Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• A013613 → x ↦ x+1 → x ↦ x/(x+1)
  1 · 7, 13 · 49, 182, 169 · 343, 1911, 3549, 2197 … Σ 1, 20, 400, 8000, 160000, 3200000, …
  [−1, 0]nonnegativerational roots
• A106246 → basis A007318 → reverse
  1 · 1, 3 · 1, 6, 6 · 1, 9, 18, 10 … Σ 1, 4, 13, 38, 104, 272, 688, 1696, …
  (−∞, 0]nonnegative
• A038221 → θ+1
  1 · 3, 6 · 9, 36, 27 · 27, 162, 243, 108 … Σ 1, 9, 72, 540, 3888, 27216, 186624, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A371080, shifted → basis A013610 → x ↦ x/(x+1)
  1 · 1, 4 · 4, 23, 28 · 28, 201, 426, 280 … Σ 1, 5, 55, 935, 21505, 623645, 21827575, …
  [−1, 0]nonnegativerational roots
• A038322 → basis A013610 → x ↦ x/(x+1)
  1 · 19, 43 · 361, 1634, 1849 … Σ 1, 62, 3844, 238328, 14776336, …
  [−1, 0]nonnegativerational roots
• A114687 → θ+1 → (θ+1)²
  1 · 1, 16 · 1, 48, 108 · 1, 96, 648, 512 … Σ 1, 17, 157, 1257, 9441, 68153, 478381, …
  (−∞, 0]nonnegative
• A121574 → x ↦ x+1 → D
  1 · 7, 2 · 35, 22, 3 · 155, 152, 45, 4 … Σ 0, 1, 9, 60, 356, 1980, 10572, 54880, …
  (−∞, 0]nonnegative
• Hadamard-factor parent of A155495 → (θ+1)²
  1, 24, 9 · 1, 60, 135, 16 · 1, 112, 630, 448, 25 … Σ 34, 212, 1216, 6592, 34304, 173056, …
  (−∞, 0]nonnegative
• A360282 → basis A007318
  1 · 1, 1 · 5, 7, 2 · 25, 43, 21, 3 … Σ 1, 2, 14, 92, 614, 4172, 28748, 200216, …
  (−∞, −1]nonnegative
• A132812 → basis A007318 → θ+1
  1 · 4, 4 · 15, 30, 9 · 56, 168, 108, 16 … Σ 1, 8, 54, 348, 2195, 13668, 84364, …
  (−∞, 0]nonnegative
• A008277 → θ+1 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 8, 10 · 1, 17, 49, 37 … Σ 1, 4, 19, 104, 641, 4380, 32803, …
  [−1, 0]nonnegative
• A232535 → D
  2 · 8, 6 · 18, 50, 12 · 32, 196, 168, 20 … Σ 0, 2, 14, 80, 416, 2048, 9728, 45056, …
  (−∞, 0]nonnegative
• A196347 → basis A013609 → D
  2 · 16, 16 · 144, 288, 144 · 1536, 4608, 4608, 1536 … Σ 0, 2, 32, 576, 12288, 307200, 8847360, …
  (−∞, 0]nonnegativerational roots
• A039755 → reverse → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 24, 38, 16, 1 … Σ 1, 3, 13, 79, 601, 5339, 53861, 607527, …
  (−∞, −1]nonnegative
• basis-map coefficients of A113214, shifted → D → D
  2 · -6, -6 · 12, 24, 12 · -20, -60, -60, -20 … Σ 0, 0, 2, -12, 48, -160, 480, -1344, …
  (−∞, 0]rational roots
• A038291 → basis A013611 → D
  4 · 80, 32 · 1200, 960, 192 … Σ 0, 4, 112, 2352, 43904, 768320, …
  (−∞, 0]nonnegativerational roots
• A143491 → basis A008277 → D
  8, 2 · 60, 30, 3 · 480, 360, 72, 4 … Σ 10, 93, 916, 9805, 114486, 1454425, …
  (−∞, 0]nonnegative
• A116412 → basis A007318 → x ↦ x+1
  1 · 5, 1 · 22, 10, 1 · 98, 69, 15, 1 … Σ 1, 6, 33, 183, 1014, 5619, 31137, …
  (−∞, −1]nonnegative
• A169803 → x ↦ x+1 → θ+1
  1 · 2, 2 · 3, 4 · 5, 10, 3 … Σ 1, 4, 7, 18, 37, 84, 179, 390, …
  (−∞, 0]nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
  100, 240, 144 · 1000, 3600, 4320, 1728 … Σ 484, 10648, 234256, 5153632, 113379904, …
  (−∞, 0)nonnegativerational roots
• A013620 → basis A013611 → x ↦ x+1
  1 · 17, 12 · 289, 408, 144 · 4913, 10404, 7344, 1728 … Σ 1, 29, 841, 24389, 707281, 20511149, …
  (−∞, −1]nonnegativerational roots
• A027467 → basis A013610 → x ↦ x/(x+1)
  1 · 16, 19 · 256, 608, 361 · 4096, 14592, 17328, 6859 … Σ 1, 35, 1225, 42875, 1500625, 52521875, …
  [−1, 0]nonnegativerational roots
• A063967 → x ↦ x−9
  1 · -8, 1 · 56, -15, 1 · -384, 160, -22, 1 … Σ 1, -7, 42, -245, 1421, -8232, 47677, …
  (4, 8]
• A225473 → D+1
  1 · 7, 4 · 49, 104, 32 · 343, 1660, 1824, 384 … Σ 1, 11, 185, 4211, 120497, 4148891, …
  (−∞, 0]nonnegative
• A225469 → x ↦ x/(x+1)
  1 · 3, 4 · 9, 28, 20 · 27, 160, 260, 128 … Σ 1, 7, 57, 575, 7249, 109463, 1892169, …
  [−1, 0]nonnegative
• narayana
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … Σ 1, 2, 5, 14, 42, 132, 429, 1430, …
  (−∞, 0)nonnegative
• basis-map coefficients of A021012 → basis A008277 → θ+1
  1 · -1, -2 · 2, 12, 6 · -6, -84, -108, -24 … Σ 1, -3, 20, -222, 3624, -80880, 2349360, …
  (−∞, 0]
• A126216 → basis A007318 → reverse
  1 · 1, 3 · 1, 7, 11 · 1, 12, 42, 45 … Σ 1, 4, 19, 100, 562, 3304, 20071, …
  (−∞, 0]nonnegative
• delannoy → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 12, 18, 8, 1 … Σ 1, 3, 11, 39, 139, 495, 1763, 6279, …
  (−∞, −1]nonnegative
• basis-map coefficients of A026729, shifted → θ+1 → D+1
  1 · -2, -2 · 0, 6, 3 · 0, 0, -12, -4 … Σ 1, -4, 9, -16, 25, -36, 49, -64, …
  (−∞, 0]rational roots
• A130534 → D²∘x (shift +1) → D
  3 · 18, 12 · 105, 120, 30 · 675, 1020, 450, 60 … Σ 0, 3, 30, 255, 2205, 20307, 201852, …
  (−∞, 0]nonnegative
• A207606 → D
  7, 4 · 16, 22, 6 · 30, 72, 45, 8 … Σ 11, 44, 155, 510, 1608, 4924, 14761, …
  (−∞, 0]nonnegative
• A207607 → basis A007318 → D+1
  1 · 5, 2 · 17, 13, 2 · 54, 63, 21, 2 … Σ 1, 7, 32, 140, 599, 2521, 10474, 43066, …
  (−∞, 0]nonnegative
• A028246 → (θ+1)²
  1 · 1, 4 · 1, 12, 18 · 1, 28, 108, 96 … Σ 1, 5, 31, 233, 2071, 21305, 249271, …
  (−∞, 0]nonnegative
• A154986 → θ+1
  1 · 1, 2 · 1, 8, 3 · 1, 22, 33, 4 … Σ 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  (−∞, 0]nonnegative
• A038255 → basis A013609 → reverse
  1 · 2, 7 · 4, 28, 49 · 8, 84, 294, 343 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, 0]nonnegativerational roots
• A178822 → x ↦ x+1 → basis A013609
  1 · 18, 12 · 189, 252, 84 · 1512, 3024, 2016, 448 … Σ 1, 30, 525, 7000, 78750, 787500, …
  (−∞, 0]nonnegativerational roots
• A114687 → reverse → x ↦ x+1
  1 · 3, 1 · 11, 8, 1 · 45, 51, 15, 1 … Σ 1, 4, 20, 112, 672, 4224, 27456, …
  (−∞, −1]nonnegative
• A169803 → Veronese 2 (j=1) → x ↦ x+1
  1 · 2 · 3 · 4 … Σ 0, 1, 2, 3, 4, 7, 14, 27, …
  (−∞, −1]nonnegative
• A102537: Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k); n, k >= 1.
  1 · 1, 3 · 1, 8, 12 · 1, 15, 55, 55 … Σ 1, 4, 21, 126, 818, 5594, 39693, …
  (−∞, 0)nonnegative
• A152198 → basis A013609 → D
  2 · 2 · 8, 8 · 8, 8 … Σ 0, 0, 2, 2, 16, 16, 96, 96, …
  (−∞, 0]nonnegativerational roots
• A008459 → basis A013610 → x ↦ x+1
  1 · 5, 3 · 33, 36, 9 · 245, 387, 189, 27 … Σ 1, 8, 78, 848, 9766, 116208, 1411404, …
  (−∞, −1]nonnegative
• A038208 → Veronese 2 → (θ+1)²
  1 · 2 · 4, 16 · 8, 96 … Σ 1, 2, 20, 104, 544, 2752, 13568, 65536, …
  (−∞, 0]nonnegative
• A013609 → D → D+1
  2 · 12, 8 · 30, 72, 24 · 56, 240, 288, 64 … Σ 0, 2, 20, 126, 648, 2970, 12636, 51030, …
  (−∞, 0]nonnegativerational roots
• A063967 → basis A013611 → x ↦ x/(x+1)
  1 · 2, 6 · 6, 32, 42 · 16, 128, 336, 288 … Σ 1, 8, 80, 768, 7424, 71680, 692224, …
  [−1, 0]nonnegative
• A395972 → reverse
  8, 16, 4 · 48, 144, 72, 8 · 384, 1536, 1152, 256, 16 … Σ 28, 272, 3344, 49472, 852928, 16758016, …
  (−∞, 0]nonnegative
• A007318 → −D²+(−1+n)D+1+n → x ↦ x+1 → D+1
  1 · 6, 2 · 28, 20, 3 · 110, 126, 42, 4 … Σ 1, 8, 51, 282, 1413, 6588, 29079, …
  (−∞, 0]nonnegative
• A086645 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 8, 40, 41 · 32, 240, 546, 365 … Σ 1, 7, 89, 1183, 15761, 210007, 2798249, …
  [−1, 0]nonnegative
• A225470 → D+1
  1 · 3, 1 · 17, 9, 1 · 146, 96, 18, 1 … Σ 1, 4, 27, 261, 3294, 51354, 953532, …
  (−∞, 0]nonnegative
• A013612 → basis A013611 → x ↦ x/(x+1)
  36, 312, 676 · 216, 2808, 12168, 17576 … Σ 1024, 32768, 1048576, 33554432, …
  [−1, 0]nonnegativerational roots