Proof methods / preserver_image

Image under a real-rootedness preserver

1,387 sequences

• A108838 → θ+1
  2 · 3, 4 · 4, 16, 6 · 5, 40, 45, 8 … Σ 2, 7, 26, 98, 372, 1419, 5434, 20878, …
  (−∞, 0]nonnegative
• A152828 → basis A007318
  1 · 1 · 1 · 2, 1 … Σ 1, 1, 1, 3, 3, 3, 9, 9, …
  (−∞, −1]nonnegativerational roots
• A172431 → reverse
  1 · 2, 1 · 3, 4, 1 · 4, 10, 6, 1 … Σ 1, 3, 8, 21, 55, 144, 377, 987, …
  (−∞, 0]nonnegative
• A007318 → −D²+(−1+n)D+1+n → D+1
  1 · 4, 2 · 11, 14, 3 · 22, 54, 30, 4 … Σ 1, 6, 28, 110, 384, 1232, 3712, 10656, …
  (−∞, 0]nonnegative
• A135838: Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
  1 · 2, 2 · 2, 4, 2 · 4, 12, 12, 4 … Σ 1, 4, 8, 32, 64, 256, 512, 2048, …
  (−∞, 0)nonnegativerational roots
• A008276: Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n…
  1 · 1, -1 · 1, -3, 2 · 1, -6, 11, -6 … Σ 1, 0, 0, 0, 0, 0, 0, 0, …
  [0, ∞)rational roots
• A081581 → D+1
  1 · 2, 1 · 9, 10, 1 · 16, 45, 18, 1 … Σ 1, 3, 20, 80, 372, 1540, 6496, 26496, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729 → basis A007318
  1 · 1, 1 · 1, 2, 1 · 1, 3, 3, 1 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A021012 → x ↦ x+1
  1 · -1, -1 · 2, 4, 2 · -6, -18, -18, -6 … Σ 1, -2, 8, -48, 384, -3840, 46080, …
  (−∞, −1]rational roots
• A143685 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 13, 13 · 1, 24, 66, 44 … Σ 1, 3, 27, 135, 891, 5103, 31347, …
  [−1, 0]nonnegative
• A249251 → D
  3 · 6, 12 · 9, 42, 36 · 12, 90, 180, 96 … Σ 0, 3, 18, 87, 378, 1548, 6102, 23421, …
  (−∞, 0]nonnegative
• A178820: Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
  1 · 4, 4 · 10, 20, 10 · 20, 60, 60, 20 … Σ 1, 8, 40, 160, 560, 1792, 5376, 15360, …
  (−∞, 0)nonnegativerational roots
• A001263 → D
  1 · 3, 2 · 6, 12, 3 · 10, 40, 30, 4 … Σ 0, 1, 5, 21, 84, 330, 1287, 5005, …
  (−∞, 0]nonnegative
• A062264 → x ↦ x+1
  1 · 6, 5 · 28, 42, 15 · 120, 252, 168, 35 … Σ 1, 11, 85, 575, 3649, 22363, 134245, …
  (−∞, −1]nonnegative
• A082985 → D+1
  1 · 4, 3 · 6, 15, 5 · 8, 35, 35, 7 … Σ 1, 7, 26, 85, 265, 804, 2393, 7019, …
  (−∞, 0]nonnegative
• A113953 → x ↦ x−9
  1 · -9, 1 · 63, -16, 1 · -405, 171, -23, 1 … Σ 1, -8, 48, -256, 1280, -6144, 28672, …
  (1, 9]
• A099097 → basis A007318
  1 · 3, 3 · 10, 19, 9 · 33, 93, 87, 27 … Σ 1, 6, 38, 240, 1516, 9576, 60488, …
  (−∞, −1]nonnegative
• A008277 → θ+2 (shift +1) → reverse
  2 · 3, 2 · 4, 9, 2 · 5, 24, 21, 2 … Σ 2, 5, 15, 52, 203, 877, 4140, 21147, …
  (−∞, 0]nonnegative
• A207606 → x ↦ x/(x+1)
  4, 15, 13 · 5, 31, 58, 34 · 6, 54, 162, 201, 89 … Σ 32, 128, 512, 2048, 8192, 32768, …
  [−1, 0)nonnegative
• A085880: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
  1 · 1, 1 · 2, 4, 2 · 5, 15, 15, 5 … Σ 1, 2, 8, 40, 224, 1344, 8448, 54912, …
  (−∞, 0)nonnegativerational roots
• A035607 → D+1
  1 · 1 · 3, 2 · 5, 8, 2 … Σ 1, 1, 5, 15, 45, 129, 361, 991, …
  (−∞, 0]nonnegative
• A135065 → θ+1
  1 · 4, 8 · 9, 36, 27 · 16, 96, 144, 64 … Σ 1, 12, 72, 320, 1200, 4032, 12544, …
  (−∞, 0]nonnegativerational roots
• A352880 → x ↦ x/(x+1)
  8 · 12, 30 · 16, 96, 112 · 20, 210, 560, 420 … Σ 8, 42, 224, 1210, 6600, 36274, 200576, …
  [−1, 0]nonnegative
• A135065 → x ↦ x/(x+1)
  1 · 4, 8 · 9, 36, 36 · 16, 96, 192, 128 … Σ 1, 12, 81, 432, 2025, 8748, 35721, …
  [−1, 0]nonnegativerational roots
• A122076 → D
  2 · 8, 4 · 30, 30, 6 · 104, 160, 72, 8 … Σ 0, 2, 12, 66, 344, 1710, 8196, 38234, …
  (−∞, ∞)nonnegative
• A118933 → x ↦ x+1
  1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 13, 61, 181, 421, …
  (−∞, −1]nonnegative
• A108838 → D
  2 · 8, 4 · 20, 30, 6 · 40, 120, 72, 8 … Σ 0, 2, 12, 56, 240, 990, 4004, 16016, …
  (−∞, 0]nonnegative
• A099097 → (θ+1)²
  1 · 0, 12 · 0, 4, 81 · 0, 0, 54, 432 … Σ 1, 12, 85, 486, 2466, 11592, 51667, …
  (−∞, 0]nonnegative
• A132393 → reverse
  1 · 1 · 1, 1 · 1, 3, 2 … Σ 1, 1, 2, 6, 24, 120, 720, 5040, …
  (−∞, 0]nonnegativerational roots
• A143495 → θ+1
  1 · 3, 2 · 9, 14, 3 · 27, 74, 36, 4 … Σ 1, 5, 26, 141, 799, 4736, 29371, …
  (−∞, 0]nonnegative
• narayana → (θ+1)²
  1 · 1, 4 · 1, 12, 9 · 1, 24, 54, 16 … Σ 1, 5, 22, 95, 406, 1722, 7260, 30459, …
  (−∞, 0]nonnegative
• A249139: Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 3, 1 · 5, 2 · 11, 7, 1 … Σ 1, 4, 7, 19, 40, 97, 217, 508, …
  (−∞, −2]nonnegative
• A111125 → x ↦ x/(x+1)
  1 · 3, 4 · 5, 15, 11 · 7, 35, 56, 29 … Σ 1, 7, 31, 127, 511, 2047, 8191, 32767, …
  [−1, 0)nonnegative
• basis-map coefficients of A113214, shifted → D+1
  1 · -2, -1 · 3, 4, 1 · -4, -9, -6, -1 … Σ 1, -3, 8, -20, 48, -112, 256, -576, …
  (−∞, 0]rational roots
• A054142 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 8, 19, 13 … Σ 1, 3, 11, 41, 153, 571, 2131, 7953, …
  [−1, 0]nonnegative
• A099097 → θ+1
  1 · 0, 6 · 0, 2, 27 · 0, 0, 18, 108 … Σ 1, 6, 29, 126, 516, 2034, 7807, 29382, …
  (−∞, 0]nonnegative
• A048854 → x ↦ −2x (shift +2)
  3, -6, 1 · -15, 45, -15, 1 · 105, -420, 210, -28, 1 … Σ -2, 16, -132, 1216, -12440, 138048, …
  [0, ∞)
• A046716 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 9, 23, 16 … Σ 1, 3, 11, 49, 261, 1631, 11743, 95901, …
  [−1, 0]nonnegative
• A081581 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 10, 10 · 1, 18, 48, 32 … Σ 1, 3, 21, 99, 549, 2835, 15093, 79299, …
  [−1, 0]nonnegative
• A081580 → θ+1
  1 · 1, 2 · 1, 14, 3 · 1, 26, 39, 4 … Σ 1, 3, 18, 70, 303, 1197, 4756, 18396, …
  (−∞, 0]nonnegative
• A064861 → θ+1
  1, 6, 6 · 1, 10, 24, 16 · 1, 12, 39, 48, 20 … Σ 13, 51, 120, 432, 972, 3348, 7344, …
  (−∞, 0]nonnegative
• type_b_eulerian → γ (scale 1/4) → D
  1 · 5 · 18, 10 · 58, 122 … Σ 0, 0, 1, 5, 28, 180, 1320, 10920, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/2) (shift +1) → D+1
  1 · 1 · 2, 1 · 5, 4 … Σ 1, 1, 3, 9, 35, 155, 791, 4529, …
  (−∞, 0]nonnegative
• A232535 → (θ+1)²
  1 · 1, 8 · 1, 32, 27 · 1, 72, 225, 64 … Σ 1, 9, 60, 362, 2032, 10848, 55808, …
  (−∞, 0]nonnegative
• A132013: T(n,j) for an iterated mixed order Laguerre transform. Coefficients of the normalized generalized Laguerre polynomials (-1)^n*n!*L(n,1-n,x).
  1 · -1, 1 · 0, -2, 1 · 0, 0, -3, 1 … Σ 1, 0, -1, -2, -3, -4, -5, -6, …
  (−∞, ∞)rational roots
• A091044 → ⊙ C(n+2,k+1) → θ+1
  1 · 1, 2 · 1, 10, 3 · 1, 28, 42, 4 … Σ 1, 3, 14, 75, 438, 2702, 17324, 114291, …
  (−∞, 0]nonnegative
• A103406 → (θ+1)²
  1 · 1, 8 · 1, 16, 27 · 1, 24, 81, 64 … Σ 1, 9, 44, 170, 576, 1792, 5248, 14688, …
  (−∞, 0]nonnegative
• A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by co…
  1 · 0, 2 · 0, 0, 4 · 0, 0, 0, 8 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  [−1, 0]nonnegativerational roots
• A169803 → x ↦ x+1
  1 · 2, 1 · 3, 2 · 5, 5, 1 … Σ 1, 3, 5, 11, 21, 43, 85, 171, …
  (−∞, −1]nonnegative
• A008292 → γ (scale 1/4) → (θ+1)²
  1 · 2 · 4, 8 · 8, 64 … Σ 1, 2, 12, 72, 512, 4144, 37632, 379008, …
  (−∞, 0]nonnegative