Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A038331 → θ+1
  144, 240, 75 · 1728, 4320, 2700, 500 … Σ 459, 9248, 181781, 3507882, 66733279, …
  (−∞, 0]nonnegativerational roots
• A103406 → x ↦ x+1
  1 · 3, 2 · 8, 10, 3 · 20, 36, 21, 4 … Σ 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  (−∞, −1]nonnegativerational roots
• A144439 → D
  1 · 8, 2 · 31, 62, 3 · 102, 652, 306, 4 … Σ 0, 1, 10, 96, 1064, 13620, 199320, …
  (−∞, 0]nonnegative
• A159854 → Veronese 2 (j=1) → D+1
  1, 1 · 2, 2 · 3, 5, 1 · 4, 12, 4 … Σ 0, 2, 4, 9, 20, 44, 96, 208, …
  (−∞, 0]nonnegative
• A027465 → basis A013611 → θ+1
  1 · 4, 8 · 16, 64, 48 · 64, 384, 576, 256 … Σ 1, 12, 128, 1280, 12288, 114688, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A371080, shifted → basis A013610 → D
  3 · 15, 18 · 117, 216, 81 · 1254, 2862, 1782, 324 … Σ 0, 3, 33, 414, 6222, 110472, 2273688, …
  (−∞, 0]nonnegative
• A111125 → basis A013610 → D
  3 · 21, 18 · 93, 180, 81 · 345, 1080, 1053, 324 … Σ 0, 3, 39, 354, 2802, 20637, 145377, …
  (−∞, 0]nonnegative
• A136011 → θ+1
  1 · 1 · 1, 2 · 1, 6 … Σ 1, 1, 3, 7, 18, 49, 142, 437, …
  (−∞, 0]nonnegative
• A114656 → θ+1 → x ↦ x+1
  1 · 4, 2 · 19, 18, 3 · 96, 132, 48, 4 … Σ 1, 6, 40, 280, 2016, 14784, 109824, …
  (−∞, −1]nonnegative
• A093375 → Veronese 2 (j=1) → D+1
  1 · 4 · 10, 1 · 24, 8 … Σ 0, 1, 4, 11, 32, 88, 232, 592, …
  (−∞, 0]nonnegative
• A038255 → x ↦ x/(x+1)
  1 · 6, 7 · 36, 84, 49 · 216, 756, 882, 343 … Σ 1, 13, 169, 2197, 28561, 371293, …
  [−1, 0]nonnegativerational roots
• A103327 → D → D
  2 · 42, 6 · 252, 216, 12 · 924, 1980, 660, 20 … Σ 0, 0, 2, 48, 480, 3584, 23040, 135168, …
  (−∞, 0]nonnegative
• A178821 → basis A007318 → (θ+1)²
  1 · 10, 20 · 60, 240, 135 · 280, 1680, 1890, 560 … Σ 1, 30, 435, 4410, 35910, 251748, …
  (−∞, 0]nonnegative
• A145597 → γ → D
  3 · 15 · 45, 18 · 105, 126 … Σ 0, 0, 3, 15, 63, 231, 798, 2646, …
  (−∞, 0]nonnegative
• A139548: Triangle T(n,k) with the coefficient of [x^k] of the polynomial (2*(x+1)^2)^n in row n, column k, 0<=k<=2n.
  1 · 2, 4, 2 · 4, 16, 24, 16, 4 … Σ 1, 8, 64, 512, 4096, 32768, 262144, …
  (−∞, 0)nonnegativerational roots
• A136124 → basis A013611 → (θ+1)²
  1 · 3, 16 · 12, 112, 144 · 60, 752, 1728, 1024 … Σ 1, 19, 268, 3564, 47800, 663528, …
  (−∞, 0]nonnegative
• A125662 → basis A013609 → reverse
  1 · 2, 4 · 4, 16, 15 · 8, 48, 92, 56 … Σ 1, 6, 35, 204, 1189, 6930, 40391, …
  (−1, 0]nonnegative
• A132166 → D+1
  378, 44, 1 · 5649, 1239, 66, 1 … Σ 423, 6955, 108134, 1635402, 24367069, …
  (−∞, 0]nonnegative
• A181543 → x ↦ x+1
  1 · 2, 1 · 10, 10, 1 · 56, 84, 30, 1 … Σ 1, 3, 21, 171, 1521, 14283, 138909, …
  (−∞, −1]nonnegative
• A056939 → x ↦ x+1 → D+1
  1 · 3, 1 · 12, 8, 1 · 55, 59, 16, 1 … Σ 1, 4, 21, 131, 909, 6798, 53737, …
  (−∞, 0]nonnegative
• A143495 → x ↦ x+1 → reverse
  1 · 1, 4 · 1, 9, 17 · 1, 15, 64, 77 … Σ 1, 5, 27, 157, 979, 6517, 46107, …
  (−∞, 0]nonnegative
• A133214 → θ+1
  1, 16, 12 · 1, 36, 108, 32 · 1, 64, 432, 512, 80 … Σ 29, 177, 1089, 6693, 41005, 250337, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729 → basis A013610 → D+1
  1 · 4, 3 · 7, 24, 9 · 10, 63, 108, 27 … Σ 1, 7, 40, 208, 1024, 4864, 22528, …
  (−∞, 0]nonnegativerational roots
• A290596 → x ↦ x/(x+1)
  1 · 2, 3 · 10, 30, 21 · 80, 360, 504, 225 … Σ 1, 5, 61, 1169, 30361, 991901, …
  [−1, 0]nonnegative
• A056939 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 13, 33, 22 … Σ 1, 3, 13, 69, 417, 2763, 19609, 146793, …
  [−1, 0]nonnegative
• A178821 → Veronese 2 → θ+1
  1 · 5 · 15, 30 · 35, 210 … Σ 1, 5, 45, 245, 1120, 4536, 16800, …
  (−∞, 0]nonnegative
• A358735 → basis A007318 → D+1
  1 · 3, 1 · 15, 12, 2 · 93, 128, 52, 6 … Σ 1, 4, 29, 279, 3293, 45802, 731673, …
  (−∞, 0]nonnegative
• A274498 → (θ+1)²
  1 · 3 · 3, 24 · 9, 72 … Σ 1, 3, 27, 81, 477, 1431, 7047, 21141, …
  (−∞, 0]nonnegative
• A113214 → basis A013610 → x ↦ x/(x+1)
  1 · 3, 6 · 4, 23, 28 · 7, 60, 162, 136 … Σ 1, 9, 55, 365, 2375, 15525, 101375, …
  [−1, 0]nonnegative
• A038321 → basis A013609 → θ+1
  324, 1008, 588 · 5832, 27216, 31752, 10976 … Σ 1920, 75776, 2883584, 106954752, …
  (−∞, 0]nonnegativerational roots
• A028338 → basis A013609 → D
  2 · 12, 8 · 88, 96, 24 · 800, 1120, 480, 64 … Σ 0, 2, 20, 208, 2464, 33408, 513792, …
  (−∞, 0]nonnegative
• A038338 → x ↦ x+1 → reverse
  144, 576, 576 · 1728, 10368, 20736, 13824 … Σ 1296, 46656, 1679616, 60466176, …
  (−∞, 0]nonnegativerational roots
• A121574 → reverse → D+1
  1 · 3, 2 · 6, 13, 4 · 9, 40, 40, 8 … Σ 1, 5, 23, 97, 390, 1518, 5773, 21575, …
  (−∞, 0]nonnegative
• A048993 → D+1 → reverse
  1 · 1, 1 · 1, 3, 1 · 1, 6, 7, 1 … Σ 1, 2, 5, 15, 52, 203, 877, 4140, …
  (−∞, 0]nonnegative
• A110608 → D
  8, 2 · 45, 36, 3 · 224, 336, 96, 4 … Σ 10, 84, 660, 5005, 37128, 271320, …
  (−∞, 0]nonnegative
• A143683 → γ → basis A007318
  1 · 1 · 9, 8 · 17, 16 … Σ 1, 1, 17, 33, 305, 833, 5713, 19041, …
  (−∞, −1]nonnegative
• A033282 → basis A013610 → D+1
  1 · 9, 6 · 56, 135, 45 · 324, 1413, 1701, 378 … Σ 1, 15, 236, 3816, 62819, 1047309, …
  (−∞, 0]nonnegative
• A038327 → basis A013611 → θ+1
  1 · 13, 8 · 169, 208, 48 · 2197, 4056, 1872, 256 … Σ 1, 21, 425, 8381, 162129, 3090277, …
  (−∞, 0]nonnegativerational roots
• A104684 → basis A013609 → x ↦ x+1
  1 · 5, 2 · 33, 24, 4 · 245, 258, 84, 8 … Σ 1, 7, 61, 595, 6145, 65527, 712909, …
  (−∞, −1]nonnegative
• A286724 → D
  1 · 8, 2 · 72, 36, 3 · 768, 576, 96, 4 … Σ 0, 1, 10, 111, 1444, 21805, 376206, …
  (−∞, 0]nonnegative
• A257621 → x ↦ x/(x+1)
  1 · 3, 6 · 9, 60, 60 · 27, 474, 1260, 840 … Σ 1, 9, 129, 2601, 67521, 2142729, …
  [−1, 0]nonnegative
• A143683 → θ+1
  1 · 1, 2 · 1, 20, 3 · 1, 38, 57, 4 … Σ 1, 3, 24, 100, 528, 2352, 11008, 48960, …
  (−∞, 0]nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.
  144, 264, 121 · 1728, 4752, 4356, 1331 … Σ 529, 12167, 279841, 6436343, 148035889, …
  (−∞, −1]nonnegativerational roots
• A038321 → basis A013611 → θ+1
  324, 2016, 2352 · 5832, 54432, 127008, 87808 … Σ 4692, 275080, 15379088, 832806816, …
  (−∞, 0]nonnegativerational roots
• A108426 → x ↦ x/(x+1)
  1 · 1, 2 · 3, 11, 10 · 12, 64, 113, 66 … Σ 1, 3, 24, 255, 3102, 40854, 566934, …
  [−1, 0]nonnegative
• A013622 → basis A007318 → D+1
  1 · 13, 5 · 144, 130, 25 · 1472, 2160, 975, 125 … Σ 1, 18, 299, 4732, 72501, 1085318, …
  (−∞, 0]nonnegativerational roots
• A120909 → (θ+1)²
  3 · 3, 24 · 3, 48, 108 · 3, 72, 324, 384 … Σ 3, 27, 159, 783, 3483, 14499, 57591, …
  (−∞, 0]nonnegative
• A030528 → basis A013611 → x ↦ x/(x+1)
  1 · 2, 6 · 3, 22, 35 · 5, 55, 191, 205 … Σ 1, 8, 60, 456, 3456, 26208, 198720, …
  [−1, 0]nonnegative
• A027465 → basis A008277 → (θ+1)²
  1 · 4, 4 · 16, 36, 9 · 64, 244, 135, 16 … Σ 1, 8, 61, 459, 3468, 26569, 207609, …
  (−∞, 0]nonnegative
• A132774 → reverse
  1 · 3, 2 · 5, 4 · 7, 6 … Σ 1, 5, 9, 13, 17, 21, 25, 29, …
  (−∞, 0]nonnegativerational roots