Proof methods / preserver_image

Image under a real-rootedness preserver

8,704 sequences

• Veronese parent of A109447 → Veronese 2 (j=1) → θ+1
  1 · 2 · 1, 6 · 4, 8 … Σ 1, 2, 7, 12, 36, 64, 176, 320, …
  (−∞, 0]nonnegative
• A200536 → basis A013609 → (θ+1)²
  1 · 6, 56, 72 · 36, 672, 2628, 3584, 1600 … Σ 1, 134, 8520, 422464, 18389504, …
  (−∞, 0]nonnegative
• A090582 → θ+1
  1 · 2, 2 · 6, 12, 3 · 24, 72, 42, 4 … Σ 1, 4, 21, 142, 1175, 11476, 129073, …
  (−∞, 0]nonnegative
• A008277 → θ+1 → D → x ↦ x+1
  2 · 12, 6 · 62, 60, 12 · 320, 450, 180, 20 … Σ 0, 2, 18, 134, 970, 7142, 54362, …
  (−∞, −1]nonnegative
• A144698 → D+1
  1 · 5, 4 · 14, 45, 16 · 33, 258, 305, 64 … Σ 1, 9, 75, 660, 6300, 65520, 740880, …
  (−∞, 0]nonnegative
• A181371 → D+1
  1 · 3 · 9, 1 · 27, 6 … Σ 1, 3, 10, 33, 108, 351, 1134, 3645, …
  (−∞, 0]nonnegative
• A156992 → basis A007318 → basis A013609
  1 · 6, 4 · 54, 72, 24 · 648, 1296, 864, 192 … Σ 1, 10, 150, 3000, 75000, 2250000, …
  (−∞, 0]nonnegativerational roots
• A131431 → θ+1
  1 · 0, 8 · 0, 0, 21 · 0, 0, 0, 40 … Σ 1, 8, 21, 40, 65, 96, 133, 176, …
  [−1, 0]nonnegativerational roots
• A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
  1 · 0, 3 · 0, 0, 6 · 0, 0, 0, 10 … Σ 1, 3, 6, 10, 15, 21, 28, 36, …
  [−1, 0]nonnegativerational roots
• A131431 → (θ+1)²
  1 · 0, 16 · 0, 0, 63 · 0, 0, 0, 160 … Σ 1, 16, 63, 160, 325, 576, 931, 1408, …
  [−1, 0]nonnegativerational roots
• A118424: Triangle read by rows: T(n,k) is the number of binary sequences of length n containing k subsequences 001 (n,k>=0).
  A102547 → x ↦ (1/8)x−1/8
  1 · 2 · 4 · 7, 1 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, 1]nonnegative
• A064861 → reverse → x ↦ x+1
  6, 5, 1 · 18, 21, 8, 1 · 36, 60, 37, 10, 1 … Σ 12, 48, 144, 576, 1728, 6912, 20736, …
  (−∞, −1]nonnegativerational roots
• A038255 → (θ+1)²
  1 · 6, 4 · 36, 48, 9 · 216, 432, 162, 16 … Σ 1, 10, 93, 826, 7105, 59682, 492205, …
  (−∞, 0]nonnegative
• A272866 → basis A013609 → θ+1
  1 · 5, 20, 12 · 25, 200, 420, 320, 80 … Σ 1, 37, 1045, 26353, 624169, 14204989, …
  (−∞, 0]nonnegative
• A013609 → Veronese 2 (j=1) → basis A007318
  2 · 4 · 14, 8 · 40, 32 … Σ 0, 2, 4, 22, 72, 298, 1100, 4286, …
  (−∞, −1]nonnegative
• A131436 → D+1
  1 · 3, 3 · 0, 14, 7 · 0, 0, 45, 15 … Σ 1, 6, 21, 60, 155, 378, 889, 2040, …
  (−∞, 0]nonnegativerational roots
• A144438 → D
  1 · 5, 2 · 14, 28, 3 · 33, 178, 99, 4 … Σ 0, 1, 7, 45, 314, 2430, 20883, 198310, …
  (−∞, 0]nonnegative
• A038333 → basis A013611 → θ+1
  361, 2128, 2352 · 6859, 60648, 134064, 87808 … Σ 4841, 289379, 16507857, 912500347, …
  (−∞, 0]nonnegativerational roots
• A038243 → basis A013609 → (θ+1)²
  1 · 6, 8 · 36, 96, 36 · 216, 864, 648, 128 … Σ 1, 14, 168, 1856, 19456, 196608, …
  (−∞, 0]nonnegative
• A154929: shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(k,n-k).
  1 · 2, 1 · 3, 4, 1 · 5, 10, 6, 1 … Σ 1, 3, 8, 22, 60, 164, 448, 1224, …
  (−∞, −1]nonnegative
• A151775 → x ↦ x/(x+1)
  1 · 1, 3 · 5, 38, 57 · 61, 845, 2827, 2763 … Σ 1, 4, 100, 6496, 795280, 156781504, …
  [−1, 0]nonnegative
• A111125 → reverse → x ↦ x+1
  1 · 4, 3 · 11, 15, 5 · 29, 56, 35, 7 … Σ 1, 7, 31, 127, 511, 2047, 8191, 32767, …
  (−∞, −1]nonnegative
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical conse…
  3 · 6, 3 · 12, 12, 3 · 24, 36, 18, 3 … Σ 3, 9, 27, 81, 243, 729, 2187, 6561, …
  (−∞, 0)nonnegativerational roots
• A038335 → basis A013609 → D+1
  1 · 39, 18 · 1197, 1404, 324 … Σ 1, 57, 2925, 141453, 6584409, …
  (−∞, 0]nonnegativerational roots
• A094638 → D+1
  1 · 2, 1 · 4, 7, 2 · 7, 28, 29, 6 … Σ 1, 3, 13, 70, 446, 3276, 27252, 253296, …
  (−∞, 0]nonnegative
• A286718 → basis A013611 → reverse
  1 · 4, 2 · 16, 28, 10 · 64, 240, 264, 80 … Σ 1, 6, 54, 648, 9720, 174960, 3674160, …
  (−∞, 0]nonnegativerational roots
• A144698 → x ↦ x/(x+1)
  1 · 1, 5 · 1, 15, 30 · 1, 35, 180, 210 … Σ 1, 6, 46, 426, 4606, 56826, 786766, …
  [−1, 0]nonnegative
• A094816 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 11, 25, 16 … Σ 1, 3, 11, 53, 345, 2947, 31411, 400437, …
  [−1, 0]nonnegative
• A133214 → (θ+1)²
  1, 32, 36 · 1, 72, 324, 128 · 1, 128, 1296, 2048, 400 … Σ 69, 525, 3873, 27753, 194197, 1332957, …
  (−∞, 0]nonnegative
• type_b_eulerian → x ↦ x+1
  1 · 2, 1 · 8, 8, 1 · 48, 72, 26, 1 … Σ 1, 3, 17, 147, 1697, 24483, 423857, …
  (−∞, −1]nonnegative
• A089503 → D+1
  1 · 5, 4 · 13, 72, 30 · 25, 360, 1176, 336 … Σ 1, 9, 115, 1897, 38421, 924121, …
  (−∞, 0]nonnegative
• A063967 → D
  1 · 3, 2 · 7, 10, 3 · 15, 32, 21, 4 … Σ 0, 1, 5, 20, 72, 244, 796, 2528, …
  (−∞, 0]nonnegative
• A086645 → D → x ↦ x+1
  1 · 8, 2 · 48, 36, 3 · 256, 320, 96, 4 … Σ 0, 1, 10, 87, 676, 4925, 34446, 234227, …
  (−∞, −1]nonnegative
• A166961 → D+1
  1 · 2, 1 · 10, 11, 1 · 60, 143, 45, 1 … Σ 1, 3, 22, 249, 4119, 91846, 2630167, …
  (−∞, 0]nonnegative
• A114655 → x ↦ x+1 → (θ+1)²
  2 · 6, 8 · 22, 64, 18 · 90, 408, 270, 32 … Σ 2, 14, 104, 800, 6272, 49728, 397056, …
  (−∞, 0]nonnegative
• basis-map coefficients of A112227 → basis A013609 → (θ+1)²
  1, 16, 36 · 1, 24, 108, 128 · 1, 32, 216, 512, 400 … Σ 53, 261, 1161, 4833, 19197, 73629, …
  (−∞, 0]nonnegative
• A088996 → basis A013611 → D+1
  1 · 5, 4 · 23, 84, 32 · 151, 936, 1552, 384 … Σ 1, 9, 139, 3023, 84489, 2880281, …
  (−∞, 0]nonnegative
• A099093 → D+1
  1 · 3, 3 · 3, 21, 9 · 0, 36, 99, 27 … Σ 1, 6, 33, 162, 756, 3402, 14931, 64314, …
  (−∞, 0]nonnegative
• A127773 → reverse
  1 · 3 · 6 · 10 … Σ 1, 3, 6, 10, 15, 21, 28, 36, …
  (−∞, −1][−1, 0]nonnegative+1
• Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(…
  1, 12, 1 · 1, 27, 27, 1 · 1, 58, 162, 58, 1 … Σ 14, 56, 280, 1680, 11760, 94080, …
  (−∞, 0)nonnegative
• A272867 → basis A013611 → reverse
  1 · 16, 24, 6 · 256, 768, 768, 288, 36 … Σ 1, 46, 2116, 97336, 4477456, 205962976, …
  (−∞, 0]nonnegative
• basis-map coefficients of A297191, shifted → basis A013609 → D+1
  1 · 5, 12, 4 · 9, 56, 120, 96, 16 … Σ 1, 21, 297, 3645, 41553, 452709, …
  (−∞, 0]nonnegativerational roots
• A188440 → θ+1
  1 · 1 · 1, 4 · 1, 4 … Σ 1, 1, 5, 5, 21, 21, 81, 81, …
  (−∞, 0]nonnegativerational roots
• A178822 → D
  6 · 42, 42 · 168, 336, 168 · 504, 1512, 1512, 504 … Σ 0, 6, 84, 672, 4032, 20160, 88704, …
  (−∞, 0]nonnegativerational roots
• A038331 → D
  120, 50 · 2160, 1800, 375 · 34560, 43200, 18000, 2500 … Σ 170, 4335, 98260, 2088025, 42595710, …
  (−∞, 0]nonnegativerational roots
• A127773 → (θ+1)² → x ↦ x+1
  1 · 12, 12 · 54, 108, 54 · 160, 480, 480, 160 … Σ 1, 24, 216, 1280, 6000, 24192, 87808, …
  (−∞, −1]nonnegativerational roots
• A249251 → θ+1
  1 · 1, 6 · 1, 12, 18 · 1, 18, 63, 48 … Σ 1, 7, 31, 130, 520, 2017, 7651, 28537, …
  (−∞, 0]nonnegative
• A085881 → D+1
  1 · 2, 1 · 9, 12, 3 · 60, 135, 90, 15 … Σ 1, 3, 24, 300, 5040, 105840, 2661120, …
  (−∞, 0]nonnegativerational roots
• A391533 → Veronese 2 → D
  1 · 5 · 13, 2 · 25, 14 … Σ 1, 5, 15, 39, 96, 228, 528, 1200, …
  (−∞, 0]nonnegative
• basis-map coefficients of A371080, shifted → basis A013610 → basis A007318
  1 · 4, 3 · 28, 33, 9 · 280, 414, 189, 27 … Σ 1, 7, 70, 910, 14560, 276640, 6086080, …
  (−∞, −1]nonnegativerational roots