Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A033282 → basis A013610 → D
  6 · 45, 90 · 279, 1134, 1134 … Σ 0, 6, 135, 2547, 45738, 806460, …
  (−∞, 0]nonnegative
• A285061 → (θ+1)²
  1 · 1, 16 · 1, 96, 144 · 1, 496, 2160, 1024 … Σ 1, 17, 241, 3681, 61473, 1117681, …
  (−∞, 0]nonnegative
• A132393 → reverse → D+1
  1 · 1 · 2, 1 · 4, 7, 2 … Σ 1, 1, 3, 13, 70, 446, 3276, 27252, …
  (−∞, 0]nonnegative
• A079638 → θ+1
  1 · 3, 2 · 14, 18, 3 · 90, 166, 54, 4 … Σ 1, 5, 35, 314, 3434, 44304, 658776, …
  (−∞, 0]nonnegative
• A038329 → basis A013611 → D
  360, 288 · 8100, 12960, 5184 … Σ 648, 26244, 944784, 31886460, …
  (−∞, 0]nonnegativerational roots
• A327997 → x ↦ x+1 → (θ+1)²
  1 · 4, 4 · 17, 36, 9 · 78, 260, 135, 16 … Σ 1, 8, 62, 489, 4027, 35150, 328052, …
  (−∞, 0]nonnegative
• A038331 → basis A013611 → D
  680, 800 · 17340, 40800, 24000 … Σ 1480, 82140, 4052240, 187416100, …
  (−∞, 0]nonnegativerational roots
• A008278 → D+1
  1 · 2, 1 · 4, 5, 1 · 7, 20, 10, 1 … Σ 1, 3, 10, 38, 161, 747, 3753, 20253, …
  (−∞, 0]nonnegative
• A178822 → basis A013611 → D+1
  1 · 36, 24 · 420, 1008, 336 … Σ 1, 60, 1764, 36288, 598752, 8491392, …
  (−∞, 0]nonnegativerational roots
• A290595 → D+1
  1 · 3, 2 · 23, 27, 4 · 250, 516, 171, 8 … Σ 1, 5, 54, 945, 22680, 688905, 25259850, …
  (−∞, 0]nonnegative
• A084950 → θ+1
  1 · 2, 2 · 6, 8 · 24, 36, 3 … Σ 1, 4, 14, 63, 339, 2140, 15544, 127885, …
  (−∞, 0]nonnegative
• A038271 → D+1
  1 · 12, 5 · 119, 120, 25 · 1078, 1785, 900, 125 … Σ 1, 17, 264, 3888, 55296, 767232, …
  (−∞, 0]nonnegativerational roots
• A352880 → reverse → D
  12 · 64, 32 · 200, 300, 60 · 480, 1440, 864, 96 … Σ 0, 12, 96, 560, 2880, 13860, 64064, …
  (−∞, 0]nonnegative
• A236076 → θ+1
  1 · 0, 4 · 0, 2, 9 · 0, 0, 9, 20 … Σ 1, 4, 11, 29, 71, 169, 391, 889, …
  (−∞, 0]nonnegative
• A105868 → D
  1 · 2, 2 · 0, 12, 3 · 0, 12, 36, 4 … Σ 0, 1, 4, 15, 52, 175, 576, 1869, …
  (−∞, 0]nonnegative
• A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
  1 · 6, 5 · 36, 60, 25 · 216, 540, 450, 125 … Σ 1, 11, 121, 1331, 14641, 161051, …
  (−∞, 0)nonnegativerational roots
• A038331 → x ↦ x+1 → D
  170, 50 · 4335, 2550, 375 · 98260, 86700, 25500, 2500 … Σ 220, 7260, 212960, 5856400, 154608960, …
  (−∞, 0]nonnegativerational roots
• A272867 → x ↦ x+1 → D+1
  1 · 12, 8, 1 · 108, 168, 84, 16, 1 … Σ 1, 21, 377, 6253, 98865, 1513733, …
  (−∞, 0]nonnegative
• A110441 → D+1
  1 · 4, 1 · 13, 8, 1 · 38, 41, 12, 1 … Σ 1, 5, 22, 92, 372, 1468, 5688, 21728, …
  (−∞, 0]nonnegative
• A135065 → D
  4 · 18, 18 · 48, 96, 48 · 100, 300, 300, 100 … Σ 0, 4, 36, 192, 800, 2880, 9408, 28672, …
  (−∞, −1]nonnegativerational roots
• A099093 → (θ+1)²
  1 · 0, 12 · 0, 12, 81 · 0, 0, 162, 432 … Σ 1, 12, 93, 594, 3402, 18144, 92043, …
  (−∞, 0]nonnegative
• A112486 → (θ+1)²
  1 · 1, 4 · 2, 20, 27 · 6, 104, 315, 240 … Σ 1, 5, 49, 665, 11365, 233077, 5564729, …
  (−∞, 0]nonnegative
• A249251: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 3 · 1, 6, 6 · 1, 9, 21, 12 … Σ 1, 4, 13, 43, 142, 469, 1549, 5116, …
  (−∞, 0)nonnegative
• A038226 → basis A013610 → D+1
  649, 1680, 576 · 10043, 46728, 60480, 13824 … Σ 2905, 131075, 5616625, 232596875, …
  (−∞, 0]nonnegativerational roots
• A249139 → basis A013609
  1 · 4, 2 · 7, 4 · 19, 18, 4 … Σ 1, 6, 11, 41, 96, 301, 781, 2286, …
  (−∞, −1]nonnegative
• A120909 → Veronese 2 (j=1) → x ↦ x+1
  6 · 12 · 42, 24 · 120, 96 … Σ 0, 6, 12, 66, 216, 894, 3300, 12858, …
  (−∞, −1]nonnegative
• A143683 → D+1
  1 · 2, 1 · 11, 12, 1 · 20, 57, 22, 1 … Σ 1, 3, 24, 100, 528, 2352, 11008, 48960, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → D
  0, 2 · 0, 0, 0, 4 · 0, 0, 0, 0, 0, 6 … Σ 0, 2, 4, 6, 8, 10, 12, 14, …
  (−∞, 0]nonnegativerational roots
• A126216 → basis A007318 → θ+1
  1 · 3, 2 · 11, 14, 3 · 45, 84, 36, 4 … Σ 1, 5, 28, 169, 1070, 6994, 46750, …
  (−∞, 0]nonnegative
• A229032 → x ↦ x+1 → x ↦ x+1
  1 · 2 · 11, 4 · 36, 16 … Σ 1, 2, 15, 52, 269, 1110, 5179, 22568, …
  (−∞, −1]nonnegative
• A059364 → basis A007318 → (θ+1)²
  1 · 3, 4 · 15, 44, 18 · 105, 488, 423, 96 … Σ 1, 7, 77, 1112, 19688, 411228, 9879156, …
  (−∞, 0]nonnegative
• A132393 → basis A008277 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 8, 7 · 6, 36, 63, 34 … Σ 1, 3, 17, 139, 1473, 19091, 291793, …
  [−1, 0]nonnegative
• A113214 → x ↦ x/(x+1)
  1 · 2, 3 · 0, 3, 4 · 0, 2, 8, 7 … Σ 1, 5, 7, 17, 31, 65, 127, 257, …
  [−1, 0]nonnegative
• A200536 → basis A007318 → (θ+1)²
  1 · 6, 28, 18 · 36, 336, 657, 448, 100 … Σ 1, 52, 1577, 39240, 876825, 18292500, …
  (−∞, 0]nonnegative
• chebyshev_t → x ↦ x+1 → x ↦ x+1
  1 · 2, 1 · 7, 8, 2 · 26, 45, 24, 4 … Σ 1, 3, 17, 99, 577, 3363, 19601, 114243, …
  (−3, −1)nonnegative
• A144439 → D+1
  1 · 2, 1 · 9, 10, 1 · 32, 93, 34, 1 … Σ 1, 3, 20, 160, 1596, 19068, 265760, …
  (−∞, 0]nonnegative
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
  1, 4, 3 · 3, 18, 27, 12 · 6, 48, 108, 96, 30 … Σ 0, 0, 8, 60, 288, 1120, 3840, 12096, …
  (−∞, 0]nonnegativerational roots
• A038329 → basis A013611 → (θ+1)²
  225, 1440, 1296 · 3375, 32400, 58320, 27648 … Σ 2961, 121743, 4625505, 166695327, …
  (−∞, 0]nonnegative
• A135065 → x ↦ x+1 → (θ+1)²
  1 · 8, 16 · 36, 144, 81 · 128, 768, 864, 256 … Σ 1, 24, 261, 2016, 12825, 71928, 369117, …
  (−∞, 0]nonnegative
• A249139 → basis A007318
  1 · 4, 1 · 7, 2 · 19, 9, 1 … Σ 1, 5, 9, 29, 65, 181, 441, 1165, …
  (−∞, −1]nonnegative
• A038255 → basis A013610 → basis A013609
  1 · 10, 6 · 100, 120, 36 · 1000, 1800, 1080, 216 … Σ 1, 16, 256, 4096, 65536, 1048576, …
  (−∞, −1]nonnegativerational roots
• A330381 → (θ+1)²
  1 · 1, 8 · 1, 20, 27 · 1, 36, 117, 64 … Σ 1, 9, 48, 218, 913, 3633, 13940, 52030, …
  (−∞, 0]nonnegative
• A116666 → Veronese 2 → θ+1
  1 · 1 · 1, 10 · 1, 30 … Σ 1, 1, 11, 31, 88, 236, 608, 1520, …
  (−∞, 0]nonnegative
• A178820 → Veronese 2 (j=1) → x ↦ x+1
  4 · 20 · 80, 20 · 280, 140 … Σ 0, 4, 20, 100, 420, 1624, 5880, 20280, …
  (−∞, −1]nonnegative
• basis-map coefficients of A112227 → x ↦ x+1 → D+1
  3, 4, 1 · 4, 9, 6, 1 · 5, 16, 18, 8, 1 … Σ 8, 20, 48, 112, 256, 576, 1280, 2816, …
  (−∞, 0]nonnegativerational roots
• A271703 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 2, 3 · 0, 6, 18, 13 … Σ 1, 1, 5, 37, 361, 4361, 62701, 1044205, …
  [−1, 0]nonnegative
• A038322 → basis A007318 → x ↦ x/(x+1)
  1 · 19, 27 · 361, 1026, 729 … Σ 1, 46, 2116, 97336, 4477456, 205962976, …
  [−1, 0]nonnegativerational roots
• A125092: Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
  1 · 1, 4 · 1, 8, 9 · 1, 12, 27, 16 … Σ 1, 5, 18, 56, 160, 432, 1120, 2816, …
  (−∞, 0)nonnegative
• A227450 → reverse → D
  2 · 12, 4 · 48, 56, 6 · 160, 400, 180, 8 … Σ 0, 2, 16, 110, 748, 5226, 38008, …
  (−∞, 0]nonnegative
• A078812 → x ↦ x+1
  1 · 3, 1 · 8, 6, 1 · 21, 25, 9, 1 … Σ 1, 4, 15, 56, 209, 780, 2911, 10864, …
  (−5, −1)nonnegative