preserver_image / derivative_polynomial, shift 0

Image under a real-rootedness preserver. All operations

410 sequences

• A099040 → D+1
  1 · 2, 2 · 2, 10, 4 · 0, 16, 32, 8 … row sums 1, 4, 16, 56, 188, 608, 1920, 5952, …
  (−∞, 0] nonnegative
• A178822 → D+1
  1 · 12, 6 · 63, 84, 21 · 224, 504, 336, 56 … row sums 1, 18, 168, 1120, 6048, 28224, 118272, …
  (−∞, 0] nonnegative rational roots
• A122542 → D+1
  1 · 1, 1 · 2, 4, 1 · 2, 10, 7, 1 … row sums 1, 2, 7, 20, 57, 158, 431, 1160, …
  (−∞, 0] nonnegative
• A039756 → D+1
  1 · 2, 1 · 5, 6, 1 · 10, 35, 16, 1 … row sums 1, 3, 12, 62, 372, 2492, 18384, 147800, …
  (−∞, 0] nonnegative
• A011973 → ⊙ C(n,k) → D
  2 · 6 · 12, 12 · 20, 60 … row sums 0, 0, 2, 6, 24, 80, 270, 882, …
  (−∞, 0] nonnegative
• A007318 → θ+1+n → D
  3 · 8, 10 · 15, 36, 21 · 24, 84, 96, 36 … row sums 0, 3, 18, 72, 240, 720, 2016, 5376, …
  (−∞, 0] nonnegative rational roots
• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
  A010054 → D∘x
  1 · 3 · 5 · 7 … row sums 1, 3, 5, 7, 9, 11, 13, 15, …
  A198954 (−∞, −1] [−1, 0] nonnegative rational roots
• A007318 → θ+1+n → D+1
  1 · 5, 3 · 11, 18, 5 · 19, 51, 39, 7 … row sums 1, 8, 34, 116, 352, 992, 2656, 6848, …
  (−∞, 0] nonnegative
• A008459 → D+1
  1 · 2, 1 · 5, 6, 1 · 10, 27, 12, 1 … row sums 1, 3, 12, 50, 210, 882, 3696, 15444, …
  (−∞, 0] nonnegative
• A007318 → θ+1 → D
  2 · 4, 6 · 6, 18, 12 · 8, 36, 48, 20 … row sums 0, 2, 10, 36, 112, 320, 864, 2240, …
  (−∞, 0] nonnegative rational roots
• A143494 → D
  1 · 5, 2 · 19, 18, 3 · 65, 110, 42, 4 … row sums 0, 1, 7, 40, 221, 1241, 7218, 43807, …
  (−∞, 0] nonnegative
• basis-map coefficients of A145596 → D
  2 · 2, 6 · -8, 16, 12 · -10, -30, 60, 20 … row sums 0, 2, 8, 20, 40, 70, 112, 168, …
  (−∞, ∞)
• A133080 → D
  1 · 0, 2 · 0, 2, 3 · 0, 0, 0, 4 … row sums 0, 1, 2, 5, 4, 9, 6, 13, …
  (−∞, 0] nonnegative rational roots
• A134082: Triangle read by rows, (n-1) zeros followed by (2n, 1).
  A023531 → 2D+1
  1 · 2, 1 · 0, 4, 1 · 0, 0, 6, 1 … row sums 1, 3, 5, 7, 9, 11, 13, 15, …
  A134082 (−∞, 0] nonnegative rational roots
• A023531 → (1/2)D²+2D+1 → D+1
  1 · 3, 1 · 5, 6, 1 · 3, 15, 9, 1 … row sums 1, 4, 12, 28, 55, 96, 154, 232, …
  (−∞, 0] nonnegative
• A137338 → D
  1 · -3, 2 · 6, -12, 3 · -9, 52, -30, 4 … row sums 0, 1, -1, -3, 17, -57, 179, -631, …
  (−∞, ∞)
• A152198 → D
  1 · 1 · 2, 2 · 2, 2 … row sums 0, 0, 1, 1, 4, 4, 12, 12, …
  (−∞, 0] nonnegative rational roots
• A063007 → D
  2 · 6, 12 · 12, 60, 60 · 20, 180, 420, 280 … row sums 0, 2, 18, 132, 900, 5910, 37926, …
  (−∞, 0] nonnegative
• A081579 → D
  1 · 6, 2 · 11, 22, 3 · 16, 92, 48, 4 … row sums 0, 1, 8, 36, 160, 640, 2496, 9408, …
  (−∞, 0] nonnegative
• A113413 → D
  1 · 4, 2 · 8, 12, 3 · 12, 36, 24, 4 … row sums 0, 1, 6, 23, 76, 233, 682, 1935, …
  (−∞, 0] nonnegative
• A172431 → D
  2 · 4, 6 · 6, 20, 12 · 8, 42, 60, 20 … row sums 0, 2, 10, 38, 130, 420, 1308, 3970, …
  (−∞, 0] nonnegative
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
  1 · 2, 1 · 4, 5, 1 · 8, 19, 9, 1 … row sums 1, 3, 10, 37, 151, 674, 3263, 17007, …
  A143494 (−∞, 0) nonnegative
• A086645 → D
  1 · 6, 2 · 15, 30, 3 · 28, 140, 84, 4 … row sums 0, 1, 8, 48, 256, 1280, 6144, 28672, …
  (−∞, 0] nonnegative
• A207606 → D+1
  11, 11, 2 · 21, 38, 17, 2 · 36, 102, 81, 23, 2 … row sums 24, 78, 244, 743, 2218, 6521, 18942, …
  (−∞, 0] nonnegative
• A008278 → x ↦ x+1 (shift +1) → D+1
  1 · 3, 1 · 10, 7, 1 · 38, 43, 13, 1 … row sums 1, 4, 18, 95, 579, 3966, 29944, 246141, …
  (−∞, 0] nonnegative
• A007318 → (1/2+(1/2)n)θ+1+n → D+1
  1 · 5, 3 · 12, 21, 6 · 22, 66, 54, 10 … row sums 1, 8, 39, 152, 520, 1632, 4816, 13568, …
  (−∞, 0] nonnegative
• A274498 → D+1
  1 · 3 · 9, 6 · 27, 18 … row sums 1, 3, 15, 45, 189, 567, 2187, 6561, …
  (−∞, 0] nonnegative rational roots
• A035607 → D
  2 · 4, 4 · 6, 16, 6 · 8, 36, 36, 8 … row sums 0, 0, 2, 8, 28, 88, 262, 752, …
  (−∞, 0] nonnegative
• A367270 → D+1
  1 · 1 · 3, 2 · 7, 12, 3 … row sums 1, 1, 5, 22, 95, 406, 1722, 7260, …
  (−∞, 0] nonnegative
• A178820 → D+1
  1 · 8, 4 · 30, 40, 10 · 80, 180, 120, 20 … row sums 1, 12, 80, 400, 1680, 6272, 21504, …
  (−∞, 0] nonnegative rational roots
• A274496 → D+1
  1 · 2 · 4, 2 · 8, 4 … row sums 1, 2, 6, 12, 32, 64, 160, 320, …
  (−∞, 0] nonnegative rational roots
• A173018 → D+1
  1 · 1 · 2, 1 · 5, 6, 1 … row sums 1, 1, 3, 12, 60, 360, 2520, 20160, …
  (−∞, 0] nonnegative
• A249252 → D
  4 · 8, 16 · 12, 64, 48 · 16, 144, 288, 128 … row sums 0, 4, 24, 124, 576, 2532, 10728, 44300, …
  (−∞, 0] nonnegative
• chebyshev_t → D
  1 · 0, 4 · -3, 0, 12 · 0, -16, 0, 32 … row sums 0, 1, 4, 9, 16, 25, 36, 49, …
  (−∞, ∞)
• A232535 → D+1
  1 · 3, 2 · 9, 14, 3 · 19, 68, 37, 4 … row sums 1, 5, 26, 128, 608, 2816, 12800, 57344, …
  (−∞, 0] nonnegative
• A133087 → D+1
  1 · 3, 1 · 3, 4, 1 · 7, 13, 7, 1 … row sums 1, 4, 8, 28, 48, 160, 256, 832, …
  (−∞, 0] nonnegative
• chebyshev_u → D
  2 · 0, 8 · -4, 0, 24 · 0, -24, 0, 64 … row sums 0, 2, 8, 20, 40, 70, 112, 168, …
  (−∞, ∞)
• A143683 → D
  1 · 10, 2 · 19, 38, 3 · 28, 236, 84, 4 … row sums 0, 1, 12, 60, 352, 1680, 8256, 38080, …
  (−∞, 0] nonnegative
• A100862 → D+1
  1 · 2, 1 · 4, 5, 1 · 7, 18, 9, 1 … row sums 1, 3, 10, 35, 129, 497, 1996, 8325, …
  (−∞, 0] nonnegative
• A007318 → (1/3+(1/2)n+(1/6)n²)θ+1+(3/2)n+(1/2)n² → D+1
  1 · 7, 4 · 22, 36, 10 · 50, 140, 110, 20 … row sums 1, 11, 68, 320, 1280, 4592, 15232, …
  (−∞, 0] nonnegative
• A013620 → D
  3 · 12, 18 · 36, 108, 81 · 96, 432, 648, 324 … row sums 0, 3, 30, 225, 1500, 9375, 56250, …
  (−∞, 0] nonnegative rational roots
• A063007 → D+1
  1 · 3, 2 · 7, 18, 6 · 13, 72, 90, 20 … row sums 1, 5, 31, 195, 1221, 7593, 46915, …
  (−∞, 0] nonnegative
• A053117 → D
  2 · 0, 8 · -4, 0, 24 · 0, -24, 0, 64 … row sums 0, 2, 8, 20, 40, 70, 112, 168, …
  (−∞, ∞)
• A123519 → D
  6, 8 · 12, 40, 24 · 20, 120, 168, 64 … row sums 14, 76, 372, 1718, 7642, 33112, 140712, …
  (−∞, 0] nonnegative
• A152198 → D+1
  1 · 1 · 2, 1 · 2, 1 … row sums 1, 1, 3, 3, 8, 8, 20, 20, …
  (−∞, 0] nonnegative rational roots
• A008292 → γ (scale 1/4) (shift +1) → D+1
  1 · 2 · 6, 2 · 24, 16 … row sums 1, 2, 8, 40, 240, 1680, 13440, 120960, …
  (−∞, 0] nonnegative
• A011973 → D+1
  1 · 1 · 2, 1 · 3, 2 … row sums 1, 1, 3, 5, 10, 18, 33, 59, …
  (−∞, 0] nonnegative
• A007318 → ×(1+(3/2)n+(1/2)n²) → D+1
  1 · 6, 3 · 18, 24, 6 · 40, 90, 60, 10 … row sums 1, 9, 48, 200, 720, 2352, 7168, 20736, …
  (−∞, 0] nonnegative rational roots
• A010054 → reverse → D
  4 · 0, 14 · 0, 0, 30 · 0, 0, 0, 52 … row sums 0, 4, 14, 30, 52, 80, 114, 154, …
  (−∞, 0] nonnegative rational roots
• A062264 → D+1
  1 · 6, 5 · 13, 42, 15 · 22, 147, 168, 35 … row sums 1, 11, 70, 372, 1815, 8437, 38038, …
  (−∞, 0] nonnegative