Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A112486 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 9, 10 · 6, 44, 105, 82 … Σ 1, 3, 21, 237, 3711, 74451, 1822557, …
  [−1, 0)nonnegative
• A008292 → γ (scale 1/4) (shift +1) → x ↦ x+1 → (θ+1)²
  1 · 2 · 6, 8 · 24, 64 … Σ 1, 2, 14, 88, 744, 7008, 76256, 925312, …
  (−∞, 0]nonnegative
• A121574 → x ↦ x+1 → D+1
  1 · 4, 1 · 17, 9, 1 · 68, 57, 14, 1 … Σ 1, 5, 27, 140, 712, 3564, 17620, 86240, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → θ+1
  0, 0, 3 · 0, 0, 0, 4 · 0, 0, 0, 0, 5 … Σ 3, 4, 5, 6, 7, 8, 9, 10, …
  [−1, 0]nonnegativerational roots
• A075497 → x ↦ x+1 → D
  1 · 8, 2 · 55, 30, 3 · 384, 332, 72, 4 … Σ 0, 1, 10, 88, 792, 7536, 76352, 823488, …
  (−∞, 0]nonnegative
• A038207 → basis A008277 → reverse
  1 · 1, 3 · 1, 7, 9 · 1, 12, 37, 27 … Σ 1, 4, 17, 77, 372, 1915, 10481, 60814, …
  (−∞, 0]nonnegative
• A038338 → basis A013609 → θ+1
  576, 2304, 1728 · 13824, 82944, 124416, 55296 … Σ 4608, 276480, 15925248, 891813888, …
  (−∞, 0]nonnegativerational roots
• A225471: Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
  1 · 3, 1 · 21, 10, 1 · 231, 131, 21, 1 … Σ 1, 4, 32, 384, 6144, 122880, 2949120, …
  (−∞, −1]nonnegativerational roots
• A038327 → D+1
  1 · 13, 1 · 168, 26, 1 · 2160, 504, 39, 1 … Σ 1, 14, 195, 2704, 37349, 514098, …
  (−∞, 0]nonnegativerational roots
• A257612 → θ+1
  1 · 2, 4 · 4, 48, 12 · 8, 368, 552, 32 … Σ 1, 6, 64, 960, 18432, 430080, 11796480, …
  (−∞, 0]nonnegative
• A113214 → basis A007318 → D
  1 · 5, 2 · 13, 14, 3 · 29, 52, 27, 4 … Σ 0, 1, 7, 30, 112, 388, 1284, 4120, …
  (−∞, 0]nonnegative
• A114687 → basis A013610 → θ+1
  1 · 3, 12 · 11, 84, 108 · 45, 504, 1296, 864 … Σ 1, 15, 203, 2709, 36365, 492819, …
  (−∞, 0]nonnegative
• A397048 → x ↦ x+1
  1 · 2, 1 · 10, 7, 1 · 77, 64, 15, 1 … Σ 1, 3, 18, 157, 1790, 25179, 421182, …
  (−∞, −1]nonnegative
• A236076 → x ↦ x/(x+1)
  1 · 0, 2 · 0, 1, 4 · 0, 0, 3, 8 … Σ 1, 2, 5, 11, 26, 59, 137, 314, …
  [−1, 0]nonnegative
• A094816 → basis A013611 → x ↦ x/(x+1)
  1 · 2, 6 · 5, 30, 41 · 16, 140, 376, 316 … Σ 1, 8, 76, 848, 10960, 161792, 2693056, …
  [−1, 0]nonnegative
• Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.
  4, 12, 4 · 8, 52, 52, 8 · 16, 196, 416, 196, 16 … Σ 20, 120, 840, 6720, 60480, 604800, …
  (−∞, 0)nonnegative
• A376827 → (θ+1)²
  1 · 1, 4 · 3, 24, 27 · 7, 84, 189, 112 … Σ 1, 5, 54, 392, 3040, 22032, 157920, …
  (−∞, 0]nonnegative
• A091044 → basis A007318 → D+1
  1 · 6, 2 · 32, 22, 3 · 160, 176, 52, 4 … Σ 1, 8, 57, 392, 2633, 17376, 113105, …
  (−∞, 0]nonnegative
• A038330 → basis A013611 → (θ+1)²
  256, 2048, 2304 · 4096, 49152, 110592, 65536 … Σ 4608, 229376, 10485760, 452984832, …
  (−∞, 0]nonnegative
• A038220 → Veronese 2 → x ↦ x+1
  1 · 3 · 13, 4 · 63, 36 … Σ 1, 3, 17, 99, 577, 3363, 19601, 114243, …
  (−∞, −1]nonnegative
• A019538 → x ↦ x+1
  1 · 3, 2 · 13, 18, 6 · 75, 158, 108, 24 … Σ 1, 5, 37, 365, 4501, 66605, 1149877, …
  (−2, −1)nonnegative
• A132393 → basis A013610 → θ+1
  1 · 1, 6 · 2, 18, 27 · 6, 66, 162, 108 … Σ 1, 7, 47, 342, 2754, 24552, 241128, …
  (−∞, 0]nonnegative
• A272866 → basis A013611 → D+1
  1 · 25, 52, 16 · 225, 1320, 2480, 1664, 256 … Σ 1, 93, 5945, 331157, 17161329, …
  (−∞, 0]nonnegative
• A114687 → θ+1 → reverse
  1 · 4, 1 · 12, 12, 1 · 32, 72, 24, 1 … Σ 1, 5, 25, 129, 681, 3653, 19825, …
  (−∞, 0]nonnegative
• A091042 → reverse → x ↦ x+1
  1 · 4, 1 · 16, 12, 1 · 64, 80, 24, 1 … Σ 1, 5, 29, 169, 985, 5741, 33461, …
  (−∞, −1]nonnegative
• A038330 → basis A007318 → basis A007318
  400, 160, 16 · 8000, 4800, 960, 64 … Σ 576, 13824, 331776, 7962624, 191102976, …
  (−∞, −1]nonnegativerational roots
• A102547 → basis A013609 → x ↦ x+1
  1 · 1 · 1 · 4, 2 … Σ 1, 1, 1, 6, 11, 16, 46, 101, …
  (−∞, −1]nonnegative
• A059364 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 19, 15 · 24, 118, 193, 105 … Σ 1, 5, 40, 440, 6160, 104720, 2094400, …
  [−1, 0]nonnegativerational roots
• A122542 → basis A013611 → D
  4 · 16, 32 · 52, 224, 192 · 160, 1024, 1920, 1024 … Σ 0, 4, 48, 468, 4128, 34388, 276048, …
  (−∞, 0]nonnegative
• A038207 → basis A008277 → x ↦ x/(x+1)
  1 · 3, 4 · 9, 25, 17 · 27, 118, 167, 77 … Σ 1, 7, 51, 389, 3121, 26423, 236403, …
  [−1, 0]nonnegative
• basis-map coefficients of A026729 → basis A013609 → reverse
  1 · 2, 1 · 4, 4, 1 · 8, 12, 6, 1 … Σ 1, 3, 9, 27, 81, 243, 729, 2187, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A297191, shifted → θ+1 → D
  0, 6 · 0, 0, 0, 20 · 0, 0, 0, 0, 0, 42 … Σ 0, 6, 20, 42, 72, 110, 156, 210, …
  (−∞, 0]nonnegativerational roots
• A038281 → D
  48, 18 · 576, 432, 81 · 6144, 6912, 2592, 324 … Σ 66, 1089, 15972, 219615, 2898918, …
  (−∞, 0]nonnegativerational roots
• A102413: Triangle read by rows: T(n,k) is the number of k-matchings in the n-sunlet graph (0 <= k <= n).
  1 · 1, 1 · 1, 4, 1 · 1, 6, 6, 1 … Σ 1, 2, 6, 14, 34, 82, 198, 478, …
  (−∞, 0)nonnegative
• basis-map coefficients of A021012 → θ+1 → D
  -2 · 0, 12 · 0, 0, -72 · 0, 0, 0, 480 … Σ 0, -2, 12, -72, 480, -3600, 30240, …
  (−∞, 0]rational roots
• A143494 → reverse → D+1
  1 · 3, 2 · 6, 13, 4 · 10, 47, 43, 8 … Σ 1, 5, 23, 108, 534, 2803, 15623, 92249, …
  (−∞, 0]nonnegative
• A136124 → x ↦ x+1 → θ+1
  1 · 3, 2 · 12, 14, 3 · 60, 94, 36, 4 … Σ 1, 5, 29, 194, 1478, 12664, 120696, …
  (−∞, 0]nonnegative
• A257619 → x ↦ x+1
  1 · 4, 2 · 52, 52, 4 · 1144, 1716, 588, 8 … Σ 1, 6, 108, 3456, 154872, 8940456, …
  (−∞, −1]nonnegative
• A081577 → basis A013611 → x ↦ x/(x+1)
  1 · 2, 6 · 6, 36, 46 · 16, 144, 400, 336 … Σ 1, 8, 88, 896, 9280, 95744, 988672, …
  [−1, 0]nonnegative
• A386789 → D
  2 · 3, 12 · 4, 40, 60 · 5, 90, 315, 280 … Σ 0, 2, 15, 104, 690, 4458, 28301, …
  (−∞, 0]nonnegative
• A272866 → basis A007318 → reverse
  1 · 1, 5, 5 · 1, 10, 35, 50, 25 … Σ 1, 11, 121, 1331, 14641, 161051, …
  (−∞, 0]nonnegative
• A114656 → θ+1
  1 · 2, 2 · 4, 12, 3 · 8, 48, 36, 4 … Σ 1, 4, 19, 96, 501, 2668, 14407, 78592, …
  (−∞, 0]nonnegative
• A038323 → basis A013610 → D
  27 · 1080, 1458 · 32400, 87480, 59049 … Σ 0, 27, 2538, 178929, 11212884, …
  (−∞, 0]nonnegativerational roots
• A257609 → x ↦ x/(x+1)
  1 · 2, 4 · 4, 24, 24 · 8, 112, 288, 192 … Σ 1, 6, 52, 600, 8656, 149856, 3026752, …
  [−1, 0]nonnegative
• A257612 → D
  2 · 24, 8 · 184, 368, 24 · 1216, 7360, 3648, 64 … Σ 0, 2, 32, 576, 12288, 307200, 8847360, …
  (−∞, 0]nonnegative
• A207823 → reverse → θ+1
  1 · 1, 8 · 1, 16, 45 · 1, 24, 138, 224 … Σ 1, 9, 62, 387, 2285, 13012, 72233, …
  (−∞, 0]nonnegative
• A225470 → basis A013610 → D+1
  1 · 6, 3 · 45, 45, 9 · 459, 621, 243, 27 … Σ 1, 9, 99, 1350, 22194, 428652, 9526572, …
  (−∞, 0]nonnegative
• A137320 → θ+1
  1 · 0, 4 · 0, 4, 12 · 0, 8, 36, 32 … Σ 1, 4, 16, 76, 428, 2808, 21096, 178848, …
  (−∞, 0]nonnegative
• A038221 → basis A013610
  1 · 6, 9 · 36, 108, 81 · 216, 972, 1458, 729 … Σ 1, 15, 225, 3375, 50625, 759375, …
  [−1, 0]nonnegativerational roots
• A122542 → basis A013609 → D+1
  1 · 3, 2 · 11, 16, 4 · 33, 82, 52, 8 … Σ 1, 5, 31, 175, 961, 5149, 27103, …
  (−∞, 0]nonnegative