preserver_image / affine, shift 0

Image under a real-rootedness preserver. All operations

318 sequences

• A105070: T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2), n >= 1.
  A011973 → x ↦ (1/2)x−1/4
  1 · 2 · 3, 2 · 4, 8 … row sums 1, 2, 5, 12, 29, 70, 169, 408, …
  A105070 (−∞, 1/2] nonnegative
• A091042 → basis A007318
  1 · 4, 3 · 16, 20, 5 · 64, 112, 56, 7 … row sums 1, 7, 41, 239, 1393, 8119, 47321, …
  (−∞, −1] nonnegative
• A063967 → x ↦ x−9
  1 · -8, 1 · 56, -15, 1 · -384, 160, -22, 1 … row sums 1, -7, 42, -245, 1421, -8232, 47677, …
  (4, 8]
• delannoy → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 12, 18, 8, 1 … row sums 1, 3, 11, 39, 139, 495, 1763, 6279, …
  (−∞, −1] nonnegative
• A113214 → x ↦ x−7
  1 · -5, 1 · 28, -11, 1 · -161, 93, -17, 1 … row sums 1, -4, 18, -84, 396, -1872, 8856, …
  (3, 7]
• A039755 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 24, 34, 12, 1 … row sums 1, 3, 13, 71, 457, 3355, 27509, 248127, …
  (−∞, −1] nonnegative
• A272866 → basis A007318
  1 · 5, 5, 1 · 25, 50, 35, 10, 1 … row sums 1, 11, 121, 1331, 14641, 161051, …
  (−∞, −1] nonnegative
• A053122 → x ↦ x+1
  1 · -1, 1 · 0, -2, 1 · 1, 1, -3, 1 … row sums 1, 0, -1, 0, 1, 0, -1, 0, …
  (−1, 3)
• basis-map coefficients of A371080, shifted → x ↦ x−1
  1 · -1, 1 · -2, 1, 1 · -10, 3, 6, 1 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  (−∞, 1] rational roots
• A099093 → x ↦ x−12
  1 · -36, 3 · 1260, -213, 9 · -44064, 11232, -954, 27 … row sums 1, -33, 1056, -33759, 1079199, …
  (32/3, 12]
• A143685 → basis A007318
  1 · 2, 1 · 13, 13, 1 · 44, 66, 24, 1 … row sums 1, 3, 27, 135, 891, 5103, 31347, …
  (−∞, −1] nonnegative
• A063967 → x ↦ x−4
  1 · -3, 1 · 6, -5, 1 · -9, 15, -7, 1 … row sums 1, -2, 2, 0, -4, 8, -8, 0, …
  (−1, 3]
• A008292 → γ (scale 1/4) (shift +1) → x ↦ x+1
  1 · 2 · 6, 2 · 24, 16 … row sums 1, 2, 8, 40, 256, 1952, 17408, 177280, …
  (−∞, −1] nonnegative
• A216973: Exponential Riordan array [x*exp(x),x].
  A007318 → D
  1 · 2, 2 · 3, 6, 3 · 4, 12, 12, 4 … row sums 0, 1, 4, 12, 32, 80, 192, 448, …
  A216973 (−∞, −1] nonnegative rational roots
• A207823: Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).
  1 · 4, 1 · 15, 8, 1 · 56, 46, 12, 1 … row sums 1, 5, 24, 115, 551, 2640, 12649, 60605, …
  A207823 (−6, −2) nonnegative
• A116414 → basis A007318
  1 · 5, 1 · 22, 10, 1 · 95, 69, 15, 1 … row sums 1, 6, 33, 180, 981, 5346, 29133, …
  (−9, −1) nonnegative
• A113214 → x ↦ x−2
  1 · 0, 1 · -2, -1, 1 · 4, -2, -2, 1 … row sums 1, 1, -2, 1, 1, -2, 1, 1, …
  (−2, 2]
• A007318 → θ+1 → x ↦ x+1
  1 · 3, 2 · 8, 10, 3 · 20, 36, 21, 4 … row sums 1, 5, 21, 81, 297, 1053, 3645, 12393, …
  (−∞, −1] nonnegative rational roots
• A063967 → x ↦ x−8
  1 · -7, 1 · 42, -13, 1 · -245, 119, -19, 1 … row sums 1, -6, 30, -144, 684, -3240, 15336, …
  (3, 7]
• A249251 → basis A007318
  1 · 4, 3 · 13, 18, 6 · 43, 87, 57, 12 … row sums 1, 7, 37, 199, 1069, 5743, 30853, …
  (−∞, −1] nonnegative
• A267482 → x ↦ x+1
  1 · 2, 1 · 1, 3, 1 · -1, 3, 4, 1 … row sums 1, 3, 5, 7, 9, 11, 13, 15, …
  (−3, 1)
• A124038 → x ↦ x+1
  1 · -1, 1 · -2, 0, 1 · -1, -3, 1, 1 … row sums 1, 0, -1, -2, -3, -4, -5, -6, …
  (−3, 2)
• A143494 → x ↦ x+1
  1 · 3, 1 · 10, 7, 1 · 37, 40, 12, 1 … row sums 1, 4, 18, 90, 494, 2946, 18926, 130066, …
  (−∞, −1] nonnegative
• A135065: A127733 * A007318 as infinite lower triangular matrices.
  A007318 → ×(1+2n+n²)
  1 · 4, 4 · 9, 18, 9 · 16, 48, 48, 16 … row sums 1, 8, 36, 128, 400, 1152, 3136, 8192, …
  A135065 (−∞, −1] nonnegative rational roots
• A063967 → x ↦ x−12
  1 · -11, 1 · 110, -21, 1 · -1089, 319, -31, 1 … row sums 1, -10, 90, -800, 7100, -63000, 559000, …
  (7, 11]
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A010054 (−∞, −1] [−1, 0] [−3−2√2, −3+2√2] nonnegative rational roots
• A391533 → x ↦ x+1
  9, 6, 1 · 18, 21, 8, 1 · 36, 60, 37, 10, 1 … row sums 16, 48, 144, 432, 1296, 3888, 11664, …
  (−∞, −1] nonnegative rational roots
• A007318 → (θ+1)² → x ↦ x+1
  1 · 5, 4 · 18, 26, 9 · 56, 114, 75, 16 … row sums 1, 9, 53, 261, 1161, 4833, 19197, …
  (−∞, −1] nonnegative
• A108838 → x ↦ x+1
  2 · 5, 2 · 14, 12, 2 · 42, 56, 21, 2 … row sums 2, 7, 28, 121, 550, 2591, 12536, 61921, …
  (−∞, −1] nonnegative
• basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
  1 · 0, 2 · -3, 3, 3 · 0, -12, 12, 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 1]
• A081577 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 16, 24, 10, 1 … row sums 1, 3, 13, 51, 205, 819, 3277, 13107, …
  (−∞, −1] nonnegative
• A113413 → basis A007318
  1 · 3, 1 · 7, 6, 1 · 17, 23, 9, 1 … row sums 1, 4, 14, 50, 178, 634, 2258, 8042, …
  (−∞, −1] nonnegative
• A081581 → basis A007318
  1 · 2, 1 · 10, 10, 1 · 32, 48, 18, 1 … row sums 1, 3, 21, 99, 549, 2835, 15093, 79299, …
  (−∞, −1] nonnegative
• A026374 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 10, 15, 7, 1 … row sums 1, 3, 11, 33, 121, 363, 1331, 3993, …
  (−∞, −1] nonnegative
• A145596 → x ↦ x+1
  1 · 4, 2 · 14, 14, 3 · 48, 72, 32, 4 … row sums 1, 6, 31, 156, 785, 3978, 20335, …
  (−∞, −1] nonnegative
• A154986 → basis A007318
  1 · 2, 1 · 6, 6, 1 · 24, 36, 14, 1 … row sums 1, 3, 13, 75, 537, 4611, 46053, 525483, …
  (−∞, −1] nonnegative
• A091044 → basis A007318
  1 · 4, 2 · 16, 16, 3 · 64, 96, 40, 4 … row sums 1, 6, 35, 204, 1189, 6930, 40391, …
  (−∞, −1] nonnegative
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
  1 · 4, 1 · 16, 8, 1 · 64, 48, 12, 1 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A038231 (−∞, −1] nonnegative rational roots
• narayana → γ (shift +1) → x ↦ x+1
  1 · 1 · 2, 1 · 4, 3 … row sums 1, 1, 3, 7, 21, 61, 191, 603, …
  (−∞, −1] nonnegative
• A113214 → x ↦ x−5
  1 · -3, 1 · 10, -7, 1 · -35, 37, -11, 1 … row sums 1, -2, 4, -8, 16, -32, 64, -128, …
  (1, 5]
• A135838 → x ↦ x+1
  1 · 4, 2 · 8, 8, 2 · 32, 48, 24, 4 … row sums 1, 6, 18, 108, 324, 1944, 5832, 34992, …
  (−∞, −1] nonnegative rational roots
• A038220 → basis A007318
  1 · 5, 2 · 25, 20, 4 · 125, 150, 60, 8 … row sums 1, 7, 49, 343, 2401, 16807, 117649, …
  (−∞, −1] nonnegative rational roots
• A035607 → x ↦ x+1
  1 · 1 · 3, 2 · 7, 8, 2 … row sums 1, 1, 5, 17, 61, 217, 773, 2753, …
  (−∞, −1] nonnegative
• A099040 → x ↦ x−2
  1 · -4, 2 · 12, -14, 4 · -32, 64, -40, 8 … row sums 1, -2, 2, 0, -4, 8, -8, 0, …
  (0, 2]
• A028326 → basis A007318
  2 · 4, 2 · 8, 8, 2 · 16, 24, 12, 2 … row sums 2, 6, 18, 54, 162, 486, 1458, 4374, …
  (−∞, −1] nonnegative rational roots
• A145597 → basis A007318
  1 · 6, 3 · 27, 27, 6 · 110, 165, 75, 10 … row sums 1, 9, 60, 360, 2061, 11529, 63728, …
  (−∞, −1] nonnegative
• A059473 → x ↦ x+1
  1 · 4, 2 · 18, 18, 4 · 80, 120, 56, 8 … row sums 1, 6, 40, 264, 1744, 11520, 76096, …
  (−∞, −1] nonnegative
• A099097 → x ↦ x−15
  1 · -45, 3 · 2010, -269, 9 · -89775, 18045, -1209, 27 … row sums 1, -42, 1750, -72912, 3037804, …
  (131/9, 15]
• A008277 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 15, 22, 9, 1 … row sums 1, 3, 11, 47, 227, 1215, 7107, 44959, …
  (−∞, −1] nonnegative
• A113953 → x ↦ x−4
  1 · -4, 1 · 8, -6, 1 · 0, 16, -8, 1 … row sums 1, -3, 3, 9, -45, 81, 27, -567, …
  (−4, 4]