Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A038305 → basis A013609 → x ↦ x/(x+1)
  169, 494, 361 · 2197, 9633, 14079, 6859 … Σ 1024, 32768, 1048576, 33554432, …
  [−1, 0]nonnegativerational roots
• A111125 → basis A013610 → (θ+1)²
  1 · 4, 12 · 11, 84, 81 · 29, 372, 810, 432 … Σ 1, 16, 176, 1643, 13957, 111448, …
  (−∞, 0]nonnegative
• A134346 → D
  3 · 14, 14 · 45, 90, 45 · 124, 372, 372, 124 … Σ 0, 3, 28, 180, 992, 5040, 24384, …
  (−∞, 0]nonnegativerational roots
• narayana → basis A013609 → reverse
  1 · 2, 2 · 4, 10, 5 · 8, 36, 42, 14 … Σ 1, 4, 19, 100, 562, 3304, 20071, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729, shifted → basis A013611
  1 · -1, -4 · 1, 8, 16 · -1, -12, -48, -64 … Σ 1, -5, 25, -125, 625, -3125, 15625, …
  [−1, 0]rational roots
• A008459 → x ↦ x+1 → θ+1
  1 · 2, 2 · 6, 12, 3 · 20, 60, 36, 4 … Σ 1, 4, 21, 120, 705, 4188, 24997, …
  (−∞, 0]nonnegative
• A094305 → D+1
  1 · 6, 3 · 18, 24, 6 · 40, 90, 60, 10 … Σ 1, 9, 48, 200, 720, 2352, 7168, 20736, …
  (−∞, 0]nonnegativerational roots
• A035607 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 6, 7 · 1, 9, 23, 17 … Σ 1, 4, 14, 50, 178, 634, 2258, 8042, …
  [−1, 0]nonnegative
• basis-map coefficients of A371080, shifted → basis A013610 → θ+1
  1 · 1, 6 · 4, 30, 27 · 28, 234, 324, 108 … Σ 1, 7, 61, 694, 9862, 168712, 3380248, …
  (−∞, 0]nonnegative
• A038255 → basis A013609 → (θ+1)²
  1 · 7, 8 · 49, 112, 36 · 343, 1176, 756, 128 … Σ 1, 15, 197, 2403, 27945, 314199, …
  (−∞, 0]nonnegative
• A137320 → x ↦ x+1
  1 · 2, 2 · 6, 10, 4 · 24, 52, 36, 8 … Σ 1, 4, 20, 120, 840, 6720, 60480, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A297191, shifted → basis A013609 → (θ+1)²
  1 · 1, 16, 36 · 1, 32, 216, 512, 400 … Σ 1, 53, 1161, 19197, 274833, 3601989, …
  (−∞, 0]nonnegative
• A064861 → x ↦ x+1 → D
  7, 4 · 33, 40, 12 · 84, 146, 84, 16 … Σ 11, 85, 330, 2100, 7425, 43875, 148500, …
  (−∞, 0]nonnegativerational roots
• A131222 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 3, 4 · 0, 14, 37, 24 … Σ 1, 1, 7, 75, 1089, 20025, 446535, …
  [−1, 0]nonnegative
• A352880: Triangle read by rows: T(n,k) = number of vertices of degree k in an origami flip graph OFG(A2n).
  8 · 12, 18 · 16, 64, 32 · 20, 150, 200, 50 … Σ 8, 30, 112, 420, 1584, 6006, 22880, …
  (−∞, 0)nonnegative
• A038279 → basis A013610
  1 · 9, 3 · 81, 54, 9 · 729, 729, 243, 27 … Σ 1, 12, 144, 1728, 20736, 248832, …
  (−∞, −1]nonnegativerational roots
• A027467 → x ↦ x+1 → D+1
  1 · 17, 1 · 288, 34, 1 · 4864, 864, 51, 1 … Σ 1, 18, 323, 5780, 103173, 1837462, …
  (−∞, 0]nonnegativerational roots
• A207606 → basis A013610 → θ+1
  13, 66, 54 · 34, 264, 459, 216 … Σ 133, 973, 6824, 46493, 310147, 2035945, …
  (−∞, 0]nonnegative
• A131431 → D
  4 · 0, 14 · 0, 0, 30 · 0, 0, 0, 52 … Σ 0, 4, 14, 30, 52, 80, 114, 154, …
  (−∞, 0]nonnegativerational roots
• A082985 → reverse → D
  1 · 5, 2 · 14, 14, 3 · 30, 54, 27, 4 … Σ 0, 1, 7, 31, 115, 390, 1254, 3893, …
  (−∞, 0]nonnegative
• A376827 → θ+1
  1 · 1, 2 · 3, 12, 9 · 7, 42, 63, 28 … Σ 1, 3, 24, 140, 912, 5712, 36096, …
  (−∞, 0]nonnegativerational roots
• A257608 → D
  1 · 20, 2 · 219, 438, 3 · 2218, 16644, 6654, 4 … Σ 0, 1, 22, 660, 25520, 1212200, …
  (−∞, 0]nonnegative
• A038325 → basis A007318 → D+1
  1 · 33, 11 · 968, 726, 121 … Σ 1, 44, 1815, 71874, 2767149, 104361048, …
  (−∞, 0]nonnegativerational roots
• A038323 → basis A013609 → x ↦ x+1
  1 · 38, 18 · 1444, 1368, 324 … Σ 1, 56, 3136, 175616, 9834496, …
  (−∞, −1]nonnegativerational roots
• A038327 → x ↦ x+1
  1 · 13, 1 · 169, 26, 1 · 2197, 507, 39, 1 … Σ 1, 14, 196, 2744, 38416, 537824, …
  (−∞, −1]nonnegativerational roots
• A062264 → D+1
  1 · 6, 5 · 13, 42, 15 · 22, 147, 168, 35 … Σ 1, 11, 70, 372, 1815, 8437, 38038, …
  (−∞, 0]nonnegative
• A128813 → D+1
  1 · 2, 1 · 5, 10, 3 · 11, 64, 81, 18 … Σ 1, 3, 18, 174, 2474, 48824, 1281104, …
  (−∞, 0]nonnegative
• A103327 → reverse → (θ+1)²
  1 · 1, 12 · 1, 40, 45 · 1, 84, 315, 112 … Σ 1, 13, 86, 512, 2848, 15104, 77312, …
  (−∞, 0]nonnegative
• A099040 → x ↦ x−6
  1 · -12, 2 · 132, -46, 4 · -1440, 768, -136, 8 … Σ 1, -10, 90, -800, 7100, -63000, 559000, …
  (4, 6]
• A071951 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 16, 13 · 8, 76, 148, 81 … Σ 1, 5, 33, 313, 4065, 67101, 1360321, …
  [−1, 0]nonnegative
• A088996 → basis A007318 → x ↦ x/(x+1)
  1 · 1, 2 · 3, 11, 10 · 15, 79, 138, 80 … Σ 1, 3, 24, 312, 5616, 129168, 3616704, …
  [−1, 0]nonnegativerational roots
• A143685: Pascal-(1,9,1) array.
  1 · 1, 1 · 1, 11, 1 · 1, 21, 21, 1 … Σ 1, 2, 13, 44, 205, 806, 3457, 14168, …
  (−∞, 0)nonnegative
• A013622 → basis A013609 → D
  10 · 160, 200 · 1920, 4800, 3000 … Σ 0, 10, 360, 9720, 233280, 5248800, …
  (−∞, 0]nonnegativerational roots
• image under θ+1 → reverse
  1 · 2 · 3 · 4 … Σ 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 0]nonnegativerational roots
• A107131 → (θ+1)²
  1 · 0, 4 · 0, 4, 9 · 0, 0, 27, 16 … Σ 1, 4, 13, 43, 139, 446, 1419, 4488, …
  (−∞, 0]nonnegative
• A290319 → basis A013610 → reverse
  1 · 3, 2 · 9, 24, 12 · 27, 162, 276, 120 … Σ 1, 5, 45, 585, 9945, 208845, 5221125, …
  (−∞, 0]nonnegativerational roots
• A027466 → basis A013609 → θ+1
  1 · 8, 4 · 64, 64, 12 · 512, 768, 288, 32 … Σ 1, 12, 140, 1600, 18000, 200000, …
  (−∞, 0]nonnegativerational roots
• A013620 → basis A013611
  1 · 5, 12 · 25, 120, 144 · 125, 900, 2160, 1728 … Σ 1, 17, 289, 4913, 83521, 1419857, …
  (−∞, 0]nonnegativerational roots
• A126216 → θ+1
  1 · 2, 2 · 5, 10, 3 · 14, 42, 27, 4 … Σ 1, 4, 18, 87, 439, 2278, 12052, 64669, …
  (−∞, 0]nonnegative
• A081578 → x ↦ x+1 → x ↦ x+1
  1 · 3, 1 · 15, 9, 1 · 63, 57, 15, 1 … Σ 1, 4, 25, 136, 769, 4300, 24121, …
  (−∞, −1]nonnegative
• A172431 → reverse → θ+1
  1 · 2, 2 · 3, 8, 3 · 4, 20, 18, 4 … Σ 1, 4, 14, 46, 145, 444, 1331, 3926, …
  (−∞, 0]nonnegative
• A038330 → θ+1
  144, 192, 48 · 1728, 3456, 1728, 256 … Σ 384, 7168, 131072, 2359296, 41943040, …
  (−∞, 0]nonnegativerational roots
• A013620 → basis A008288 → reverse
  1 · 3, 5 · 9, 39, 25 · 27, 189, 333, 125 … Σ 1, 8, 73, 674, 6229, 57572, 532117, …
  (−∞, 0]nonnegative
• basis-map coefficients of A021012 → θ+1 → reverse
  1 · -2 · 6 · -24 … Σ 1, -2, 6, -24, 120, -720, 5040, -40320, …
  (−∞, 0]rational roots
• A008955 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 7, 10 · 1, 17, 80, 100 … Σ 1, 3, 18, 198, 3564, 96228, 3656664, …
  [−1, 0]nonnegativerational roots
• A062190 → x ↦ x+1
  1 · 7, 6 · 36, 56, 21 · 165, 360, 252, 56 … Σ 1, 13, 113, 833, 5641, 36365, 227305, …
  (−∞, −1]nonnegative
• theta_plus_one of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
  1 · 0, 2 · 0, 0, 3 · 0, 0, 0, 4 … Σ 1, 2, 3, 4, 5, 6, 7, 8, …
  [−1, 0]nonnegativerational roots
• narayana → basis A013611 → θ+1
  1 · 2, 8 · 5, 40, 48 · 14, 168, 432, 256 … Σ 1, 10, 93, 870, 8266, 79636, 775765, …
  (−∞, 0]nonnegative
• A257180 → D
  24, 18 · 141, 282, 81 · 726, 2820, 2178, 324 … Σ 42, 504, 6048, 75600, 997920, 13970880, …
  (−∞, 0]nonnegative
• A033282 → θ+1
  1 · 1, 4 · 1, 10, 15 · 1, 18, 63, 56 … Σ 1, 5, 26, 138, 743, 4043, 22180, …
  (−∞, 0]nonnegative