Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A114687 → basis A013611 → x ↦ x/(x+1)
  1 · 3, 11 · 11, 78, 131 · 45, 471, 1575, 1661 … Σ 1, 14, 220, 3752, 67792, 1277024, …
  [−1, 0]nonnegative
• A124324 → (θ+1)²
  1 · 1 · 1, 4 · 1, 16 … Σ 1, 1, 5, 17, 72, 330, 1639, 8755, …
  (−∞, 0]nonnegative
• A143495 → reverse → (θ+1)²
  1 · 1, 12 · 1, 28, 81 · 1, 48, 333, 432 … Σ 1, 13, 110, 814, 5771, 40779, 292751, …
  (−∞, 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
• A038319 → x ↦ x+1
  256, 160, 25 · 4096, 3840, 1200, 125 … Σ 441, 9261, 194481, 4084101, 85766121, …
  (−∞, −1]nonnegativerational roots
• A185410 → reverse → x ↦ x/(x+1)
  1 · 0, 1 · 0, 2, 3 · 0, 4, 18, 15 … Σ 1, 1, 5, 37, 377, 4921, 78365, 1473277, …
  [−1, 0]nonnegative
• A136572 → basis A013611 → basis A007318
  1 · 5, 4 · 50, 80, 32 · 750, 1800, 1440, 384 … Σ 1, 9, 162, 4374, 157464, 7085880, …
  (−∞, −1]nonnegativerational roots
• A147716 → reverse
  1 · 1, 14 · 1, 28, 196 · 1, 42, 588, 2744 … Σ 1, 15, 225, 3375, 50625, 759375, …
  (−∞, 0]nonnegativerational roots
• 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
• hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).
  1, 6, 1 · 1, 15, 15, 1 · 1, 28, 70, 28, 1 … Σ 8, 32, 128, 512, 2048, 8192, 32768, …
  (−∞, 0)nonnegative
• A075498 → θ+1
  1 · 3, 2 · 9, 18, 3 · 27, 126, 54, 4 … Σ 1, 5, 30, 211, 1691, 15126, 148975, …
  (−∞, 0]nonnegative
• A375853 → x ↦ x/(x+1)
  2 · 8, 16 · 20, 96, 96 · 40, 336, 768, 512 … Σ 2, 24, 212, 1656, 12102, 84816, 577576, …
  [−1, 0]nonnegative
• A207606 → basis A013609 → D+1
  35, 38, 8 · 122, 224, 116, 16 … Σ 81, 478, 2721, 15101, 82242, 441425, …
  (−∞, 0]nonnegative
• theta_plus_one of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between k …
  1, 2 · 1, 6, 9, 4 · 1, 14, 48, 64, 35, 6 … Σ 3, 20, 168, 1728, 21120, 299520, …
  (−∞, 0]nonnegative
• A120909 → basis A013610 → θ+1
  3 · 9, 36 · 27, 216, 324 · 81, 972, 2916, 2592 … Σ 3, 45, 567, 6561, 72171, 767637, …
  (−∞, 0]nonnegativerational roots
• A099093 → θ+1 → reverse
  1 · 6 · 27, 6 · 108, 54 … Σ 1, 6, 33, 162, 756, 3402, 14931, 64314, …
  (−∞, 0]nonnegative
• A046716 → basis A013609 → x ↦ x/(x+1)
  1 · 2, 4 · 5, 20, 19 · 16, 98, 192, 118 … Σ 1, 6, 44, 424, 5520, 94304, 2010304, …
  [−1, 0]nonnegative
• A110552 → D
  2 · 5, 8 · 10, 34, 24 · 19, 102, 147, 64 … Σ 0, 2, 13, 68, 332, 1568, 7248, 32960, …
  (−∞, 0]nonnegativerational roots
• A125092 → reverse → x ↦ x/(x+1)
  1 · 4, 5 · 9, 26, 18 · 16, 75, 114, 56 … Σ 1, 9, 53, 261, 1161, 4833, 19197, …
  [−1, 0]nonnegative
• A272866 → D → θ+1
  3, 4 · 6, 44, 54, 16 · 9, 120, 405, 480, 225, 36 … Σ 0, 7, 120, 1275, 11000, 84375, 600000, …
  (−∞, 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
• A038305 → basis A013609 → (θ+1)²
  169, 624, 324 · 2197, 12168, 12636, 3456 … Σ 1117, 30457, 780121, 19143469, …
  (−∞, 0]nonnegative
• A008459 → x ↦ x+1 → D+1
  1 · 3, 1 · 12, 8, 1 · 50, 54, 15, 1 … Σ 1, 4, 21, 120, 705, 4188, 24997, …
  (−∞, 0]nonnegative
• A106246 → reverse → (θ+1)²
  1 · 1, 8 · 1, 16, 9 · 1, 24, 27 … Σ 1, 9, 26, 52, 87, 131, 184, 246, …
  (−∞, 0]nonnegative
• A013611 → basis A013611 → basis A007318
  1 · 21, 16 · 441, 672, 256 · 9261, 21168, 16128, 4096 … Σ 1, 37, 1369, 50653, 1874161, 69343957, …
  (−∞, −1]nonnegativerational roots
• A038335 → basis A013610 → θ+1
  1 · 21, 54 · 441, 2268, 2187 … Σ 1, 75, 4896, 297216, 17252352, …
  (−∞, 0]nonnegativerational roots
• A178820 → basis A013611 → (θ+1)²
  1 · 8, 64 · 40, 640, 1440 · 160, 3840, 17280, 20480 … Σ 1, 72, 2120, 41760, 650160, 8660736, …
  (−∞, 0]nonnegative
• A038303 → basis A013611 → D+1
  1 · 15, 4 · 209, 120, 16 · 2783, 2508, 720, 64 … Σ 1, 19, 345, 6075, 104625, 1771875, …
  (−∞, 0]nonnegativerational roots
• A193685 → (θ+1)²
  1 · 5, 4 · 25, 44, 9 · 125, 364, 162, 16 … Σ 1, 9, 78, 667, 5703, 49150, 429187, …
  (−∞, 0]nonnegative
• A136124 → basis A013610 → D+1
  1 · 6, 3 · 33, 39, 9 · 201, 357, 189, 27 … Σ 1, 9, 81, 774, 7974, 88812, 1067652, …
  (−∞, 0]nonnegative
• A081580 → basis A007318 → x ↦ x/(x+1)
  1 · 2, 3 · 9, 27, 19 · 28, 126, 184, 87 … Σ 1, 5, 55, 425, 3775, 31625, 271375, …
  [−1, 0]nonnegative
• A111577 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 7, 7 · 1, 24, 57, 35 … Σ 1, 3, 15, 117, 1161, 13419, 176823, …
  [−1, 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
• A257608: 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) = 9*x + 1.
  1 · 1, 1 · 1, 20, 1 · 1, 219, 219, 1 … Σ 1, 2, 22, 440, 12760, 484880, 22789360, …
  (−∞, 0)nonnegative
• A207824 → basis A013609 → θ+1
  1 · 6, 4 · 35, 48, 12 · 204, 424, 216, 32 … Σ 1, 10, 95, 876, 7905, 70174, 614975, …
  (−∞, 0]nonnegative
• A001263 → basis A013611 → reverse
  1 · 4, 2 · 16, 20, 5 · 64, 144, 84, 14 … Σ 1, 6, 41, 306, 2426, 20076, 171481, …
  (−∞, 0]nonnegative
• A086645 → γ → (θ+1)²
  1 · 1 · 1, 16 · 1, 48 … Σ 1, 1, 17, 49, 241, 881, 3425, 12545, …
  (−∞, 0]nonnegative
• A114656 → (θ+1)²
  1 · 2, 4 · 4, 24, 9 · 8, 96, 108, 16 … Σ 1, 6, 37, 228, 1401, 8578, 52333, …
  (−∞, 0]nonnegative
• A008277 → θ+1 → θ+1
  1 · 1, 4 · 1, 12, 9 · 1, 28, 54, 16 … Σ 0, 1, 5, 22, 99, 471, 2386, 12867, …
  (−∞, 0]nonnegative
• A038336 → D+1
  384, 440, 100 · 6048, 11520, 6600, 1000 … Σ 924, 25168, 660176, 16866432, …
  (−∞, 0]nonnegativerational roots
• A038319 → (θ+1)²
  121, 440, 225 · 1331, 7260, 7425, 2000 … Σ 786, 18016, 388096, 8011776, 160301056, …
  (−∞, 0]nonnegative
• A100861 → D+1
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 22, 66, 226, 778, …
  (−∞, 0]nonnegative
• A038319 → basis A013609 → D+1
  576, 520, 100 · 11776, 17280, 7800, 1000 … Σ 1196, 37856, 1160016, 34730176, …
  (−∞, 0]nonnegativerational roots
• A080248 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 25, 36, 13, 1 … Σ 1, 3, 13, 75, 549, 4931, 52973, 667451, …
  (−∞, −1]nonnegative
• A038322 → basis A007318 → (θ+1)²
  1 · 19, 32 · 361, 1216, 576 … Σ 1, 51, 2153, 82539, 2980881, 103316067, …
  (−∞, 0]nonnegative
• A225471 → θ+1
  1 · 3, 2 · 21, 20, 3 · 231, 262, 63, 4 … Σ 1, 5, 44, 560, 9344, 193024, 4755456, …
  (−∞, 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]
• A008459 → basis A013611 → D+1
  1 · 6, 4 · 30, 56, 16 · 140, 504, 384, 64 … Σ 1, 10, 102, 1092, 11910, 130476, …
  (−∞, 0]nonnegative
• A038271 → basis A007318 → (θ+1)²
  1 · 12, 20 · 144, 480, 225 · 1728, 8640, 8100, 2000 … Σ 1, 32, 849, 20468, 465001, 10140432, …
  (−∞, 0]nonnegative
• A145596 → (θ+1)²
  1 · 2, 8 · 3, 32, 27 · 4, 80, 180, 64 … Σ 1, 10, 62, 328, 1605, 7502, 34034, …
  (−∞, 0]nonnegative