preserver_image / affine, shift 0

Image under a real-rootedness preserver. All operations

318 sequences

• A130850: Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.
  A028246 → reverse
  1 · 1, 1 · 2, 3, 1 · 6, 12, 7, 1 … row sums 1, 2, 6, 26, 150, 1082, 9366, 94586, …
  A130850 (−∞, −1] nonnegative
• A008292 → γ (scale 1/4) → x ↦ x+1
  1 · 2 · 6, 2 · 24, 16 … row sums 1, 2, 8, 40, 256, 1952, 17408, 177280, …
  (−∞, −1] nonnegative
• A123519 → x ↦ x+1
  11, 14, 4 · 41, 76, 44, 8 · 153, 372, 324, 120, 16 … row sums 29, 169, 985, 5741, 33461, 195025, …
  (−3, −1) nonnegative
• A007318 → D+1 → x ↦ x+1
  1 · 3, 1 · 8, 6, 1 · 20, 24, 9, 1 … row sums 1, 4, 15, 54, 189, 648, 2187, 7290, …
  (−∞, −1] nonnegative rational roots
• A099093 → x ↦ x−6
  1 · -18, 3 · 306, -105, 9 · -5184, 2700, -468, 27 … row sums 1, -15, 210, -2925, 40725, -567000, …
  (14/3, 6]
• A304236 → basis A007318
  1 · 1 · 4, 3 · 7, 6 … row sums 1, 1, 7, 13, 55, 133, 463, 1261, …
  (−∞, −1] nonnegative
• A188440 → basis A007318
  1 · 1 · 3, 2 · 3, 2 … row sums 1, 1, 5, 5, 25, 25, 125, 125, …
  (−∞, −1] nonnegative rational roots
• A099093 → x ↦ x−3
  1 · -9, 3 · 72, -51, 9 · -567, 621, -225, 27 … row sums 1, -6, 30, -144, 684, -3240, 15336, …
  (5/3, 3]
• A113214 → x ↦ x−4
  1 · -2, 1 · 4, -5, 1 · -8, 18, -8, 1 … row sums 1, -1, 0, 3, -9, 18, -27, 27, …
  (0, 4]
• A081580 → basis A007318
  1 · 2, 1 · 9, 9, 1 · 28, 42, 16, 1 … row sums 1, 3, 19, 87, 451, 2223, 11179, 55767, …
  (−∞, −1] nonnegative
• A063967 → x ↦ x−2
  1 · -1, 1 · 0, -1, 1 · 1, -1, -1, 1 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  (−3, 1]
• A023531 → x ↦ x−1
  1 · -1, 1 · 1, -2, 1 · -1, 3, -3, 1 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  (−∞, 1] [0, ∞) rational roots
• A063967 → x ↦ x−7
  1 · -6, 1 · 30, -11, 1 · -144, 84, -16, 1 … row sums 1, -5, 20, -75, 275, -1000, 3625, …
  (2, 6]
• A118424: Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 001 (n,k>=0).
  A102547 → x ↦ (1/8)x−1/8
  1 · 2 · 4 · 7, 1 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A118424 (−∞, 1] nonnegative
• A030528 → basis A007318
  1 · 2, 1 · 3, 4, 1 · 5, 10, 6, 1 … row sums 1, 3, 8, 22, 60, 164, 448, 1224, …
  (−∞, −1] nonnegative
• A113953 → x ↦ x−5
  1 · -5, 1 · 15, -8, 1 · -25, 35, -11, 1 … row sums 1, -4, 8, 0, -64, 256, -512, 0, …
  (−3, 5]
• A108838 → ⊙ C(n+1,k+1) (shift +1) → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 22, 33, 13, 1 … row sums 1, 3, 13, 69, 417, 2763, 19609, 146793, …
  (−∞, −1] nonnegative
• A126216: Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A126216 (−∞, −1] nonnegative
• A008459 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 20, 30, 12, 1 … row sums 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  (−∞, −1] nonnegative
• A178822 → x ↦ x+1
  1 · 12, 6 · 84, 84, 21 · 448, 672, 336, 56 … row sums 1, 18, 189, 1512, 10206, 61236, 336798, …
  (−∞, −1] nonnegative rational roots
• A023531 → (1/2)D²+2D+1 → x ↦ x+1
  1 · 3, 1 · 6, 6, 1 · 10, 18, 9, 1 … row sums 1, 4, 13, 38, 104, 272, 688, 1696, …
  (−∞, −1] nonnegative
• A334218 → x ↦ x+1
  1 · 1 · 2, 2 · 6, 9, 3 … row sums 1, 1, 4, 18, 104, 750, 6492, 65562, …
  (−∞, −1] nonnegative
• A023531 → 2D+1+2n → x ↦ x+1
  1 · 5, 3 · 9, 14, 5 · 13, 33, 27, 7 … row sums 1, 8, 28, 80, 208, 512, 1216, 2816, …
  (−∞, −1] nonnegative rational roots
• A345013: Triangle read by rows, related to clusters of type D.
  A062196 → x ↦ x+1
  1 · 4, 3 · 15, 20, 6 · 56, 105, 60, 10 … row sums 1, 7, 41, 231, 1289, 7183, 40081, …
  A345013 (−∞, −1] nonnegative
• A099040 → x ↦ x−1
  1 · -2, 2 · 2, -6, 4 · 0, 8, -16, 8 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  (−1, 1]
• A094646: Triangle of generalized Stirling numbers of the first kind read by rows: T(n, k) = (-1)^(n+k)*Sum_{m=0..n} Stirling1(n, m) * binomial(m, k) * 2^(m-k), where Stirling1 is A048994.
  A028338 → x ↦ 2x−5
  1 · -2, 1 · 2, -3, 1 · 0, 2, -3, 1 … row sums 1, -1, 0, 0, 0, 0, 0, 0, …
  A094646 (−∞, 5/2] rational roots
• A249248 → x ↦ x+1
  1 · 4, 3 · 11, 13, 3 · 38, 65, 31, 3 … row sums 1, 7, 27, 137, 681, 3687, 20595, …
  (−∞, −1] nonnegative
• type_b_eulerian → γ (scale 1/4) → x ↦ x+1
  1 · 1 · 2, 1 · 6, 5 … row sums 1, 1, 3, 11, 57, 361, 2763, 24611, …
  (−∞, −1] nonnegative
• A178820 → basis A007318
  1 · 8, 4 · 40, 40, 10 · 160, 240, 120, 20 … row sums 1, 12, 90, 540, 2835, 13608, 61236, …
  (−∞, −1] nonnegative rational roots
• chebyshev_t → x ↦ x+1
  1 · 1, 1 · 1, 4, 2 · 1, 9, 12, 4 … row sums 1, 2, 7, 26, 97, 362, 1351, 5042, …
  (−2, 0) nonnegative
• A102662 → x ↦ x+1
  1 · 4, 3 · 9, 11, 3 · 22, 38, 20, 3 … row sums 1, 7, 23, 83, 295, 1051, 3743, 13331, …
  (−∞, −1] nonnegative
• A007318 → basis A049020 → x ↦ x+1
  1 · 3, 1 · 11, 7, 1 · 47, 43, 12, 1 … row sums 1, 4, 19, 103, 622, 4117, 29521, …
  (−∞, −1] nonnegative
• A052553 → x ↦ x+1
  1 · 1 · 2, 1 · 3, 2 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  (−∞, −1] nonnegative
• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
  A010054 → D∘x
  1 · 3 · 5 · 7 … row sums 1, 3, 5, 7, 9, 11, 13, 15, …
  A198954 (−∞, −1] [−1, 0] nonnegative rational roots
• A102547 → reverse → x ↦ x+1
  1, 1 · 1, 2, 1 · 1, 3, 3, 1 · 2, 7, 9, 5, 1 … row sums 2, 4, 8, 24, 64, 160, 416, 1088, …
  (−∞, −1] nonnegative
• A060920 → x ↦ x+1
  1 · 3, 1 · 11, 7, 1 · 43, 41, 12, 1 … row sums 1, 4, 19, 97, 508, 2683, 14209, 75316, …
  (−∞, ∞) nonnegative
• A007318 → ×(1+2n+n²) → x ↦ x+1
  1 · 8, 4 · 36, 36, 9 · 128, 192, 96, 16 … row sums 1, 12, 81, 432, 2025, 8748, 35721, …
  (−∞, −1] nonnegative rational roots
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x+1
  4, 4, 1 · 24, 36, 18, 3 · 96, 192, 144, 48, 6 … row sums 0, 0, 9, 81, 486, 2430, 10935, 45927, …
  (−∞, −1] nonnegative rational roots
• A111125 → x ↦ x+1
  1 · 4, 1 · 11, 7, 1 · 29, 31, 10, 1 … row sums 1, 5, 19, 71, 265, 989, 3691, 13775, …
  (−5, −1) nonnegative
• A152842 → x ↦ x+1
  1 · 4, 3 · 8, 10, 3 · 32, 64, 42, 9 … row sums 1, 7, 21, 147, 441, 3087, 9261, 64827, …
  (−∞, −1] nonnegative rational roots
• A023531 → 2D+1 → x ↦ x+1
  1 · 3, 1 · 5, 6, 1 · 7, 15, 9, 1 … row sums 1, 4, 12, 32, 80, 192, 448, 1024, …
  (−∞, −1] nonnegative rational roots
• A066325 → x ↦ x+1
  1 · 1, 1 · 0, 2, 1 · -2, 0, 3, 1 … row sums 1, 2, 3, 2, -5, -18, -11, 86, …
  (−∞, ∞)
• A114656 → basis A007318
  1 · 3, 1 · 11, 8, 1 · 45, 51, 15, 1 … row sums 1, 4, 20, 112, 672, 4224, 27456, …
  (−∞, −1] nonnegative
• A100862 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … row sums 1, 3, 11, 45, 201, 963, 4899, 26253, …
  (−∞, −1] nonnegative
• A110441 → basis A007318
  1 · 4, 1 · 14, 8, 1 · 48, 44, 12, 1 … row sums 1, 5, 23, 105, 479, 2185, 9967, 45465, …
  (−7, −1) nonnegative
• A099040 → x ↦ x−12
  1 · -24, 2 · 552, -94, 4 · -12672, 3264, -280, 8 … row sums 1, -22, 462, -9680, 202796, -4248552, …
  (10, 12]
• A102547 → x ↦ x+1
  1 · 1 · 1 · 2, 1 … row sums 1, 1, 1, 3, 5, 7, 13, 23, …
  (−∞, −1] nonnegative
• A030528 → x ↦ x−15
  1 · -14, 1 · 195, -28, 1 · -2715, 586, -42, 1 … row sums 1, -13, 168, -2170, 28028, -362012, …
  (11, 15]
• A172431 → ⊙ C(n,k+1) (shift -1) → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 21, 32, 13, 1 … row sums 1, 3, 13, 67, 377, 2243, 13877, 88371, …
  (−∞, −1] nonnegative
• narayana → basis A007318
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  (−∞, −1] nonnegative