 0)nonnegativerational roots
• A008955 → D
  1 · 5, 8 · 14, 98, 108 · 30, 546, 2460, 2304 … Σ 0, 1, 13, 220, 5340, 181340, 8300780, …
  (−∞, 0]nonnegative
• A085478 → reverse → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 13, 20, 9, 1 … Σ 1, 3, 11, 43, 171, 683, 2731, 10923, …
  (−∞, −1]nonnegative
• A038271 → basis A013609 → D
  10 · 240, 200 · 4320, 7200, 3000 … Σ 0, 10, 440, 14520, 425920, 11712800, …
  (−∞, 0]nonnegativerational roots
• A156992 → basis A013611 → D+1
  1 · 12, 8 · 120, 288, 96 · 1344, 5760, 6912, 1536 … Σ 1, 20, 504, 15552, 570240, 24261120, …
  (−∞, 0]nonnegativerational roots
• A013610 → D → (θ+1)²
  3 · 6, 72 · 9, 216, 729 · 12, 432, 2916, 5184 … Σ 0, 3, 78, 954, 8544, 64320, 433152, …
  (−∞, 0]nonnegative
• A078812: Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
  1 · 2, 1 · 3, 4, 1 · 4, 10, 6, 1 … Σ 1, 3, 8, 21, 55, 144, 377, 987, …
  (−4, 0)nonnegative
• A075503 → (θ+1)²
  64, 96, 9 · 512, 1792, 432, 16 … Σ 169, 2752, 50521, 1024988, 22713041, …
  (−∞, 0]nonnegative
• A089231 → x ↦ x+1 → reverse
  1 · 2, 3 · 6, 18, 13 · 24, 108, 156, 73 … Σ 1, 5, 37, 361, 4361, 62701, 1044205, …
  (−∞, 0]nonnegative
• A085881 → basis A007318 → basis A007318
  1 · 3, 1 · 27, 18, 3 · 405, 405, 135, 15 … Σ 1, 4, 48, 960, 26880, 967680, 42577920, …
  (−∞, −1]nonnegativerational roots
• A359363 → reverse
  1 · 1 · 1, 1 · 1, 4, 1 … Σ 1, 1, 2, 6, 22, 92, 422, 2074, …
  (−∞, 0]nonnegative
• A282629 → reverse
  1 · 3, 1 · 9, 15, 1 · 27, 108, 63, 1 … Σ 1, 4, 25, 199, 1876, 20257, 245017, …
  (−∞, 0]nonnegative
• A114687 → basis A007318 → (θ+1)²
  1 · 3, 8 · 11, 56, 36 · 45, 336, 432, 128 … Σ 1, 11, 103, 941, 8581, 78375, 716955, …
  (−∞, 0]nonnegative
• A136394 → θ+1
  1 · 1 · 1, 2 · 1, 10 … Σ 1, 1, 3, 11, 50, 274, 1764, 13068, …
  (−∞, 0]nonnegative
• A053123 → D
  -2 · -4, 6 · -6, 20, -12 · -8, 42, -60, 20 … Σ 0, -2, 2, 2, -6, 4, 4, -10, …
  (−∞, ∞)
• A136124 → x ↦ x+1 → reverse
  1 · 1, 3 · 1, 7, 12 · 1, 12, 47, 60 … Σ 1, 4, 20, 120, 840, 6720, 60480, …
  (−∞, 0]nonnegativerational roots
• A209599 → x ↦ x+1
  1 · 2 · 4, 1 · 8, 3 … Σ 1, 2, 5, 11, 26, 59, 137, 314, …
  (−∞, −1]nonnegative
• A091044 → D
  2 · 10, 6 · 28, 56, 12 · 60, 252, 180, 20 … Σ 0, 2, 16, 96, 512, 2560, 12288, 57344, …
  (−∞, 0]nonnegative
• A143491 → basis A013611 → D
  28, 32 · 188, 384, 192 · 1368, 3808, 3456, 1024 … Σ 60, 764, 9656, 126376, 1738272, …
  (−∞, 0]nonnegative
• A100861: Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
  1 · 1 · 1, 1 · 1, 3 … Σ 1, 1, 2, 4, 10, 26, 76, 232, …
  (−∞, 0)nonnegativeA144299
• A090452 → D
  3, 4 · 7, 32, 45, 20 · 12, 102, 315, 456, 315, 84 … Σ 0, 7, 104, 1284, 15040, 172720, …
  (−∞, 0]nonnegative
• A286718 → basis A013610 → basis A007318
  1 · 5, 3 · 40, 39, 9 · 440, 549, 216, 27 … Σ 1, 8, 88, 1232, 20944, 418880, 9634240, …
  (−∞, −1]nonnegativerational roots