Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A128099 → x ↦ x+1 → D
  2 · 4 · 14, 8 · 32, 24 … Σ 0, 0, 2, 4, 22, 56, 202, 556, …
  (−∞, −1]nonnegative
• A391533 → x ↦ x+1 → reverse
  1, 6, 9 · 1, 8, 21, 18 · 1, 10, 37, 60, 36 … Σ 16, 48, 144, 432, 1296, 3888, 11664, …
  (−∞, 0]nonnegativerational roots
• A038303 → θ+1
  1 · 10, 2 · 100, 40, 3 · 1000, 600, 90, 4 … Σ 1, 12, 143, 1694, 19965, 234256, …
  (−∞, 0]nonnegativerational roots
• A331333 → x ↦ x+1
  1 · 3, 2 · 14, 16, 4 · 86, 132, 60, 8 … Σ 1, 5, 34, 286, 2840, 32344, 414160, …
  (−∞, −1]nonnegative
• A038226 → basis A013611 → D+1
  825, 2752, 1024 · 12947, 79200, 132096, 32768 … Σ 4601, 257011, 13595697, 694016603, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A026729, shifted → D+1
  1 · -1, -1 · 0, 2, 1 · 0, 0, -3, -1 … Σ 1, -2, 3, -4, 5, -6, 7, -8, …
  (−∞, 0]rational roots
• A026729 → basis A013609 → x ↦ x+1
  1 · 3, 2 · 12, 14, 4 · 45, 78, 44, 8 … Σ 1, 5, 30, 175, 1025, 6000, 35125, …
  (−∞, −1]nonnegative
• A008957 → θ+1
  1 · 1, 2 · 1, 10, 3 · 1, 28, 63, 4 … Σ 1, 3, 14, 96, 847, 9335, 125211, …
  (−∞, 0]nonnegative
• A038305 → x ↦ x+1 → x ↦ x/(x+1)
  169, 416, 256 · 2197, 8112, 9984, 4096 … Σ 841, 24389, 707281, 20511149, …
  [−1, 0]nonnegativerational roots
• A049403 → (θ+1)²
  0, 12, 9 · 0, 12, 54, 16 · 0, 0, 135, 160, 25 … Σ 21, 82, 320, 1266, 5110, 21176, 89892, …
  (−∞, 0]nonnegative
• A078812 → basis A013609 → D+1
  1 · 5, 2 · 20, 20, 4 · 71, 122, 60, 8 … Σ 1, 7, 44, 261, 1491, 8296, 45269, …
  (−∞, 0]nonnegative
• A178822 → Veronese 2 (j=1) → D+1
  6 · 42 · 224, 56 · 1008, 504 … Σ 0, 6, 42, 280, 1512, 7056, 29568, …
  (−∞, 0]nonnegative
• A038243 → basis A013611
  1 · 6, 4 · 36, 48, 16 · 216, 432, 288, 64 … Σ 1, 10, 100, 1000, 10000, 100000, …
  (−∞, −1]nonnegativerational roots
• A008955 → θ+1
  1 · 1, 2 · 1, 10, 12 · 1, 28, 147, 144 … Σ 1, 3, 23, 320, 7040, 225540, 9936180, …
  (−∞, 0]nonnegative
• A099093 → reverse → (θ+1)²
  1 · 3 · 9, 12 · 27, 72 … Σ 1, 3, 21, 99, 486, 2268, 10395, 46737, …
  (−∞, 0]nonnegative
• A257621 → θ+1
  1 · 3, 6 · 9, 84, 27 · 27, 786, 1179, 108 … Σ 1, 9, 120, 2100, 45360, 1164240, …
  (−∞, 0]nonnegative
• A327997 → θ+1
  1 · 3, 2 · 9, 14, 3 · 27, 76, 36, 4 … Σ 1, 5, 26, 143, 845, 5437, 38466, …
  (−∞, 0]nonnegative
• A035607 → x ↦ x+1
  1 · 1 · 3, 2 · 7, 8, 2 … Σ 1, 1, 5, 17, 61, 217, 773, 2753, …
  (−∞, −1]nonnegative
• A143491 → basis A008277 → reverse
  1, 8, 12 · 1, 15, 60, 60 · 1, 24, 180, 480, 360 … Σ 21, 136, 1045, 9276, 93289, 1047376, …
  (−∞, 0]nonnegative
• A121574 → basis A013609 → D
  2 · 14, 8 · 70, 88, 24 · 310, 608, 360, 64 … Σ 0, 2, 22, 182, 1342, 9284, 61684, …
  (−∞, 0]nonnegative
• A132393 → reverse → D
  1 · 3, 4 · 6, 22, 18 · 10, 70, 150, 96 … Σ 0, 0, 1, 7, 46, 326, 2556, 22212, …
  (−∞, 0]nonnegative
• A358735 → reverse
  1 · 1, 1 · 2, 4, 1 · 6, 16, 10, 1 … Σ 1, 2, 7, 33, 191, 1304, 10241, 90865, …
  (−∞, 0]nonnegative
• A046739 → θ+1
  1, 14, 3 · 1, 42, 63, 4 · 1, 102, 483, 204, 5 … Σ 18, 110, 795, 6489, 59332, 600732, …
  (−∞, 0]nonnegative
• A038207 → basis A008277
  1 · 3, 1 · 9, 7, 1 · 27, 37, 12, 1 … Σ 1, 4, 17, 77, 372, 1915, 10481, 60814, …
  (−∞, 0]nonnegative
• A099040 → x ↦ x−2
  1 · -4, 2 · 12, -14, 4 · -32, 64, -40, 8 … Σ 1, -2, 2, 0, -4, 8, -8, 0, …
  (0, 2]
• A059364 → basis A013610 → reverse
  1 · 3, 3 · 18, 33, 15 · 162, 423, 366, 105 … Σ 1, 6, 66, 1056, 22176, 576576, …
  (−∞, 0]nonnegativerational roots
• A114656 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 14, 11 · 8, 48, 84, 45 … Σ 1, 5, 29, 185, 1257, 8925, 65445, …
  [−1, 0]nonnegative
• A038301 → reverse
  121, 198, 81 · 1331, 3267, 2673, 729 … Σ 400, 8000, 160000, 3200000, 64000000, …
  (−∞, 0]nonnegativerational roots
• A130534 → Veronese 2 (j=1) → (θ+1)²
  1 · 3 · 11, 4 · 50, 40 … Σ 0, 1, 3, 15, 90, 623, 4893, 43058, …
  (−∞, 0]nonnegative
• delannoy → x ↦ x+1 → D+1
  1 · 3, 1 · 10, 7, 1 · 30, 34, 11, 1 … Σ 1, 4, 18, 76, 314, 1272, 5078, 20036, …
  (−∞, 0]nonnegative
• A152198: Triangle read by rows, A007318 rows repeated.
  1 · 1 · 1, 1 · 1, 1 … Σ 1, 1, 2, 2, 4, 4, 8, 8, …
  (−∞, 0]nonnegativerational roots+1
• A136572 → basis A013610
  1 · 1, 3 · 2, 12, 18 · 6, 54, 162, 162 … Σ 1, 4, 32, 384, 6144, 122880, 2949120, …
  [−1, 0]nonnegativerational roots
• A196347 → x ↦ x+1 → (θ+1)²
  1 · 2, 4 · 8, 32, 18 · 48, 288, 324, 96 … Σ 1, 6, 58, 756, 12312, 239760, 5423760, …
  (−∞, 0]nonnegative
• A103327 → D → D+1
  1 · 12, 2 · 77, 48, 3 · 336, 468, 120, 4 … Σ 0, 1, 14, 128, 928, 5888, 34304, …
  (−∞, 0]nonnegative
• A027465 → basis A008277
  1 · 4, 1 · 16, 9, 1 · 64, 61, 15, 1 … Σ 1, 5, 26, 141, 799, 4736, 29371, …
  (−∞, 0]nonnegative
• A166972 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 12, 12 · 1, 86, 210, 126 … Σ 1, 3, 25, 423, 10045, 326931, 14166385, …
  [−1, 0]nonnegative
• A102547 → basis A013610 → x ↦ x+1
  1 · 1 · 1 · 5, 3 … Σ 1, 1, 1, 8, 15, 22, 78, 183, …
  (−∞, −1]nonnegative
• A367270 → basis A013611 → D+1
  1 · 1 · 11, 8 · 58, 144, 48 … Σ 1, 1, 19, 250, 2979, 34206, 385646, …
  (−∞, 0]nonnegative
• A257623: Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.
  1 · 3, 3 · 9, 48, 9 · 27, 501, 501, 27 … Σ 1, 6, 66, 1056, 22176, 576576, …
  (−∞, 0)nonnegative
• A166344 → D
  6, 2 · 15, 30, 3 · 34, 180, 102, 4 … Σ 8, 48, 320, 2400, 20160, 188160, …
  (−∞, 0]nonnegative
• A086645 → basis A013611 → θ+1
  1 · 2, 8 · 8, 64, 48 · 32, 384, 864, 256 … Σ 1, 10, 120, 1536, 19328, 237056, …
  (−∞, 0]nonnegative
• A209414 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 10, 26, 18 … Σ 1, 3, 13, 55, 233, 987, 4181, 17711, …
  [−1, 0]nonnegative
• basis-map coefficients of A026729 → basis A013609 → D+1
  1 · 3, 2 · 5, 12, 4 · 7, 30, 36, 8 … Σ 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  (−∞, 0]nonnegativerational roots
• A028326 → basis A007318
  2 · 4, 2 · 8, 8, 2 · 16, 24, 12, 2 … Σ 2, 6, 18, 54, 162, 486, 1458, 4374, …
  (−∞, −1]nonnegativerational roots
• A257609: Triangle read by rows: 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) = 2*x + 2.
  1 · 2, 2 · 4, 16, 4 · 8, 88, 88, 8 … Σ 1, 4, 24, 192, 1920, 23040, 322560, …
  (−∞, 0)nonnegative
• A013620 → basis A008288 → x ↦ x+1
  1 · 8, 3 · 73, 57, 9 · 674, 792, 270, 27 … Σ 1, 11, 139, 1763, 22363, 283667, …
  (−∞, −1]nonnegative
• A200536 → basis A013610 → D+1
  1 · 27, 57, 18 · 288, 1566, 2925, 2052, 324 … Σ 1, 102, 7155, 437400, 24877125, …
  (−∞, 0]nonnegative
• A126216 → basis A013610 → D
  3 · 21, 18 · 126, 216, 81 · 726, 1872, 1458, 324 … Σ 0, 3, 39, 423, 4380, 44670, 453213, …
  (−∞, 0]nonnegative
• A038327 → basis A013611 → x ↦ x+1
  1 · 17, 4 · 289, 136, 16 · 4913, 3468, 816, 64 … Σ 1, 21, 441, 9261, 194481, 4084101, …
  (−∞, −1]nonnegativerational roots
• A106246 → θ+1
  1 · 2, 2 · 1, 8, 3 · 0, 6, 18, 4 … Σ 1, 4, 12, 28, 55, 96, 154, 232, …
  (−∞, 0]nonnegative