Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A038271 → basis A013610 → (θ+1)²
  1 · 12, 60 · 144, 1440, 2025 … Σ 1, 72, 3609, 154548, 6042681, …
  (−∞, 0]nonnegative
• Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.
  1, 3, 2 · 1, 5, 8, 4 · 1, 6, 13, 12, 4 … Σ 6, 18, 36, 108, 216, 648, 1296, 3888, …
  (−∞, 0]nonnegativerational roots
• A334218 → x ↦ x+1
  1 · 1 · 2, 2 · 6, 9, 3 … Σ 1, 1, 4, 18, 104, 750, 6492, 65562, …
  (−∞, −1]nonnegative
• A116666 → Veronese 2 (j=1) → x ↦ x+1
  3 · 6 · 16, 7 · 40, 28 … Σ 0, 3, 6, 23, 68, 199, 562, 1555, …
  (−∞, −1]nonnegative
• A132774 → basis A007318
  1 · 5, 3 · 9, 14, 5 · 13, 33, 27, 7 … Σ 1, 8, 28, 80, 208, 512, 1216, 2816, …
  (−∞, −1]nonnegativerational roots
• A137597 → D+1 → D
  1 · 7, 2 · 40, 24, 3 · 221, 206, 54, 4 … Σ 0, 1, 9, 67, 485, 3571, 27181, 215195, …
  (−∞, 0]nonnegative
• A091044 → basis A013610 → basis A007318
  1 · 10, 6 · 91, 102, 27 · 820, 1332, 684, 108 … Σ 1, 16, 220, 2944, 39184, 520960, …
  (−∞, −1]nonnegative
• A075505 → D+1
  130, 32, 1 · 1700, 820, 63, 1 … Σ 163, 2584, 47905, 1016756, 24101267, …
  (−∞, 0]nonnegative
• A345013: Triangle read by rows, related to clusters of type D.
  1 · 4, 3 · 15, 20, 6 · 56, 105, 60, 10 … Σ 1, 7, 41, 231, 1289, 7183, 40081, …
  (−∞, −1]nonnegative
• A038329 → basis A013609 → (θ+1)²
  225, 720, 324 · 3375, 16200, 14580, 3456 … Σ 1269, 37611, 1051785, 28255311, …
  (−∞, 0]nonnegative
• A105278 → D+1
  1 · 3, 1 · 12, 8, 1 · 60, 60, 15, 1 … Σ 1, 4, 21, 136, 1045, 9276, 93289, …
  (−∞, 0]nonnegative
• A033282 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 11 · 1, 12, 42, 45 … Σ 1, 4, 19, 100, 562, 3304, 20071, …
  [−1, 0)nonnegative
• A111125 → D+1
  1 · 4, 1 · 10, 7, 1 · 21, 28, 10, 1 … Σ 1, 5, 18, 60, 191, 589, 1775, 5257, …
  (−∞, 0]nonnegative
• A064861 → x ↦ x+1 → reverse
  2, 7, 6 · 4, 20, 33, 18 · 4, 28, 73, 84, 36 … Σ 15, 75, 225, 1125, 3375, 16875, 50625, …
  (−∞, 0]nonnegativerational roots
• A119900 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 1, 4 · 0, 0, 4, 8 … Σ 1, 2, 5, 12, 29, 70, 169, 408, …
  [−1, 0]nonnegative
• A281260 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 2, 5 · 0, 2, 12, 14 … Σ 1, 2, 7, 28, 121, 550, 2591, 12536, …
  [−1, 0]nonnegative
• A128100 → θ+1
  1 · 1 · 2, 2 · 3, 4 … Σ 1, 1, 4, 7, 18, 37, 84, 179, …
  (−∞, 0]nonnegative
• A120910 → D → D
  6 · 36, 18 · 144, 144, 36 · 480, 720, 360, 60 … Σ 0, 0, 6, 54, 324, 1620, 7290, 30618, …
  (−∞, 0]nonnegativerational roots
• A091042 → basis A013610 → (θ+1)²
  1 · 4, 36 · 16, 240, 405 · 64, 1344, 4536, 3024 … Σ 1, 40, 661, 8968, 112225, 1336864, …
  (−∞, 0]nonnegative
• A132812 → γ → x ↦ x+1
  1 · 2 · 6, 3 · 16, 12 … Σ 1, 2, 9, 28, 105, 366, 1337, 4824, …
  (−∞, −1]nonnegative
• A304334 → D
  1 · 1, 12 · 1, 60, 180 · 1, 252, 2520, 3360 … Σ 0, 1, 13, 241, 6133, 203881, 8518093, …
  (−∞, 0]nonnegative
• A120910 → basis A013610 → θ+1
  3 · 9, 18 · 27, 108, 81 · 81, 486, 729, 324 … Σ 3, 27, 216, 1620, 11664, 81648, 559872, …
  (−∞, 0]nonnegativerational roots
• A267006 → θ+1
  1 · 0, 2, 3 · 0, 0, 0, 4, 5 · 0, 0, 0, 0, 0, 6, 7 … Σ 1, 5, 9, 13, 17, 21, 25, 29, …
  (−∞, 0]nonnegativerational roots
• A060920 → Veronese 2 → D+1
  1 · 2 · 6, 1 · 22, 9 … Σ 1, 2, 7, 31, 139, 615, 2691, 11679, …
  (−∞, 0]nonnegative
• A062196 → D → θ+1
  3 · 8, 24 · 15, 120, 90 · 24, 360, 720, 240 … Σ 0, 3, 32, 225, 1344, 7350, 38016, …
  (−∞, 0]nonnegative
• A136572 → Veronese 2 (j=1) → (θ+1)²
  1 · 0, 24 · 0, 0, 1080 · 0, 0, 0, 80640 … Σ 0, 1, 0, 24, 0, 1080, 0, 80640, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A297191, shifted → basis A008277 → θ+1
  1 · 1, 6, 3 · 1, 30, 75, 40, 5 … Σ 1, 10, 151, 3263, 94828, 3535027, …
  (−∞, 0]nonnegative
• A249252 → basis A007318 → x ↦ x+1
  1 · 9, 4 · 49, 40, 8 · 281, 332, 128, 16 … Σ 1, 13, 97, 757, 5881, 45709, 355249, …
  (−∞, −1]nonnegative
• A099093 → θ+1
  1 · 0, 6 · 0, 6, 27 · 0, 0, 54, 108 … Σ 1, 6, 33, 162, 756, 3402, 14931, 64314, …
  (−∞, 0]nonnegative
• A049218 → D
  0, 2 · -8, 0, 3 · 0, -40, 0, 4 · 184, 0, -120, 0, 5 … Σ 2, -5, -36, 69, 1294, -1609, -80168, …
  (−∞, ∞)
• A081580 → basis A013611 → D
  4 · 36, 32 · 168, 512, 192 · 808, 3968, 4416, 1024 … Σ 0, 4, 68, 872, 10216, 111900, 1181724, …
  (−∞, 0]nonnegative
• A013622 → basis A013609 → basis A013609
  1 · 18, 20 · 324, 720, 400 · 5832, 19440, 21600, 8000 … Σ 1, 38, 1444, 54872, 2085136, 79235168, …
  (−∞, 0]nonnegativerational roots
• A003506 → Veronese 2 → x ↦ x+1
  1 · 2 · 6, 3 · 16, 12 … Σ 0, 1, 2, 9, 28, 85, 246, 693, …
  (−∞, −1]nonnegative
• A185410 → reverse → θ+1
  1 · 0, 2 · 0, 4, 3 · 0, 8, 30, 4 … Σ 1, 2, 7, 42, 345, 3570, 44415, 644490, …
  (−∞, 0]nonnegative
• A064861 → Veronese 2 → θ+1
  1, 4 · 1, 16 · 1, 26, 12 · 1, 50, 84 … Σ 5, 17, 39, 135, 297, 999, 2160, 7128, …
  (−∞, 0]nonnegative
• A038314 → x ↦ x+1 → (θ+1)²
  484, 2112, 1296 · 10648, 69696, 85536, 27648 … Σ 3892, 193528, 8993680, 399171424, …
  (−∞, 0]nonnegative
• A257617: 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) = 7*x + 2.
  1 · 2, 2 · 4, 36, 4 · 8, 388, 388, 8 … Σ 1, 4, 44, 792, 19800, 633600, 24710400, …
  (−∞, 0)nonnegative
• A099040 → x ↦ x−1
  1 · -2, 2 · 2, -6, 4 · 0, 8, -16, 8 … Σ 1, 0, 0, 0, 0, 0, 0, 0, …
  (−1, 1]
• A008288 → basis A013611 → x ↦ x+1
  1 · 6, 4 · 41, 52, 16 · 276, 520, 320, 64 … Σ 1, 10, 109, 1180, 12781, 138430, …
  (−∞, −1]nonnegative
• A094646: Triangle of generalized Stirling numbers of the first kind read by rows: T(n, k) = (-1)^(n+k)*Sum_{m=0..n} Stirling1(n, m) * binomial(m, k) * 2^(m-k),…
  1 · -2, 1 · 2, -3, 1 · 0, 2, -3, 1 … Σ 1, -1, 0, 0, 0, 0, 0, 0, …
  (−∞, 5/2]rational roots
• A119808 → θ+1
  1 · 2, 2 · 4, 10 · 8, 34, 6 … Σ 1, 4, 14, 48, 162, 540, 1782, 5832, …
  (−∞, 0]nonnegative
• A274498 → θ+1
  1 · 3 · 3, 12 · 9, 36 … Σ 1, 3, 15, 45, 189, 567, 2187, 6561, …
  (−∞, 0]nonnegativerational roots
• A204420 → basis A007318 → (θ+1)²
  1 · 1, 4 · 9, 48, 27 · 225, 1380, 1215, 240 … Σ 1, 5, 84, 3060, 195720, 19459440, …
  (−∞, 0]nonnegative
• A136124 → basis A008277 → x ↦ x+1
  1 · 4, 1 · 21, 10, 1 · 136, 93, 18, 1 … Σ 1, 5, 32, 248, 2248, 23272, 270400, …
  (−∞, −1]nonnegative
• A113413 → basis A013611 → D+1
  1 · 7, 4 · 31, 56, 16 · 109, 380, 336, 64 … Σ 1, 11, 103, 889, 7305, 58091, 451247, …
  (−∞, 0]nonnegative
• A156365 → D+1
  9, 16, 4 · 23, 110, 68, 8 · 53, 580, 888, 272, 16 … Σ 29, 209, 1809, 18269, 210853, 2737769, …
  (−∞, 0]nonnegative
• A154227 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 10, 10 · 1, 22, 60, 40 … Σ 1, 3, 21, 123, 999, 8163, 80433, …
  [−1, 0]nonnegative
• A156992 → basis A007318 → basis A008277
  1 · 6, 2 · 54, 42, 6 · 648, 888, 288, 24 … Σ 1, 8, 102, 1848, 44640, 1378800, …
  (−∞, 0]nonnegative
• A272866 → basis A007318 → θ+1
  1 · 5, 10, 3 · 25, 100, 105, 40, 5 … Σ 1, 18, 275, 3872, 51909, 673486, …
  (−∞, 0]nonnegative
• A038301 → basis A013609 → D
  880, 968 · 26400, 58080, 31944 … Σ 1848, 116424, 6519744, 342286560, …
  (−∞, 0]nonnegativerational roots