Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• narayana → basis A007318 → θ+1
  1 · 2, 2 · 5, 10, 3 · 14, 42, 27, 4 … Σ 1, 4, 18, 87, 439, 2278, 12052, 64669, …
  (−∞, 0]nonnegative
• A081577 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 16, 24, 10, 1 … Σ 1, 3, 13, 51, 205, 819, 3277, 13107, …
  (−∞, −1]nonnegative
• A104684 → basis A013611 → reverse
  1 · 4, 3 · 16, 32, 13 · 64, 240, 228, 63 … Σ 1, 7, 61, 595, 6145, 65527, 712909, …
  (−∞, 0]nonnegative
• A169803 → D
  1 · 2 · 3, 2 · 4, 6 … Σ 0, 1, 2, 5, 10, 20, 38, 71, …
  (−∞, 0]nonnegative
• A113413 → x ↦ x+1
  1 · 3, 1 · 7, 6, 1 · 17, 23, 9, 1 … Σ 1, 4, 14, 50, 178, 634, 2258, 8042, …
  (−∞, −1]nonnegative
• A088996 → basis A013611 → θ+1
  1 · 1, 8 · 3, 40, 96 · 15, 272, 1200, 1536 … Σ 1, 9, 139, 3023, 84489, 2880281, …
  (−∞, 0]nonnegative
• A038303 → basis A013610
  1 · 11, 3 · 121, 66, 9 · 1331, 1089, 297, 27 … Σ 1, 14, 196, 2744, 38416, 537824, …
  (−∞, −1]nonnegativerational roots
• A034870 → θ+1
  1 · 1, 4, 3 · 1, 8, 18, 16, 5 … Σ 1, 8, 48, 256, 1280, 6144, 28672, …
  (−∞, 0]nonnegativerational roots
• A089231 → θ+1
  1 · 1, 4 · 1, 12, 18 · 1, 24, 108, 96 … Σ 1, 5, 31, 229, 1961, 19081, 207775, …
  (−∞, 0]nonnegative
• A038322 → basis A013609 → θ+1
  1 · 19, 32 · 361, 1216, 768 … Σ 1, 51, 2345, 101675, 4244625, …
  (−∞, 0]nonnegativerational roots
• A081581 → basis A007318
  1 · 2, 1 · 10, 10, 1 · 32, 48, 18, 1 … Σ 1, 3, 21, 99, 549, 2835, 15093, 79299, …
  (−∞, −1]nonnegative
• A038207 → basis A008277 → D
  1 · 7, 2 · 37, 24, 3 · 175, 194, 54, 4 … Σ 0, 1, 9, 64, 427, 2821, 18890, 129683, …
  (−∞, 0]nonnegative
• A143494 → reverse → D
  2 · 5, 8 · 9, 38, 24 · 14, 110, 195, 64 … Σ 0, 2, 13, 71, 383, 2129, 12360, 75242, …
  (−∞, 0]nonnegative
• A038336 → basis A013609 → basis A007318
  1764, 1680, 400 · 74088, 105840, 50400, 8000 … Σ 3844, 238328, 14776336, 916132832, …
  (−∞, −1]nonnegativerational roots
• A008277 → D → D
  2 · 12, 6 · 50, 60, 12 · 180, 390, 180, 20 … Σ 0, 0, 2, 18, 122, 770, 4832, 30940, …
  (−∞, 0]nonnegative
• A257609 → (θ+1)²
  1 · 2, 8 · 4, 64, 36 · 8, 352, 792, 128 … Σ 1, 10, 104, 1280, 18240, 295680, …
  (−∞, 0]nonnegative
• A085880 → D+1
  1 · 2, 1 · 6, 8, 2 · 20, 45, 30, 5 … Σ 1, 3, 16, 100, 672, 4704, 33792, …
  (−∞, 0]nonnegativerational roots
• A185410 → reverse → (θ+1)²
  1 · 0, 4 · 0, 8, 9 · 0, 16, 90, 16 … Σ 1, 4, 17, 122, 1173, 13920, 195405, …
  (−∞, 0]nonnegative
• A038322 → basis A007318 → x ↦ x+1
  1 · 27, 8 · 729, 432, 64 · 19683, 17496, 5184, 512 … Σ 1, 35, 1225, 42875, 1500625, 52521875, …
  (−∞, −1]nonnegativerational roots
• A154233 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 9, 9 · 1, 22, 60, 40 … Σ 1, 3, 19, 123, 1091, 10407, 120683, …
  [−1, 0]nonnegative
• A084950 → D
  1 · 4 · 18, 2 · 96, 18 … Σ 0, 1, 4, 20, 114, 747, 5568, 46684, …
  (−∞, 0]nonnegative
• A207824 → x ↦ x+1 → x ↦ x+1
  1 · 7, 1 · 48, 14, 1 · 329, 145, 21, 1 … Σ 1, 8, 63, 496, 3905, 30744, 242047, …
  (−9, −5)nonnegative
• A038271 → basis A013611 → D+1
  1 · 32, 20 · 624, 1280, 400 … Σ 1, 52, 2304, 94208, 3670016, 138412032, …
  (−∞, 0]nonnegativerational roots
• A106246 → θ+1 → (θ+1)²
  1 · 2, 8 · 1, 32, 27 · 0, 24, 162, 64 … Σ 1, 10, 60, 250, 799, 2106, 4810, 9850, …
  (−∞, 0]nonnegative
• A026374 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 10, 15, 7, 1 … Σ 1, 3, 11, 33, 121, 363, 1331, 3993, …
  (−∞, −1]nonnegative
• A038281 → θ+1
  64, 96, 27 · 512, 1152, 648, 108 … Σ 187, 2420, 30613, 380666, 4670479, …
  (−∞, 0]nonnegativerational roots
• A241171 → θ+1
  1 · 1, 12 · 1, 60, 270 · 1, 252, 3780, 10080 … Σ 1, 13, 331, 14113, 910111, 82795813, …
  (−∞, 0]nonnegative
• A033282 → D+1
  1 · 3, 2 · 6, 15, 5 · 10, 51, 63, 14 … Σ 1, 5, 26, 138, 743, 4043, 22180, …
  (−∞, 0]nonnegative
• A145596 → x ↦ x+1
  1 · 4, 2 · 14, 14, 3 · 48, 72, 32, 4 … Σ 1, 6, 31, 156, 785, 3978, 20335, …
  (−∞, −1]nonnegative
• A110441 → reverse → D+1
  1 · 4, 3 · 7, 20, 7 · 10, 55, 68, 15 … Σ 1, 7, 34, 148, 612, 2452, 9608, 37024, …
  (−∞, 0]nonnegative
• A027467 → basis A013611 → D
  4 · 128, 32 · 3072, 1536, 192 … Σ 0, 4, 160, 4800, 128000, 3200000, …
  (−∞, 0]nonnegativerational roots
• A026729 → basis A008277 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 8, 7 · 3, 22, 43, 25 … Σ 1, 3, 17, 93, 595, 4183, 32249, 269277, …
  [−1, 0]nonnegative
• A131222 → (θ+1)²
  1 · 0, 4 · 0, 12, 9 · 0, 56, 81, 16 … Σ 1, 4, 21, 153, 1420, 15992, 211700, …
  (−∞, 0]nonnegative
• A154986 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 24, 36, 14, 1 … Σ 1, 3, 13, 75, 537, 4611, 46053, 525483, …
  (−∞, −1]nonnegative
• A290595 → (θ+1)²
  1 · 1, 8 · 4, 76, 36 · 28, 888, 1323, 128 … Σ 1, 9, 116, 2367, 64800, 2208465, …
  (−∞, 0]nonnegative
• A178820 → basis A007318 → θ+1
  1 · 8, 8 · 40, 80, 30 · 160, 480, 360, 80 … Σ 1, 16, 150, 1080, 6615, 36288, 183708, …
  (−∞, 0]nonnegativerational roots
• A124932 → basis A007318 → x ↦ x/(x+1)
  1 · 5, 8 · 18, 57, 45 · 56, 264, 414, 216 … Σ 1, 13, 120, 950, 6875, 46875, 306250, …
  [−1, 0]nonnegativerational roots
• A008277: Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
  1 · 1, 1 · 1, 3, 1 · 1, 7, 6, 1 … Σ 1, 2, 5, 15, 52, 203, 877, 4140, …
  (−∞, 0)nonnegative
• A269944 → x ↦ x+1 → θ+1
  1 · 1, 2 · 2, 6, 3 · 10, 34, 24, 4 … Σ 1, 3, 11, 72, 828, 15300, 420228, …
  (−∞, 0]nonnegative
• A249250 → reverse
  1 · 3, 1 · 3, 6, 1 · 3, 15, 9, 1 … Σ 1, 4, 10, 28, 76, 208, 568, 1552, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → basis A013609 → (θ+1)²
  1 · -2, -8 · 4, 32, 36 · -8, -96, -216, -128 … Σ 1, -10, 72, -448, 2560, -13824, 71680, …
  (−∞, 0]
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0
  1 · 0, -1 · 0, 0, 1 · 0, 0, 0, -1 … Σ 1, -1, 1, -1, 1, -1, 1, -1, …
  [−1, 0]rational roots
• A304236 → D+1
  1 · 1 · 4, 3 · 7, 6 … Σ 1, 1, 7, 13, 46, 106, 301, 739, …
  (−∞, 0]nonnegative
• A145324 → basis A007318 → (θ+1)²
  1 · 3, 8 · 12, 68, 54 · 60, 532, 882, 384 … Σ 1, 11, 134, 1858, 29173, 513451, …
  (−∞, 0]nonnegative
• A114656 → basis A013610 → reverse
  1 · 3, 3 · 9, 24, 11 · 27, 135, 153, 45 … Σ 1, 6, 44, 360, 3152, 28896, 273856, …
  (−∞, 0]nonnegative
• A113953 → basis A013609 → D+1
  1 · 3, 2 · 11, 16, 4 · 27, 78, 52, 8 … Σ 1, 5, 31, 165, 867, 4401, 21951, …
  (−∞, 0]nonnegative
• A088617 → basis A007318
  1 · 2, 1 · 6, 7, 2 · 22, 41, 25, 5 … Σ 1, 3, 15, 93, 645, 4791, 37275, 299865, …
  (−∞, −1]nonnegative
• A245962 → D+1
  1 · 1 · 5, 2 · 7, 3 … Σ 1, 1, 7, 10, 29, 56, 131, 274, …
  (−∞, 0]nonnegative
• A116414 → reverse → θ+1
  1 · 1, 8 · 1, 16, 39 · 1, 24, 126, 160 … Σ 1, 9, 56, 311, 1635, 8308, 41221, …
  (−∞, 0]nonnegative
• reversed rows → θ+1
  1 · 1, 4 · 1, 6 · 1, 8, 6 … Σ 1, 5, 7, 15, 26, 48, 85, 151, …
  (−∞, 0]nonnegative