preserver_image / affine, shift 0

Image under a real-rootedness preserver. All operations

318 sequences

• A099040 → basis A007318
  1 · 2, 2 · 6, 10, 4 · 16, 40, 32, 8 … row sums 1, 4, 20, 96, 464, 2240, 10816, 52224, …
  (−∞, −1] nonnegative
• A113953 → x ↦ x−3
  1 · -3, 1 · 3, -4, 1 · 9, 3, -5, 1 … row sums 1, -2, 0, 8, -16, 0, 64, -128, …
  (−5, 3]
• A132812 → x ↦ x+1
  1 · 4, 2 · 15, 15, 3 · 56, 84, 36, 4 … row sums 1, 6, 33, 180, 985, 5418, 29953, …
  (−∞, −1] nonnegative
• A082985 → x ↦ x+1
  1 · 4, 3 · 11, 15, 5 · 29, 56, 35, 7 … row sums 1, 7, 31, 127, 511, 2047, 8191, 32767, …
  (−∞, −1] nonnegative
• basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1).
  1 · 0, 2 · -2, 2, 3 · 0, -8, 8, 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 1]
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
  1 · 3, 2 · 9, 12, 4 · 27, 54, 36, 8 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A038220 (−∞, −1] nonnegative rational roots
• A063007 → x ↦ x+1
  1 · 3, 2 · 13, 18, 6 · 63, 132, 90, 20 … row sums 1, 5, 37, 305, 2641, 23525, 213445, …
  (−2, −1) nonnegative
• A113953 → basis A007318
  1 · 1, 1 · 3, 4, 1 · 5, 11, 7, 1 … row sums 1, 2, 8, 24, 80, 256, 832, 2688, …
  (−∞, −1] nonnegative
• A013620 → basis A007318
  1 · 5, 3 · 25, 30, 9 · 125, 225, 135, 27 … row sums 1, 8, 64, 512, 4096, 32768, 262144, …
  (−∞, −1] nonnegative rational roots
• A033282 → x ↦ x+1
  1 · 3, 2 · 11, 15, 5 · 45, 93, 63, 14 … row sums 1, 5, 31, 215, 1597, 12425, 99955, …
  (−2, −1) nonnegative
• A034870: Even-numbered rows of Pascal's triangle.
  1 · 1, 2, 1 · 1, 4, 6, 4, 1 · 1, 6, 15, 20, 15, 6, 1 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A034870 (−∞, −1] nonnegative rational roots
• A099097 → x ↦ x−2
  1 · -6, 3 · 34, -35, 9 · -192, 300, -156, 27 … row sums 1, -3, 8, -21, 55, -144, 377, -987, …
  (14/9, 2]
• A081579 → basis A007318
  1 · 2, 1 · 8, 8, 1 · 24, 36, 14, 1 … row sums 1, 3, 17, 75, 361, 1683, 7937, 37275, …
  (−∞, −1] nonnegative
• A039756 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 24, 38, 16, 1 … row sums 1, 3, 13, 79, 601, 5339, 53861, 607527, …
  (−∞, −1] nonnegative
• A099097 → x ↦ x−8
  1 · -24, 3 · 568, -143, 9 · -13440, 5088, -642, 27 … row sums 1, -21, 434, -8967, 185269, -3827880, …
  (68/9, 8]
• A128099 → x ↦ x+1
  1 · 1 · 3, 2 · 5, 4 … row sums 1, 1, 5, 9, 29, 65, 181, 441, …
  (−∞, −1] nonnegative
• A051379: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 7.
  A028338 → x ↦ −2x+15
  1 · -8, 1 · 72, -17, 1 · -720, 242, -27, 1 … row sums 1, -7, 56, -504, 5040, -55440, 665280, …
  A051379 [15/2, ∞) rational roots
• A113214 → x ↦ x−9
  1 · -7, 1 · 54, -15, 1 · -423, 173, -23, 1 … row sums 1, -6, 40, -272, 1856, -12672, 86528, …
  (5, 9]
• A113214 → x ↦ x−10
  1 · -8, 1 · 70, -17, 1 · -620, 222, -26, 1 … row sums 1, -7, 54, -423, 3321, -26082, 204849, …
  (6, 10]
• A145596: basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x -> x + 1
  1 · 2, 2 · 3, 8, 3 · 4, 20, 20, 4 … row sums 1, 4, 14, 48, 165, 572, 2002, 7072, …
  A145596 (−∞, 0) nonnegative
• A104684 → basis A007318
  1 · 3, 1 · 13, 8, 1 · 63, 57, 15, 1 … row sums 1, 4, 22, 136, 886, 5944, 40636, …
  (−∞, −1] nonnegative
• A318773: Triangle T(n,k) = 3*T(n-1,k) + T(n-4,k-1) for k = 0..floor(n/4), with T(0,0) = 1 and T(n,k) = 0 for n or k < 0, read by rows.
  A318772 → x ↦ (1/243)x
  1 · 3 · 9 · 27 … row sums 1, 3, 9, 27, 82, 249, 756, 2295, …
  A318773 (−∞, 0] nonnegative
• A100861 → reverse → x ↦ x+1
  1 · 1, 1 · 2, 3, 1 · 4, 9, 6, 1 … row sums 1, 2, 6, 20, 76, 312, 1384, 6512, …
  (−∞, −1] nonnegative
• A059344 → ⊙ 1/k! (shift -1) → x ↦ x+1
  1 · 1 · 3, 2 · 7, 6 … row sums 1, 1, 5, 13, 49, 161, 581, 2045, …
  (−∞, −1] nonnegative
• A116412 → basis A007318
  1 · 4, 1 · 13, 8, 1 · 43, 42, 12, 1 … row sums 1, 5, 22, 98, 436, 1940, 8632, 38408, …
  (−∞, −1] nonnegative
• A062145 → x ↦ x+1
  1 · 5, 4 · 21, 30, 10 · 84, 168, 105, 20 … row sums 1, 9, 61, 377, 2241, 13073, 75517, …
  (−∞, −1] nonnegative
• A008278 → x ↦ x+1 (shift +1) → x ↦ x+1
  1 · 3, 1 · 11, 7, 1 · 49, 46, 13, 1 … row sums 1, 4, 19, 109, 742, 5815, 51193, …
  (−∞, −1] nonnegative
• A008292 → γ (scale 1/2) (shift +1) → x ↦ x+1
  1 · 1 · 2, 1 · 5, 4 … row sums 1, 1, 3, 9, 39, 189, 1107, 7281, …
  (−∞, −1] nonnegative
• A099040 → x ↦ x−9
  1 · -18, 2 · 306, -70, 4 · -5184, 1800, -208, 8 … row sums 1, -16, 240, -3584, 53504, -798720, …
  (7, 9]
• A274498 → x ↦ x+1
  1 · 3 · 9, 6 · 27, 18 … row sums 1, 3, 15, 45, 225, 675, 3375, 10125, …
  (−∞, −1] nonnegative rational roots
• A318772 → basis A007318
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 7, 13, 19, 25, …
  (−∞, −1] nonnegative
• A099097 → x ↦ x−10
  1 · -30, 3 · 890, -179, 9 · -26400, 7980, -804, 27 … row sums 1, -27, 720, -19197, 511839, -13646880, …
  (86/9, 10]
• A051339: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 6.
  A028338 → x ↦ −2x+13
  1 · -7, 1 · 56, -15, 1 · -504, 191, -24, 1 … row sums 1, -6, 42, -336, 3024, -30240, 332640, …
  A051339 [13/2, ∞) rational roots
• A227450 → x ↦ x+1
  2 · 6, 4 · 22, 28, 8 · 94, 172, 96, 16 … row sums 2, 10, 58, 378, 2714, 21178, 177818, …
  (−∞, −1] nonnegative
• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
  1 · 0, 1 · 0, 0, 1 · 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0] [0, ∞) nonnegative rational roots
• A011973 → ⊙ C(n,k) → x ↦ x+1
  1 · 1 · 3, 2 · 7, 6 … row sums 1, 1, 5, 13, 49, 161, 581, 2045, …
  (−∞, −1] nonnegative
• A099093 → basis A007318
  1 · 3, 3 · 12, 21, 9 · 45, 117, 99, 27 … row sums 1, 6, 42, 288, 1980, 13608, 93528, …
  (−∞, −1] nonnegative
• A181371: Triangle read by rows: T(n,k) is the number of ternary words (i.e., finite sequences of 0's, 1's and 2's) of length n having k occurrences of 01's (0 <= k <= floor(n/2)).
  A011973 → x ↦ (1/9)x−1/9
  1 · 3 · 8, 1 · 21, 6 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A181371 (−∞, 1] nonnegative
• A249252 → basis A007318
  1 · 5, 4 · 17, 24, 8 · 61, 124, 80, 16 … row sums 1, 9, 49, 281, 1601, 9129, 52049, …
  (−∞, −1] nonnegative
• A249250 → x ↦ x+1
  1 · 4, 3 · 10, 12, 3 · 28, 48, 24, 3 … row sums 1, 7, 25, 103, 409, 1639, 6553, 26215, …
  (−∞, −1] nonnegative
• A159854 → x ↦ x+1
  1, 2, 1 · 2, 5, 4, 1 · 4, 12, 13, 6, 1 … row sums 4, 12, 36, 108, 324, 972, 2916, 8748, …
  (−∞, −1] nonnegative rational roots
• A113214 → x ↦ x−6
  1 · -4, 1 · 18, -9, 1 · -84, 62, -14, 1 … row sums 1, -3, 10, -35, 125, -450, 1625, -5875, …
  (2, 6]
• A099040 → x ↦ x−15
  1 · -30, 2 · 870, -118, 4 · -25200, 5160, -352, 8 … row sums 1, -28, 756, -20384, 549584, -14817600, …
  (13, 15]
• A049460: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 4.
  A028338 → x ↦ −2x+9
  1 · -5, 1 · 30, -11, 1 · -210, 107, -18, 1 … row sums 1, -4, 20, -120, 840, -6720, 60480, …
  A049460 [9/2, ∞) rational roots
• A122076 → x ↦ x+1
  2 · 5, 2 · 17, 12, 2 · 65, 66, 21, 2 … row sums 2, 7, 31, 154, 799, 4207, 22258, …
  (−∞, ∞) nonnegative
• A063967 → x ↦ x−10
  1 · -9, 1 · 72, -17, 1 · -567, 207, -25, 1 … row sums 1, -8, 56, -384, 2624, -17920, 122368, …
  (5, 9]
• A114655 → x ↦ x+1
  2 · 6, 2 · 22, 16, 2 · 90, 102, 30, 2 … row sums 2, 8, 40, 224, 1344, 8448, 54912, …
  (−∞, −1] nonnegative
• A049444: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 1.
  A028338 → x ↦ −2x+3
  1 · -2, 1 · 6, -5, 1 · -24, 26, -9, 1 … row sums 1, -1, 2, -6, 24, -120, 720, -5040, …
  A049444 [3/2, ∞) rational roots
• basis-map coefficients of A026729 → basis A056242 → x ↦ x+1
  1 · 3, 2 · 10, 13, 4 · 34, 65, 40, 8 … row sums 1, 5, 27, 147, 801, 4365, 23787, …
  (−∞, −1] nonnegative
• A030528 → x ↦ x−1
  1 · 0, 1 · -1, 0, 1 · 1, -2, 0, 1 … row sums 1, 1, 0, 0, 0, 0, 0, 0, …
  (−3, 1]