Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A232535 → basis A007318 → x ↦ x+1
  1 · 5, 2 · 29, 20, 3 · 169, 166, 49, 4 … Σ 1, 7, 52, 388, 2896, 21616, 161344, …
  (−∞, −1]nonnegative
• A048854 → x ↦ −2x (shift +2)
  3, -6, 1 · -15, 45, -15, 1 · 105, -420, 210, -28, 1 … Σ -2, 16, -132, 1216, -12440, 138048, …
  [0, ∞)
• A046716 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 9, 23, 16 … Σ 1, 3, 11, 49, 261, 1631, 11743, 95901, …
  [−1, 0]nonnegative
• A038331 → basis A013611 → x ↦ x/(x+1)
  289, 1258, 1369 · 4913, 32079, 69819, 50653 … Σ 2916, 157464, 8503056, 459165024, …
  [−1, 0]nonnegativerational roots
• A391533 → Pₙ + Pₙ₊₁ → x ↦ x/(x+1)
  8, 36, 54, 27 · 8, 52, 126, 135, 54 … Σ 125, 375, 1125, 3375, 10125, 30375, …
  [−1, 0]nonnegativerational roots
• A094816 → basis A013610 → reverse
  1 · 3, 2 · 9, 15, 5 · 27, 81, 69, 16 … Σ 1, 5, 29, 193, 1457, 12341, 116125, …
  (−∞, 0]nonnegative
• delannoy → basis A013611 → D
  4 · 20, 32 · 72, 256, 192 · 232, 1280, 2112, 1024 … Σ 0, 4, 52, 520, 4648, 39036, 315084, …
  (−∞, 0]nonnegative
• A356546 → basis A013609 → D+1
  1 · 8, 4 · 72, 96, 24 · 640, 1440, 960, 160 … Σ 1, 12, 192, 3200, 53760, 903168, …
  (−∞, 0]nonnegativerational roots
• A038326 → basis A013609 → θ+1
  1 · 23, 48 · 529, 2208, 1728 … Σ 1, 71, 4465, 262871, 14846689, …
  (−∞, 0]nonnegativerational roots
• A038243 → basis A013611 → (θ+1)²
  1 · 6, 16 · 36, 192, 144 · 216, 1728, 2592, 1024 … Σ 1, 22, 372, 5560, 77200, 1020000, …
  (−∞, 0]nonnegative
• A046716 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 5, 25, 29 · 16, 123, 297, 217 … Σ 1, 7, 59, 653, 9941, 203107, 5256799, …
  [−1, 0]nonnegative
• A142459 → D
  1 · 10, 2 · 59, 118, 3 · 308, 2124, 924, 4 … Σ 0, 1, 12, 180, 3360, 75600, 1995840, …
  (−∞, 0]nonnegative
• A081581 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 10, 10 · 1, 18, 48, 32 … Σ 1, 3, 21, 99, 549, 2835, 15093, 79299, …
  [−1, 0]nonnegative
• A256890 → D+1
  16, 20, 4 · 60, 156, 76, 8 · 212, 1028, 1004, 260, 16 … Σ 40, 300, 2520, 23520, 241920, 2721600, …
  (−∞, 0]nonnegative
• one_plus_derivative 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 betwe…
  2, 1 · 4, 9, 6, 1 · 8, 39, 64, 44, 12, 1 … Σ 3, 20, 168, 1728, 21120, 299520, …
  (−∞, 0]nonnegative
• A081580 → θ+1
  1 · 1, 2 · 1, 14, 3 · 1, 26, 39, 4 … Σ 1, 3, 18, 70, 303, 1197, 4756, 18396, …
  (−∞, 0]nonnegative
• A120910 → Veronese 2 (j=1) → (θ+1)²
  3 · 12 · 36, 12 · 96, 96 … Σ 0, 3, 12, 48, 192, 747, 2820, 10380, …
  (−∞, 0]nonnegative
• A038335 → basis A013609 → x ↦ x+1
  1 · 39, 18 · 1521, 1404, 324 … Σ 1, 57, 3249, 185193, 10556001, …
  (−∞, −1]nonnegativerational roots
• A008517 → x ↦ x+1
  1 · 3, 2 · 15, 20, 6 · 105, 210, 130, 24 … Σ 1, 5, 41, 469, 6889, 123605, 2620169, …
  (−∞, −1]nonnegative
• A257616 → θ+1
  1 · 2, 4 · 4, 64, 12 · 8, 624, 936, 32 … Σ 1, 6, 80, 1600, 42240, 1379840, …
  (−∞, 0]nonnegative
• chebyshev_u → x ↦ x+1 → D+1
  1 · 4, 2 · 11, 16, 4 · 24, 68, 48, 8 … Σ 1, 6, 31, 148, 673, 2962, 12735, 53800, …
  (−∞, 0]nonnegative
• A088996 → D+1
  1 · 1, 1 · 1, 5, 2 · 2, 16, 25, 6 … Σ 1, 2, 8, 49, 403, 4152, 51342, 740211, …
  (−∞, 0]nonnegative
• A035607 → basis A007318 → θ+1
  1 · 1 · 3, 4 · 7, 16, 6 … Σ 1, 1, 7, 29, 123, 505, 2039, 8117, …
  (−∞, 0]nonnegative
• A064861 → θ+1
  1, 6, 6 · 1, 10, 24, 16 · 1, 12, 39, 48, 20 … Σ 13, 51, 120, 432, 972, 3348, 7344, …
  (−∞, 0]nonnegative
• A257611 → x ↦ x/(x+1)
  1 · 3, 6 · 9, 48, 48 · 27, 294, 720, 480 … Σ 1, 9, 105, 1521, 26385, 533529, …
  [−1, 0]nonnegative
• A128099 → basis A013609 → basis A007318
  1 · 1 · 7, 4 · 13, 8 … Σ 1, 1, 11, 21, 131, 341, 1651, 5061, …
  (−∞, −1]nonnegative
• A059473 → γ → θ+1
  1 · 2 · 4, 4 · 8, 16 … Σ 1, 2, 8, 24, 76, 232, 704, 2112, …
  (−∞, 0]nonnegative
• A028338 → basis A013609 → reverse
  1 · 2, 2 · 4, 12, 8 · 8, 48, 88, 48 … Σ 1, 4, 24, 192, 1920, 23040, 322560, …
  (−∞, 0]nonnegativerational roots
• A094816 → x ↦ x+1 → (θ+1)²
  1 · 2, 4 · 5, 20, 9 · 16, 92, 81, 16 … Σ 1, 6, 34, 205, 1363, 10088, 83040, …
  (−∞, 0]nonnegative
• A304236 → basis A007318 → D
  3 · 6 · 27, 18 · 66, 54 … Σ 0, 0, 3, 6, 45, 120, 555, 1674, …
  (−∞, 0]nonnegative
• A144697 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 12, 20 · 1, 28, 120, 120 … Σ 1, 5, 33, 269, 2601, 29045, 367473, …
  [−1, 0]nonnegative
• A038301 → basis A013611 → basis A007318
  4096, 5632, 1936 · 262144, 540672, 371712, 85184 … Σ 11664, 1259712, 136048896, 14693280768, …
  (−∞, −1]nonnegativerational roots
• A330381 → D
  2 · 5, 6 · 9, 26, 12 · 14, 70, 78, 20 … Σ 0, 2, 11, 47, 182, 668, 2369, 8201, …
  (−∞, 0]nonnegative
• type_b_eulerian → γ (scale 1/4) → D
  1 · 5 · 18, 10 · 58, 122 … Σ 0, 0, 1, 5, 28, 180, 1320, 10920, …
  (−∞, 0]nonnegative
• A147716 → x ↦ x/(x+1)
  1 · 14, 15 · 196, 420, 225 · 2744, 8820, 9450, 3375 … Σ 1, 29, 841, 24389, 707281, 20511149, …
  [−1, 0]nonnegativerational roots
• A038255 → basis A013611
  1 · 7, 4 · 49, 56, 16 · 343, 588, 336, 64 … Σ 1, 11, 121, 1331, 14641, 161051, …
  (−∞, −1]nonnegativerational roots
• A356546 → basis A013611 → x ↦ x/(x+1)
  1 · 4, 12 · 24, 144, 216 · 160, 1440, 4320, 4320 … Σ 1, 16, 384, 10240, 286720, 8257536, …
  [−1, 0]nonnegativerational roots
• A008292 → γ (scale 1/2) (shift +1) → D+1
  1 · 1 · 2, 1 · 5, 4 … Σ 1, 1, 3, 9, 35, 155, 791, 4529, …
  (−∞, 0]nonnegative
• A232535 → (θ+1)²
  1 · 1, 8 · 1, 32, 27 · 1, 72, 225, 64 … Σ 1, 9, 60, 362, 2032, 10848, 55808, …
  (−∞, 0]nonnegative
• A143491 → basis A013611 → basis A007318
  56, 60, 16 · 504, 764, 384, 64 … Σ 132, 1716, 24024, 360360, 5765760, …
  (−∞, −1]nonnegativerational roots
• A013622 → D
  5 · 30, 50 · 135, 450, 375 · 540, 2700, 4500, 2500 … Σ 0, 5, 80, 960, 10240, 102400, 983040, …
  (−∞, 0]nonnegativerational roots
• A038331 → basis A013610 → x ↦ x/(x+1)
  289, 1088, 1024 · 4913, 27744, 52224, 32768 … Σ 2401, 117649, 5764801, 282475249, …
  [−1, 0]nonnegativerational roots
• A090452 → (θ+1)²
  1 · 1, 12, 18 · 1, 28, 144, 240, 125 … Σ 1, 31, 538, 7992, 110592, 1470416, …
  (−∞, 0]nonnegative
• A159854 → Veronese 2 → D+1
  1, 1 · 1, 1 · 1, 3, 1 · 1, 7, 3 … Σ 2, 2, 5, 11, 24, 52, 112, 240, …
  (−∞, 0]nonnegative
• A257619 → (θ+1)²
  1 · 2, 8 · 4, 176, 36 · 8, 2256, 5076, 128 … Σ 1, 10, 216, 7468, 332176, 18018000, …
  (−∞, 0]nonnegative
• A161198 → basis A013611 → D+1
  1 · 11, 8 · 79, 192, 64 · 673, 2488, 2496, 512 … Σ 1, 19, 335, 6169, 122033, 2610347, …
  (−∞, 0]nonnegative
• A132013: T(n,j) for an iterated mixed order Laguerre transform. Coefficients of the normalized generalized Laguerre polynomials (-1)^n*n!*L(n,1-n,x).
  1 · -1, 1 · 0, -2, 1 · 0, 0, -3, 1 … Σ 1, 0, -1, -2, -3, -4, -5, -6, …
  (−∞, ∞)rational roots
• A103327 → basis A013609 → θ+1
  1 · 4, 4 · 16, 48, 12 · 64, 320, 288, 32 … Σ 1, 8, 76, 704, 6288, 54656, 465600, …
  (−∞, 0]nonnegative
• A119308 → D
  1 · 5, 2 · 14, 18, 3 · 30, 80, 42, 4 … Σ 0, 1, 7, 35, 156, 660, 2717, 11011, …
  (−∞, 0]nonnegative
• A085881 → x ↦ x/(x+1)
  1 · 1, 2 · 3, 12, 12 · 15, 90, 180, 120 … Σ 1, 3, 27, 405, 8505, 229635, 7577955, …
  [−1, 0]nonnegativerational roots