Proof methods / preserver_image

Image under a real-rootedness preserver

1,387 sequences

• A033282 → θ+1
  1 · 1, 4 · 1, 10, 15 · 1, 18, 63, 56 … Σ 1, 5, 26, 138, 743, 4043, 22180, …
  (−∞, 0]nonnegative
• A003506: Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.
  1 · 2, 2 · 3, 6, 3 · 4, 12, 12, 4 … Σ 1, 4, 12, 32, 80, 192, 448, 1024, …
  (−∞, 0)nonnegativerational roots
• basis-map coefficients of A132812 → D
  2 · 3, 6 · -12, 24, 12 · -20, -40, 90, 20 … Σ 0, 2, 9, 24, 50, 90, 147, 224, …
  (−∞, ∞)
• A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
  1 · 5, 1 · 25, 10, 1 · 125, 75, 15, 1 … Σ 1, 6, 36, 216, 1296, 7776, 46656, …
  (−∞, −1]nonnegativerational roots
• A091044 → θ+1
  1 · 2, 4 · 3, 20, 9 · 4, 56, 84, 16 … Σ 1, 6, 32, 160, 768, 3584, 16384, 73728, …
  (−∞, 0]nonnegative
• A145596 → D
  2 · 8, 6 · 20, 40, 12 · 40, 150, 120, 20 … Σ 0, 2, 14, 72, 330, 1430, 6006, 24752, …
  (−∞, 0]nonnegative
• A055356 → θ+1
  1 · 1 · 1, 2 · 1, 8, 6 … Σ 1, 1, 3, 15, 101, 851, 8603, 101423, …
  (−∞, 0]nonnegative
• A118884: Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0011 (n,k>=0).
  A318772 → x ↦ (1/48)x−1/48
  1 · 2 · 4 · 8 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, 1]nonnegative
• A003506 → x ↦ x+1
  1 · 4, 2 · 12, 12, 3 · 32, 48, 24, 4 … Σ 0, 1, 6, 27, 108, 405, 1458, 5103, …
  (−∞, −1]nonnegativerational roots
• A099040 → x ↦ x−4
  1 · -8, 2 · 56, -30, 4 · -384, 320, -88, 8 … Σ 1, -6, 30, -144, 684, -3240, 15336, …
  (2, 4]
• A128100: Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= f…
  1 · 1 · 2, 1 · 3, 2 … Σ 1, 1, 3, 5, 11, 21, 43, 85, …
  (−∞, −1]nonnegative
• A173018 → reverse (shift +1) → reverse
  1 · 1 · 1, 1 · 1, 4, 1 … Σ 1, 1, 2, 6, 24, 120, 720, 5040, …
  (−∞, 0]nonnegative
• A007318 → ×(−(1/2)n+(1/2)n²) → D
  2, 2 · 9, 18, 9 · 24, 72, 72, 24 … Σ 0, 0, 4, 36, 192, 800, 2880, 9408, …
  (−∞, 0]nonnegativerational roots
• A132774 → x ↦ x/(x+1)
  1 · 2, 5 · 0, 4, 9 · 0, 0, 6, 13 … Σ 1, 7, 13, 19, 25, 31, 37, 43, …
  [−1, 0]nonnegativerational roots
• A113413 → x ↦ x/(x+1)
  1 · 2, 3 · 2, 8, 7 · 2, 14, 28, 17 … Σ 1, 5, 17, 61, 217, 773, 2753, 9805, …
  [−1, 0]nonnegative
• A075497 → reverse
  1 · 1, 2 · 1, 6, 4 · 1, 12, 28, 8 … Σ 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  (−∞, 0]nonnegative
• A039756 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 12, 34, 24 … Σ 1, 3, 13, 71, 457, 3355, 27509, 248127, …
  [−1, 0]nonnegative
• A075497 → x ↦ x+1
  1 · 3, 1 · 11, 8, 1 · 49, 55, 15, 1 … Σ 1, 4, 20, 120, 832, 6496, 56128, …
  (−∞, −1]nonnegative
• A130123 → (θ+1)²
  1 · 0, 8 · 0, 0, 36 · 0, 0, 0, 128 … Σ 1, 8, 36, 128, 400, 1152, 3136, 8192, …
  (−∞, 0]nonnegativerational roots
• A094305: Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).
  1 · 3, 3 · 6, 12, 6 · 10, 30, 30, 10 … Σ 1, 6, 24, 80, 240, 672, 1792, 4608, …
  (−∞, −1]nonnegativerational roots
• A114655 → D+1
  2 · 6, 2 · 20, 16, 2 · 64, 96, 30, 2 … Σ 2, 8, 38, 192, 1002, 5336, 28814, …
  (−∞, 0]nonnegative
• A114655 → θ+1
  2 · 4, 4 · 8, 24, 6 · 16, 96, 72, 8 … Σ 2, 8, 38, 192, 1002, 5336, 28814, …
  (−∞, 0]nonnegative
• A099097 → x ↦ x−12
  1 · -36, 3 · 1284, -215, 9 · -45792, 11520, -966, 27 … Σ 1, -33, 1078, -35211, 1150105, …
  (104/9, 12]
• A122431 → x ↦ x/(x+1)
  1, 6, 12, 8 · 0, 1, 6, 12, 8 · 0, 0, 1, 6, 12, 8 … Σ 27, 27, 27, 27, 27, 27, 27, 27, …
  [−1, 0]nonnegativerational roots
• A152842 → D+1
  1 · 4, 3 · 5, 10, 3 · 8, 37, 42, 9 … Σ 1, 7, 18, 96, 224, 1088, 2432, 11264, …
  (−∞, 0]nonnegative
• A059364: Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.
  1 · 2, 1 · 6, 7, 2 · 24, 46, 29, 6 … Σ 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, −1]nonnegativerational roots
• A059344 → D
  2 · 6 · 12, 24 · 20, 120 … Σ 0, 0, 2, 6, 36, 140, 750, 3402, …
  (−∞, 0]nonnegative
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewher…
  1 · 1, 2 · 1, 4, 3 · 1, 6, 9, 4 … Σ 1, 3, 8, 20, 48, 112, 256, 576, …
  (−∞, 0)nonnegativerational roots
• A249252 → (θ+1)²
  1 · 1, 16 · 1, 32, 72 · 1, 48, 288, 256 … Σ 1, 17, 105, 593, 3049, 14801, 68873, …
  (−∞, 0]nonnegative
• A133087 → θ+1
  1 · 2, 2 · 1, 4, 3 · 2, 10, 12, 4 … Σ 1, 4, 8, 28, 48, 160, 256, 832, …
  (−∞, 0]nonnegative
• A178822 → (θ+1)²
  1 · 6, 24 · 21, 168, 189 · 56, 672, 1512, 896 … Σ 1, 30, 378, 3136, 20160, 108864, …
  (−∞, 0]nonnegative
• A121574 → D
  1 · 5, 2 · 16, 16, 3 · 44, 74, 33, 4 … Σ 0, 1, 7, 35, 155, 642, 2550, 9841, …
  (−∞, 0]nonnegative
• A060920 → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 9, 20, 13 … Σ 1, 3, 11, 43, 171, 683, 2731, 10923, …
  (−∞, ∞)nonnegative
• A008288: Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
  1 · 1, 1 · 1, 3, 1 · 1, 5, 5, 1 … Σ 1, 2, 5, 12, 29, 70, 169, 408, …
  [−3−2√2, −3+2√2]nonnegative
• A104684 → D+1
  1 · 3, 1 · 12, 8, 1 · 50, 54, 15, 1 … Σ 1, 4, 21, 120, 705, 4188, 24997, …
  (−∞, 0]nonnegative
• A100861 → basis A007318
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 25, 81, 331, 1303, …
  (−∞, −1]nonnegative
• A062190 → reverse
  1 · 6, 1 · 21, 14, 1 · 56, 84, 24, 1 … Σ 1, 7, 36, 165, 715, 3003, 12376, 50388, …
  (−∞, 0]nonnegative
• A132774 → D+1
  1 · 5, 3 · 4, 14, 5 · 0, 12, 27, 7 … Σ 1, 8, 23, 46, 77, 116, 163, 218, …
  (−∞, 0]nonnegative
• A023531 → D+1
  1 · 1, 1 · 0, 2, 1 · 0, 0, 3, 1 … Σ 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 0]nonnegativerational roots
• A059473 → (θ+1)²
  1 · 2, 8 · 4, 40, 36 · 8, 128, 288, 128 … Σ 1, 10, 80, 552, 3508, 21080, 121664, …
  (−∞, 0]nonnegative
• A030528 → x ↦ x−3
  1 · -2, 1 · 3, -4, 1 · -3, 10, -6, 1 … Σ 1, -1, 0, 2, -4, 4, 0, -8, …
  (−1, 3]
• A008292 → θ+1 (shift +2)
  1, 8, 3 · 1, 22, 33, 4 · 1, 52, 198, 104, 5 … Σ 12, 60, 360, 2520, 20160, 181440, …
  (−∞, 0]nonnegative
• A062196 → D
  3 · 8, 12 · 15, 60, 30 · 24, 180, 240, 60 … Σ 0, 3, 20, 105, 504, 2310, 10296, 45045, …
  (−∞, 0]nonnegative
• A056939 → D+1
  1 · 2, 1 · 5, 6, 1 · 11, 30, 13, 1 … Σ 1, 3, 12, 55, 276, 1477, 8296, 48393, …
  (−∞, 0]nonnegative
• A066325 → D+1
  1 · 1, 1 · -1, 2, 1 · -3, -3, 3, 1 … Σ 1, 2, 2, -2, -10, -4, 52, 92, …
  (−∞, ∞)
• A125092 → D+1
  1 · 5, 4 · 9, 26, 9 · 13, 66, 75, 16 … Σ 1, 9, 44, 170, 576, 1792, 5248, 14688, …
  (−∞, 0]nonnegative
• A133087 → D
  1 · 2, 2 · 5, 8, 3 · 4, 12, 12, 4 … Σ 0, 1, 4, 16, 32, 112, 192, 640, …
  (−∞, 0]nonnegativerational roots
• A030528 → θ+1
  1 · 1, 2 · 0, 4, 3 · 0, 2, 9, 4 … Σ 1, 3, 7, 15, 30, 58, 109, 201, …
  (−∞, 0]nonnegative
• A126216 → basis A007318
  1 · 3, 1 · 11, 7, 1 · 45, 42, 12, 1 … Σ 1, 4, 19, 100, 562, 3304, 20071, …
  (−∞, −1]nonnegative
• delannoy → D+1
  1 · 2, 1 · 4, 5, 1 · 6, 15, 8, 1 … Σ 1, 3, 10, 30, 87, 245, 676, 1836, …
  (−∞, 0]nonnegative