preserver_image / affine, shift 0

Image under a real-rootedness preserver. All operations

318 sequences

• A063967 → x ↦ x−15
  1 · -14, 1 · 182, -27, 1 · -2352, 532, -40, 1 … row sums 1, -13, 156, -1859, 22139, -263640, …
  (10, 14]
• A063967 → x ↦ x−6
  1 · -5, 1 · 20, -9, 1 · -75, 55, -13, 1 … row sums 1, -4, 12, -32, 80, -192, 448, -1024, …
  (1, 5]
• A030528 → x ↦ x−7
  1 · -6, 1 · 35, -12, 1 · -203, 106, -18, 1 … row sums 1, -5, 24, -114, 540, -2556, 12096, …
  (3, 7]
• A232535 → x ↦ x+1
  1 · 3, 2 · 12, 14, 3 · 48, 80, 37, 4 … row sums 1, 5, 29, 169, 985, 5741, 33461, …
  (−∞, −1] nonnegative
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0
  1 · 0, 0, 1 · 0, 0, 0, 0, 1 · 0, 0, 0, 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0] [0, ∞) nonnegative rational roots
• A099093 → x ↦ x−5
  1 · -15, 3 · 210, -87, 9 · -2925, 1845, -387, 27 … row sums 1, -12, 132, -1440, 15696, -171072, …
  (11/3, 5]
• A229032: Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).
  A011973 → x ↦ x−1/4
  1 · 2 · 3, 4 · 4, 16 … row sums 1, 2, 7, 20, 61, 182, 547, 1640, …
  A229032 (−∞, 0] nonnegative
• A013611 → basis A007318
  1 · 5, 4 · 25, 40, 16 · 125, 300, 240, 64 … row sums 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, −1] nonnegative rational roots
• A085478 → x ↦ x+1
  1 · 1 · 2, 1 · 5, 5, 1 … row sums 1, 1, 3, 11, 41, 153, 571, 2131, …
  (−5, −1) nonnegative
• A113214 → x ↦ x−12
  1 · -10, 1 · 108, -21, 1 · -1176, 338, -32, 1 … row sums 1, -9, 88, -869, 8591, -84942, 839861, …
  (8, 12]
• A063967 → x ↦ x−5
  1 · -4, 1 · 12, -7, 1 · -32, 32, -10, 1 … row sums 1, -3, 6, -9, 9, 0, -27, 81, …
  (0, 4]
• A038208: Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
  1 · 2, 2 · 4, 8, 4 · 8, 24, 24, 8 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A038208 (−∞, −1] nonnegative rational roots
• A113214 → x ↦ x−1
  1 · 1, 1 · -2, 1, 1 · 1, -3, 1, 1 … row sums 1, 2, 0, 0, 0, 0, 0, 0, …
  (−3, 1]
• A274496 → x ↦ x+1
  1 · 2 · 4, 2 · 8, 4 … row sums 1, 2, 6, 12, 36, 72, 216, 432, …
  (−∞, −1] nonnegative rational roots
• A007318 → ⊙ C(n+1,k+1) (shift +2) → x ↦ x+1
  8 · 30, 18 · 112, 128, 32 · 420, 700, 350, 50 … row sums 8, 48, 272, 1520, 8472, 47264, 264224, …
  (−∞, −1] nonnegative
• A131689 → reverse
  1 · 1 · 2, 1 · 6, 6, 1 … row sums 1, 1, 3, 13, 75, 541, 4683, 47293, …
  (−∞, −1] nonnegative
• A133087: A133080 * A007318.
  1 · 2, 1 · 1, 2, 1 · 2, 5, 4, 1 … row sums 1, 3, 4, 12, 16, 48, 64, 192, …
  A133087 (−∞, −1] nonnegative rational roots
• A007318 → 2θ+1 → x ↦ x+1
  1 · 4, 3 · 12, 16, 5 · 32, 60, 36, 7 … row sums 1, 7, 33, 135, 513, 1863, 6561, 22599, …
  (−∞, −1] nonnegative rational roots
• A113953 → x ↦ x−7
  1 · -7, 1 · 35, -12, 1 · -147, 91, -17, 1 … row sums 1, -6, 24, -72, 144, 0, -1728, 10368, …
  (−1, 7]
• A099040 → x ↦ x−10
  1 · -20, 2 · 380, -78, 4 · -7200, 2240, -232, 8 … row sums 1, -18, 306, -5184, 87804, -1487160, …
  (8, 10]
• basis-map coefficients of A112227 → x ↦ x+1
  1, 2, 1 · 1, 3, 3, 1 · 1, 4, 6, 4, 1 … row sums 4, 8, 16, 32, 64, 128, 256, 512, …
  (−∞, −1] nonnegative rational roots
• A207606 → basis A007318
  13, 11, 2 · 34, 44, 17, 2 · 89, 155, 93, 23, 2 … row sums 26, 97, 362, 1351, 5042, 18817, 70226, …
  (−5, −1) nonnegative
• A039991 → x ↦ x+1
  1 · 1 · 1, -2, -1 · 1, -6, -3 … row sums 1, 1, -2, -8, -8, 16, 64, 64, …
  (−∞, ∞)
• A099040 → x ↦ x−8
  1 · -16, 2 · 240, -62, 4 · -3584, 1408, -184, 8 … row sums 1, -14, 182, -2352, 30380, -392392, …
  (6, 8]
• A108299 → x ↦ x+1
  1 · 0, -1 · -1, -3, -1 · -1, -2, 1, 1 … row sums 1, -1, -5, -1, 19, 23, -53, -145, …
  (−∞, ∞)
• A099093 → x ↦ x−10
  1 · -30, 3 · 870, -177, 9 · -25200, 7740, -792, 27 … row sums 1, -27, 702, -18225, 473121, -12282192, …
  (26/3, 10]
• A261711: Triangle read by rows: T(n,k) is the number of words over alphabet {0,1,2,3} having exactly k occurrences of the string 01, where n>=0 and k>=0.
  A011973 → x ↦ (1/16)x−1/16
  1 · 4 · 15, 1 · 56, 8 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A261711 (−∞, 1] nonnegative
• A113214 → x ↦ x−8
  1 · -6, 1 · 40, -13, 1 · -272, 130, -20, 1 … row sums 1, -5, 28, -161, 931, -5390, 31213, …
  (4, 8]
• A099093 → x ↦ x−4
  1 · -12, 3 · 132, -69, 9 · -1440, 1152, -306, 27 … row sums 1, -9, 72, -567, 4455, -34992, 274833, …
  (8/3, 4]
• A011973 → x ↦ x−1/4 → x ↦ x+1
  1 · 2 · 7, 4 · 20, 16 … row sums 1, 2, 11, 36, 149, 550, 2143, 8136, …
  (−∞, −1] nonnegative
• A133087 → x ↦ x+1
  1 · 3, 1 · 4, 4, 1 · 12, 16, 7, 1 … row sums 1, 4, 9, 36, 81, 324, 729, 2916, …
  (−∞, −1] nonnegative rational roots
• A011973 → x ↦ x+2 (shift +1) → basis A007318
  1 · 4, 1 · 7, 2 · 19, 9, 1 … row sums 1, 5, 9, 29, 65, 181, 441, 1165, …
  (−∞, −1] nonnegative
• A078812 → basis A007318
  1 · 3, 1 · 8, 6, 1 · 21, 25, 9, 1 … row sums 1, 4, 15, 56, 209, 780, 2911, 10864, …
  (−5, −1) nonnegative
• A013610 → basis A007318
  1 · 4, 3 · 16, 24, 9 · 64, 144, 108, 27 … row sums 1, 7, 49, 343, 2401, 16807, 117649, …
  (−∞, −1] nonnegative rational roots
• A008292 → γ (shift +1) → x ↦ x+1
  1 · 1 · 3, 2 · 9, 8 … row sums 1, 1, 5, 17, 109, 649, 5285, 44513, …
  (−∞, −1] nonnegative
• A344565 → basis A007318
  3 · 30, 15 · 180, 180, 45 · 840, 1260, 630, 105 … row sums 3, 45, 405, 2835, 17010, 91854, 459270, …
  (−∞, −1] nonnegative rational roots
• A113953 → x ↦ x−15
  1 · -15, 1 · 195, -28, 1 · -2475, 555, -41, 1 … row sums 1, -14, 168, -1960, 22736, -263424, …
  (7, 15]
• A059374: Triangle read by rows, T(n, k) = Sum_{i=0..n} L'(n, n-i) * binomial(i, k), for k = 0..n-1.
  A089231 → x ↦ x+1
  1 · 3, 2 · 13, 18, 6 · 73, 156, 108, 24 … row sums 1, 5, 37, 361, 4361, 62701, 1044205, …
  A059374 (−∞, −1] nonnegative
• A008277 → θ+2 (shift +1) → x ↦ x+1
  2 · 5, 3 · 15, 17, 4 · 52, 84, 39, 5 … row sums 2, 8, 36, 180, 988, 5892, 37852, …
  (−∞, −1] nonnegative
• A099097 → x ↦ x−9
  1 · -27, 3 · 720, -161, 9 · -19197, 6453, -723, 27 … row sums 1, -24, 568, -13440, 318016, -7524864, …
  (77/9, 9]
• A105793: Expansion of e.g.f. (1 + y)^(1 + x).
  A028338 → x ↦ −2x−3
  1 · 1, 1 · 0, 1, 1 · 0, -1, 0, 1 … row sums 1, 2, 2, 0, 0, 0, 0, 0, …
  A105793 [−3/2, ∞) rational roots
• A122542 → x ↦ x+1
  1 · 1, 1 · 3, 4, 1 · 7, 13, 7, 1 … row sums 1, 2, 8, 28, 100, 356, 1268, 4516, …
  (−∞, −1] nonnegative
• A023531 → basis A049020 → x ↦ x+1
  1 · 2, 1 · 6, 5, 1 · 22, 25, 9, 1 … row sums 1, 3, 12, 57, 309, 1866, 12351, 88563, …
  (−∞, −1] nonnegative
• A113953 → x ↦ x−6
  1 · -6, 1 · 24, -10, 1 · -72, 60, -14, 1 … row sums 1, -5, 15, -25, -25, 375, -1625, 4375, …
  (−2, 6]
• A091044 → ⊙ C(n+2,k+1) → x ↦ x+1
  1 · 2, 1 · 7, 7, 1 · 30, 45, 17, 1 … row sums 1, 3, 15, 93, 653, 4983, 40339, 341337, …
  (−∞, −1] nonnegative
• A131689 → x ↦ x+1
  1 · 1, 1 · 3, 5, 2 · 13, 31, 24, 6 … row sums 1, 2, 10, 74, 730, 9002, 133210, …
  (−2, −1] nonnegative
• A130534: Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, …
  1 · 1, 1 · 2, 3, 1 · 6, 11, 6, 1 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A130534 (−∞, −1] nonnegative rational roots
• A010054 → reverse → x ↦ x+1
  1 · 4, 4 · 7, 14, 7 · 10, 30, 30, 10 … row sums 1, 8, 28, 80, 208, 512, 1216, 2816, …
  (−∞, −1] nonnegative rational roots
• A051380: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 8.
  A028338 → x ↦ −2x+17
  1 · -9, 1 · 90, -19, 1 · -990, 299, -30, 1 … row sums 1, -8, 72, -720, 7920, -95040, 1235520, …
  A051380 [17/2, ∞) rational roots
• A039683: Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).
  A028338 → x ↦ −x+1
  1 · -2, 1 · 8, -6, 1 · -48, 44, -12, 1 … row sums 1, -1, 3, -15, 105, -945, 10395, …
  A039683 [1, ∞) rational roots