preserver_image / affine, shift 0

Image under a real-rootedness preserver. All operations

318 sequences

• basis-map coefficients of A026729, shifted → basis A007318
  1 · -1, -1 · 1, 2, 1 · -1, -3, -3, -1 … row sums 1, -2, 4, -8, 16, -32, 64, -128, …
  (−∞, −1] rational roots
• A051338: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 5.
  A028338 → x ↦ −2x+11
  1 · -6, 1 · 42, -13, 1 · -336, 146, -21, 1 … row sums 1, -5, 30, -210, 1680, -15120, 151200, …
  A051338 [11/2, ∞) rational roots
• A063967 → x ↦ x−3
  1 · -2, 1 · 2, -3, 1 · 0, 4, -4, 1 … row sums 1, -1, 0, 1, -1, 0, 1, -1, …
  (−2, 2]
• A132812: Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
  1 · 2, 2 · 3, 9, 3 · 4, 24, 24, 4 … row sums 1, 4, 15, 56, 210, 792, 3003, 11440, …
  A132812 (−∞, 0) nonnegative
• A113953 → x ↦ x−8
  1 · -8, 1 · 48, -14, 1 · -256, 128, -20, 1 … row sums 1, -7, 35, -147, 539, -1715, 4459, …
  (0, 8]
• A391533 → Pₙ + Pₙ₊₁
  8, 12, 6, 1 · 8, 20, 18, 7, 1 · 8, 28, 38, 25, 8, 1 … row sums 27, 54, 108, 216, 432, 864, 1728, 3456, …
  (−∞, −1] nonnegative rational roots
• A053123 → x ↦ x+1
  1 · -1, -2 · 0, 2, 3 · 1, 2, -2, -4 … row sums 1, -3, 5, -3, -11, 45, -91, 93, …
  (−∞, ∞)
• A081582 → x ↦ x+1
  1 · 2, 1 · 11, 11, 1 · 36, 54, 20, 1 … row sums 1, 3, 23, 111, 655, 3519, 19727, …
  (−∞, −1] nonnegative
• A120910 → basis A007318
  3 · 9, 3 · 27, 18, 3 · 81, 81, 27, 3 … row sums 3, 12, 48, 192, 768, 3072, 12288, …
  (−∞, −1] nonnegative rational roots
• chebyshev_u → x ↦ x+1
  1 · 2, 2 · 3, 8, 4 · 4, 20, 24, 8 … row sums 1, 4, 15, 56, 209, 780, 2911, 10864, …
  (−2, 0) nonnegative
• A317497: Triangle T(n,k) = 3*T(n-1,k) + T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.
  A102547 → x ↦ (1/27)x
  1 · 3 · 9 · 27, 1 … row sums 1, 3, 9, 28, 87, 270, 838, 2601, …
  A317497 (−∞, 0] nonnegative
• A152828 → basis A007318
  1 · 1 · 1 · 2, 1 … row sums 1, 1, 1, 3, 3, 3, 9, 9, …
  (−∞, −1] nonnegative rational roots
• basis-map coefficients of A026729 → basis A007318
  1 · 1, 1 · 1, 2, 1 · 1, 3, 3, 1 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, −1] nonnegative rational roots
• basis-map coefficients of A021012 → x ↦ x+1
  1 · -1, -1 · 2, 4, 2 · -6, -18, -18, -6 … row sums 1, -2, 8, -48, 384, -3840, 46080, …
  (−∞, −1] rational roots
• A062264 → x ↦ x+1
  1 · 6, 5 · 28, 42, 15 · 120, 252, 168, 35 … row sums 1, 11, 85, 575, 3649, 22363, 134245, …
  (−∞, −1] nonnegative
• A113953 → x ↦ x−9
  1 · -9, 1 · 63, -16, 1 · -405, 171, -23, 1 … row sums 1, -8, 48, -256, 1280, -6144, 28672, …
  (1, 9]
• A099097 → basis A007318
  1 · 3, 3 · 10, 19, 9 · 33, 93, 87, 27 … row sums 1, 6, 38, 240, 1516, 9576, 60488, …
  (−∞, −1] nonnegative
• A118933 → x ↦ x+1
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 13, 61, 181, 421, …
  (−∞, −1] nonnegative
• A169803 → x ↦ x+1
  1 · 2, 1 · 3, 2 · 5, 5, 1 … row sums 1, 3, 5, 11, 21, 43, 85, 171, …
  (−∞, −1] nonnegative
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
  1 · 6, 3 · 24, 24, 6 · 80, 120, 60, 10 … row sums 1, 9, 54, 270, 1215, 5103, 20412, …
  (−∞, −1] nonnegative rational roots
• A094645: 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), where Stirling1 is A048994.
  A028338 → x ↦ 2x−3
  1 · -1, 1 · 0, -1, 1 · 0, -1, 0, 1 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  A094645 (−∞, 3/2] rational roots
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
  1 · 2, 1 · 4, 4, 1 · 8, 12, 6, 1 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A038207 (−∞, −1] nonnegative rational roots
• A099040 → x ↦ x−5
  1 · -10, 2 · 90, -38, 4 · -800, 520, -112, 8 … row sums 1, -8, 56, -384, 2624, -17920, 122368, …
  (3, 5]
• A007318 → −D²+(−1+n)D+1+n → x ↦ x+1
  1 · 4, 2 · 14, 14, 3 · 44, 66, 30, 4 … row sums 1, 6, 31, 144, 621, 2538, 9963, 37908, …
  (−∞, −1] nonnegative
• A099097 → x ↦ x−1
  1 · -3, 3 · 8, -17, 9 · -21, 69, -75, 27 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  (5/9, 1]
• A053117 → x ↦ x+1
  1 · 2, 2 · 3, 8, 4 · 4, 20, 24, 8 … row sums 1, 4, 15, 56, 209, 780, 2911, 10864, …
  (−2, 0) nonnegative
• A118931 → basis A007318
  1 · 1 · 1 · 3, 2 … row sums 1, 1, 1, 5, 17, 41, 241, 1261, …
  (−∞, −1] nonnegative
• A099093 → x ↦ x−8
  1 · -24, 3 · 552, -141, 9 · -12672, 4896, -630, 27 … row sums 1, -21, 420, -8379, 167139, -3333960, …
  (20/3, 8]
• A317496 → x ↦ x+1
  1 · 1 · 1 · 4, 3 … row sums 1, 1, 1, 7, 13, 19, 61, 139, …
  (−∞, −1] nonnegative
• A099093 → x ↦ x−7
  1 · -21, 3 · 420, -123, 9 · -8379, 3717, -549, 27 … row sums 1, -18, 306, -5184, 87804, -1487160, …
  (17/3, 7]
• A103633 → basis A007318
  1 · 1, 1 · 2, 3, 1 · 2, 5, 4, 1 … row sums 1, 2, 6, 12, 36, 72, 216, 432, …
  (−∞, −1] nonnegative rational roots
• A099040 → x ↦ x−3
  1 · -6, 2 · 30, -22, 4 · -144, 168, -64, 8 … row sums 1, -4, 12, -32, 80, -192, 448, -1024, …
  (1, 3]
• A143683 → x ↦ x+1
  1 · 2, 1 · 12, 12, 1 · 40, 60, 22, 1 … row sums 1, 3, 25, 123, 769, 4275, 25129, …
  (−∞, −1] nonnegative
• A114687 → basis A007318
  1 · 3, 2 · 11, 14, 4 · 45, 84, 48, 8 … row sums 1, 5, 29, 185, 1257, 8925, 65445, …
  (−∞, −1] nonnegative
• A137338 → x ↦ x+1
  1 · 0, 1 · -2, -1, 1 · 4, -3, -3, 1 … row sums 1, 1, -2, -1, 10, -25, 40, -25, …
  (−∞, ∞)
• A121574 → basis A007318
  1 · 3, 1 · 10, 7, 1 · 33, 35, 11, 1 … row sums 1, 4, 18, 80, 356, 1584, 7048, 31360, …
  (−∞, −1] nonnegative
• A081578 → x ↦ x+1
  1 · 2, 1 · 7, 7, 1 · 20, 30, 12, 1 … row sums 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  (−∞, −1] nonnegative
• A143495 → x ↦ x+1
  1 · 4, 1 · 17, 9, 1 · 77, 64, 15, 1 … row sums 1, 5, 27, 157, 979, 6517, 46107, …
  (−∞, −1] nonnegative
• A049020: Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.
  A023531 → basis A049020
  1 · 1, 1 · 2, 3, 1 · 5, 10, 6, 1 … row sums 1, 2, 6, 22, 94, 454, 2430, 14214, …
  A049020 (−∞, −1] nonnegative
• A086645 → x ↦ x+1
  1 · 2, 1 · 8, 8, 1 · 32, 48, 18, 1 … row sums 1, 3, 17, 99, 577, 3363, 19601, 114243, …
  (−∞, −1] nonnegative
• A207607 → basis A007318
  1 · 3, 2 · 8, 9, 2 · 21, 33, 15, 2 … row sums 1, 5, 19, 71, 265, 989, 3691, 13775, …
  (−5, −1) nonnegative
• A367270 → x ↦ x+1
  1 · 1 · 3, 2 · 10, 12, 3 … row sums 1, 1, 5, 25, 129, 681, 3653, 19825, …
  (−∞, −1] nonnegative
• A207824: Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).
  1 · 5, 1 · 24, 10, 1 · 115, 73, 15, 1 … row sums 1, 6, 35, 204, 1189, 6930, 40391, …
  A207824 (−7, −3) nonnegative
• A063967 → basis A007318
  1 · 2, 1 · 6, 5, 1 · 16, 20, 8, 1 … row sums 1, 3, 12, 45, 171, 648, 2457, 9315, …
  (−∞, −1] nonnegative
• A106800 → x ↦ x+1
  1 · 1 · 2, 1 · 5, 5, 1 … row sums 1, 1, 3, 11, 49, 257, 1539, 10299, …
  (−∞, −1] nonnegative
• A063967: Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
  1 · 1, 1 · 2, 3, 1 · 3, 7, 5, 1 … row sums 1, 2, 6, 16, 44, 120, 328, 896, …
  A063967 (−5, −1] nonnegative
• A099093 → x ↦ x−1
  1 · -3, 3 · 6, -15, 9 · -9, 45, -63, 27 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  (−1/3, 1]
• A007318: Binomial
  1 · 1, 1 · 1, 2, 1 · 1, 3, 3, 1 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A007318 (−∞, −1] nonnegative rational roots
• A102547 → x ↦ (1/27)x → x ↦ x+1
  1 · 3 · 9 · 28, 1 … row sums 1, 3, 9, 29, 93, 297, 949, 3033, …
  (−∞, −1] nonnegative
• A196347: Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
  1 · 1, 1 · 2, 4, 2 · 6, 18, 18, 6 … row sums 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  A196347 (−∞, −1] nonnegative rational roots