preserver_image / affine, shift 0

Image under a real-rootedness preserver. All operations

318 sequences

• A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
  1 · 0, 1 · 0, 0, 1 · 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A023531 [−1, 0] [0, ∞) nonnegative rational roots
• A051523: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 9.
  A028338 → x ↦ −2x+19
  1 · -10, 1 · 110, -21, 1 · -1320, 362, -33, 1 … row sums 1, -9, 90, -990, 11880, -154440, …
  A051523 [19/2, ∞) rational roots
• A152830 → x ↦ x+1
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 3, 3, 3, 3, …
  (−∞, −1] nonnegative rational roots
• A064861 → x ↦ x+1
  6, 7, 2 · 18, 33, 20, 4 · 36, 84, 73, 28, 4 … row sums 15, 75, 225, 1125, 3375, 16875, 50625, …
  (−∞, −1] nonnegative rational roots
• A007318 → (1/3+(1/2)n+(1/6)n²)θ+1+(3/2)n+(1/2)n² → x ↦ x+1
  1 · 7, 4 · 32, 36, 10 · 120, 200, 110, 20 … row sums 1, 11, 78, 450, 2295, 10773, 47628, …
  (−∞, −1] nonnegative rational roots
• A172431 → basis A007318
  1 · 3, 2 · 8, 10, 3 · 21, 38, 22, 4 … row sums 1, 5, 21, 85, 341, 1365, 5461, 21845, …
  (−∞, −1] nonnegative
• basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
  1 · -1, -1 · 1, 2, 1 · -1, -3, -3, -1 … row sums 1, -2, 4, -8, 16, -32, 64, -128, …
  (−∞, −1] rational roots
• A007318 → θ+1+n → x ↦ x+1
  1 · 5, 3 · 16, 18, 5 · 44, 72, 39, 7 … row sums 1, 8, 39, 162, 621, 2268, 8019, 27702, …
  (−∞, −1] nonnegative rational roots
• A099093 → x ↦ x−9
  1 · -27, 3 · 702, -159, 9 · -18225, 6237, -711, 27 … row sums 1, -24, 552, -12672, 290880, -6676992, …
  (23/3, 9]
• A008292 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 24, 36, 14, 1 … row sums 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  (−∞, −1] nonnegative
• A113953 → x ↦ x−2
  1 · -2, 1 · 0, -2, 1 · 8, -4, -2, 1 … row sums 1, -1, -1, 3, -1, -5, 7, 3, …
  (−6, 2]
• A008278 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 15, 23, 10, 1 … row sums 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  (−∞, −1] nonnegative
• A137320 → x ↦ x+1
  1 · 2, 2 · 6, 10, 4 · 24, 52, 36, 8 … row sums 1, 4, 20, 120, 840, 6720, 60480, …
  (−∞, −1] nonnegative rational roots
• A099040 → x ↦ x−6
  1 · -12, 2 · 132, -46, 4 · -1440, 768, -136, 8 … row sums 1, -10, 90, -800, 7100, -63000, 559000, …
  (4, 6]
• A062190 → x ↦ x+1
  1 · 7, 6 · 36, 56, 21 · 165, 360, 252, 56 … row sums 1, 13, 113, 833, 5641, 36365, 227305, …
  (−∞, −1] nonnegative
• A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
  1 · 5, 1 · 25, 10, 1 · 125, 75, 15, 1 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …
  A038243 (−∞, −1] nonnegative rational roots
• A118884: Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 0011 (n,k>=0).
  A318772 → x ↦ (1/48)x−1/48
  1 · 2 · 4 · 8 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A118884 (−∞, 1] nonnegative
• A038207 → D
  1 · 4, 2 · 12, 12, 3 · 32, 48, 24, 4 … row sums 0, 1, 6, 27, 108, 405, 1458, 5103, …
  (−∞, −1] nonnegative rational roots
• A099040 → x ↦ x−4
  1 · -8, 2 · 56, -30, 4 · -384, 320, -88, 8 … row sums 1, -6, 30, -144, 684, -3240, 15336, …
  (2, 4]
• A128100: Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).
  A011973 → basis A007318
  1 · 1 · 2, 1 · 3, 2 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  A128100 (−∞, −1] nonnegative
• A075497 → x ↦ x+1
  1 · 3, 1 · 11, 8, 1 · 49, 55, 15, 1 … row sums 1, 4, 20, 120, 832, 6496, 56128, …
  (−∞, −1] nonnegative
• A094305: Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).
  A007318 → ×(1+(3/2)n+(1/2)n²)
  1 · 3, 3 · 6, 12, 6 · 10, 30, 30, 10 … row sums 1, 6, 24, 80, 240, 672, 1792, 4608, …
  A094305 (−∞, −1] nonnegative rational roots
• A099097 → x ↦ x−12
  1 · -36, 3 · 1284, -215, 9 · -45792, 11520, -966, 27 … row sums 1, -33, 1078, -35211, 1150105, …
  (104/9, 12]
• A059364: Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.
  1 · 2, 1 · 6, 7, 2 · 24, 46, 29, 6 … row sums 1, 3, 15, 105, 945, 10395, 135135, …
  A059364 (−∞, −1] nonnegative rational roots
• A100861 → basis A007318
  1 · 1 · 2, 1 · 4, 3 … row sums 1, 1, 3, 7, 25, 81, 331, 1303, …
  (−∞, −1] nonnegative
• A030528 → x ↦ x−3
  1 · -2, 1 · 3, -4, 1 · -3, 10, -6, 1 … row sums 1, -1, 0, 2, -4, 4, 0, -8, …
  (−1, 3]
• A126216 → basis A007318
  1 · 3, 1 · 11, 7, 1 · 45, 42, 12, 1 … row sums 1, 4, 19, 100, 562, 3304, 20071, …
  (−∞, −1] nonnegative
• A008288 → basis A007318
  1 · 2, 1 · 5, 5, 1 · 12, 18, 8, 1 … row sums 1, 3, 11, 39, 139, 495, 1763, 6279, …
  (−∞, −1] nonnegative
• A102365 → x ↦ x+1
  1 · 1 · 2, 1 · 7, 7, 1 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, −1] nonnegative
• A113953 → x ↦ x−1
  1 · -1, 1 · -1, 0, 1 · 3, -5, 1, 1 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  (−7, 1]
• A113214 → x ↦ x−3
  1 · -1, 1 · 0, -3, 1 · 3, 5, -5, 1 … row sums 1, 0, -2, 4, -4, 0, 8, -16, …
  (−1, 3]
• A103327 → basis A007318
  1 · 4, 1 · 16, 12, 1 · 64, 80, 24, 1 … row sums 1, 5, 29, 169, 985, 5741, 33461, …
  (−∞, −1] nonnegative
• A059344 → basis A007318
  1 · 1 · 3, 2 · 7, 6 … row sums 1, 1, 5, 13, 73, 281, 1741, 8485, …
  (−∞, −1] nonnegative
• A049458: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 2.
  A028338 → x ↦ −2x+5
  1 · -3, 1 · 12, -7, 1 · -60, 47, -12, 1 … row sums 1, -2, 6, -24, 120, -720, 5040, -40320, …
  A049458 [5/2, ∞) rational roots
• A099093 → x ↦ x−15
  1 · -45, 3 · 1980, -267, 9 · -87075, 17685, -1197, 27 … row sums 1, -42, 1722, -70560, 2891196, …
  (41/3, 15]
• A094816 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 16, 23, 9, 1 … row sums 1, 3, 11, 49, 261, 1631, 11743, 95901, …
  (−∞, −1] nonnegative
• A049459: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 3.
  A028338 → x ↦ −2x+7
  1 · -4, 1 · 20, -9, 1 · -120, 74, -15, 1 … row sums 1, -3, 12, -60, 360, -2520, 20160, …
  A049459 [7/2, ∞) rational roots
• A113214 → x ↦ x−15
  1 · -13, 1 · 180, -27, 1 · -2505, 557, -41, 1 … row sums 1, -12, 154, -1988, 25676, -331632, …
  (11, 15]
• A034870 → x ↦ x+1
  1 · 4, 4, 1 · 16, 32, 24, 8, 1 … row sums 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, −1] nonnegative rational roots
• A152198 → x ↦ x+1
  1 · 1 · 2, 1 · 2, 1 … row sums 1, 1, 3, 3, 9, 9, 27, 27, …
  (−∞, −1] nonnegative rational roots
• A055356 → x ↦ x+1
  1 · 1 · 2, 1 · 7, 8, 2 … row sums 1, 1, 3, 17, 143, 1597, 22243, 371241, …
  (−∞, −1] nonnegative
• A085880 → x ↦ x+1
  1 · 2, 1 · 8, 8, 2 · 40, 60, 30, 5 … row sums 1, 3, 18, 135, 1134, 10206, 96228, …
  (−∞, −1] nonnegative rational roots
• A027465: Cube of lower triangular normalized binomial matrix.
  1 · 3, 1 · 9, 6, 1 · 27, 27, 9, 1 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A027465 (−∞, −1] nonnegative rational roots
• A008277 → θ+1 → x ↦ x+1
  1 · 3, 2 · 10, 12, 3 · 37, 62, 30, 4 … row sums 0, 1, 5, 25, 133, 761, 4677, 30745, …
  (−∞, −1] nonnegative
• A318772 → x ↦ (1/243)x → x ↦ x+1
  1 · 3 · 9 · 27 … row sums 1, 3, 9, 27, 83, 255, 783, 2403, …
  (−∞, −1] nonnegative
• A118394 → basis A007318
  1 · 1 · 1 · 7, 6 … row sums 1, 1, 1, 13, 49, 121, 1681, 10501, …
  (−∞, −1] nonnegative
• A136124: Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).
  1 · 2, 1 · 6, 5, 1 · 24, 26, 9, 1 … row sums 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  A136124 (−∞, −1] nonnegative rational roots
• A113953 → x ↦ x−12
  1 · -12, 1 · 120, -22, 1 · -1152, 336, -32, 1 … row sums 1, -11, 99, -847, 7139, -59895, 501787, …
  (4, 12]
• A099097 → x ↦ x−5
  1 · -15, 3 · 220, -89, 9 · -3225, 1965, -399, 27 … row sums 1, -12, 140, -1632, 19024, -221760, …
  (41/9, 5]
• A099097 → x ↦ x−4
  1 · -12, 3 · 140, -71, 9 · -1632, 1248, -318, 27 … row sums 1, -9, 78, -675, 5841, -50544, 437373, …
  (32/9, 4]