Proof methods / preserver_image

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8,704 sequences

• A156992 → basis A013610 → D+1
  1 · 10, 6 · 96, 180, 54 · 1056, 3456, 3240, 648 … Σ 1, 16, 330, 8400, 255000, 9000000, …
  (−∞, 0]nonnegativerational roots
• A178821 → basis A013610 → x ↦ x+1
  1 · 25, 15 · 375, 450, 135 · 4375, 7875, 4725, 945 … Σ 1, 40, 960, 17920, 286720, 4128768, …
  (−∞, −1]nonnegativerational roots
• A099097 → basis A007318 → D
  3 · 19, 18 · 93, 174, 81 · 407, 1136, 1053, 324 … Σ 0, 3, 37, 348, 2920, 23004, 174108, …
  (−∞, 0]nonnegative
• A166962 → θ+1
  1 · 1, 2 · 1, 24, 3 · 1, 206, 207, 4 … Σ 1, 3, 28, 418, 9448, 293072, 11811692, …
  (−∞, 0]nonnegative
• A038301 → reverse → θ+1
  121, 396, 243 · 1331, 6534, 8019, 2916 … Σ 760, 18800, 448000, 10400000, …
  (−∞, 0]nonnegativerational roots
• A152198 → x ↦ x+1
  1 · 1 · 2, 1 · 2, 1 … Σ 1, 1, 3, 3, 9, 9, 27, 27, …
  (−∞, −1]nonnegativerational roots
• A375853 → x ↦ x+1
  2 · 16, 8 · 96, 96, 20 · 512, 768, 336, 40 … Σ 2, 24, 212, 1656, 12102, 84816, 577576, …
  (−∞, −1]nonnegative
• A199478 → (θ+1)²
  1, 4 · 1, 12 · 1, 20, 18 · 1, 28, 72 … Σ 5, 13, 39, 101, 263, 653, 1599, 3829, …
  (−∞, 0]nonnegative
• A103633 → basis A007318 → θ+1
  1 · 1, 2 · 2, 6, 3 · 2, 10, 12, 4 … Σ 1, 3, 11, 28, 96, 228, 756, 1728, …
  (−∞, 0]nonnegative
• A108838 → reverse → x ↦ x+1
  2 · 5, 3 · 14, 16, 4 · 42, 70, 35, 5 … Σ 2, 8, 34, 152, 706, 3376, 16514, 82256, …
  (−∞, −1]nonnegative
• A177267 → x ↦ x+1
  6, 1 · 24, 10 · 120, 86, 8 · 720, 756, 168 … Σ 7, 34, 214, 1644, 14973, 157318, …
  (−∞, −1]nonnegative
• A120910 → D+1
  3 · 9, 3 · 24, 18, 3 · 60, 72, 27, 3 … Σ 3, 12, 45, 162, 567, 1944, 6561, 21870, …
  (−∞, 0]nonnegativerational roots
• A053124 → D
  4 · -16, 32 · 40, -192, 192 · -80, 672, -1536, 1024 … Σ 0, 4, 16, 40, 80, 140, 224, 336, …
  (−∞, ∞)
• A122431 → θ+1
  1, 6, 9, 4 · 0, 2, 9, 12, 5 · 0, 0, 3, 12, 15, 6 … Σ 20, 28, 36, 44, 52, 60, 68, 76, …
  (−∞, 0]nonnegativerational roots
• A114687: Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of se…
  1 · 1, 2 · 1, 6, 4 · 1, 12, 24, 8 … Σ 1, 3, 11, 45, 197, 903, 4279, 20793, …
  (−∞, 0)nonnegative
• A033282 → θ+1 → D
  4 · 10, 30 · 18, 126, 168 · 28, 336, 1008, 840 … Σ 0, 4, 40, 312, 2212, 14920, 97632, …
  (−∞, 0]nonnegative
• A055356 → x ↦ x+1
  1 · 1 · 2, 1 · 7, 8, 2 … Σ 1, 1, 3, 17, 143, 1597, 22243, 371241, …
  (−∞, −1]nonnegative
• A103327 → basis A013609 → D+1
  1 · 6, 2 · 40, 32, 4 · 224, 352, 120, 8 … Σ 1, 8, 76, 704, 6288, 54656, 465600, …
  (−∞, 0]nonnegative
• A069466 → basis A007318 → D
  2 · 36, 12 · 600, 480, 60 · 9800, 12600, 4200, 280 … Σ 0, 2, 48, 1140, 26880, 631260, …
  (−∞, 0]nonnegative
• A178820 → Veronese 2 (j=1) → (θ+1)²
  4 · 20 · 60, 80 · 140, 560 … Σ 0, 4, 20, 140, 700, 3024, 11760, 42240, …
  (−∞, 0]nonnegative
• A103633 → D
  1 · 1, 2 · 0, 2, 3 · 0, 2, 6, 4 … Σ 0, 1, 3, 5, 12, 16, 36, 44, …
  (−∞, 0]nonnegativerational roots
• A038255 → basis A013610 → θ+1
  1 · 7, 6 · 49, 84, 27 · 343, 882, 567, 108 … Σ 1, 13, 160, 1900, 22000, 250000, …
  (−∞, 0]nonnegativerational roots
• A159854 → basis A013610 → θ+1
  1, 12, 27 · 2, 30, 108, 108 · 4, 72, 351, 648, 405 … Σ 40, 248, 1480, 8600, 49000, 275000, …
  (−∞, 0]nonnegative
• A358735 → D
  1 · 4, 4 · 10, 32, 18 · 20, 140, 228, 96 … Σ 0, 1, 8, 60, 484, 4283, 41604, 441840, …
  (−∞, 0]nonnegative
• A085880 → x ↦ x+1
  1 · 2, 1 · 8, 8, 2 · 40, 60, 30, 5 … Σ 1, 3, 18, 135, 1134, 10206, 96228, …
  (−∞, −1]nonnegativerational roots
• A120910 → basis A007318 → D+1
  3 · 12, 3 · 45, 24, 3 · 162, 135, 36, 3 … Σ 3, 15, 72, 336, 1536, 6912, 30720, …
  (−∞, 0]nonnegativerational roots
• A161198 → basis A007318 → reverse
  1 · 2, 3 · 4, 16, 15 · 8, 60, 142, 105 … Σ 1, 5, 35, 315, 3465, 45045, 675675, …
  (−∞, 0]nonnegativerational roots
• A063967 → (θ+1)²
  1 · 1, 4 · 2, 12, 9 · 3, 28, 45, 16 … Σ 1, 5, 23, 92, 346, 1240, 4296, 14496, …
  (−∞, 0]nonnegative
• A013611 → D
  4 · 8, 32 · 12, 96, 192 · 16, 192, 768, 1024 … Σ 0, 4, 40, 300, 2000, 12500, 75000, …
  (−∞, 0]nonnegativerational roots
• A154233 → x ↦ x+1
  1 · 2, 1 · 9, 9, 1 · 40, 60, 22, 1 … Σ 1, 3, 19, 123, 1091, 10407, 120683, …
  (−∞, −1]nonnegative
• A088996 → basis A013610 → D
  3 · 15, 36 · 102, 450, 486 · 894, 5598, 11502, 7776 … Σ 0, 3, 51, 1038, 25770, 762120, …
  (−∞, 0]nonnegative
• A109692 → basis A007318 → D+1
  1 · 3, 1 · 18, 16, 3 · 148, 236, 113, 15 … Σ 1, 4, 37, 512, 9297, 207828, 5504229, …
  (−∞, 0]nonnegative
• A008277 → θ+1 → D+1
  1 · 3, 2 · 7, 12, 3 · 15, 50, 30, 4 … Σ 0, 1, 5, 22, 99, 471, 2386, 12867, …
  (−∞, 0]nonnegative
• A143685 → D
  1 · 11, 2 · 21, 42, 3 · 31, 282, 93, 4 … Σ 0, 1, 13, 66, 410, 2015, 10371, 49588, …
  (−∞, 0]nonnegative
• A110441 → (θ+1)²
  1 · 3, 4 · 7, 24, 9 · 15, 92, 81, 16 … Σ 1, 7, 40, 204, 968, 4372, 19048, 80736, …
  (−∞, 0]nonnegative
• A147716 → basis A013610 → x ↦ x/(x+1)
  1 · 15, 18 · 225, 540, 324 · 3375, 12150, 14580, 5832 … Σ 1, 33, 1089, 35937, 1185921, 39135393, …
  [−1, 0]nonnegativerational roots
• A027465: Cube of lower triangular normalized binomial matrix.
  1 · 3, 1 · 9, 6, 1 · 27, 27, 9, 1 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  (−∞, −1]nonnegativerational roots
• A038221: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
  1 · 3, 3 · 9, 18, 9 · 27, 81, 81, 27 … Σ 1, 6, 36, 216, 1296, 7776, 46656, …
  (−∞, 0)nonnegativerational roots
• A081582 → γ → D
  7 · 14 · 21, 98 · 28, 294 … Σ 0, 0, 7, 14, 119, 322, 1652, 5138, …
  (−∞, 0]nonnegative
• A075502 → reverse
  1, 21, 49 · 1, 42, 343, 343 · 1, 70, 1225, 5145, 2401 … Σ 71, 729, 8842, 125399, 2026249, …
  (−∞, 0]nonnegative
• A144699 → x ↦ x/(x+1)
  1 · 1, 6 · 1, 18, 42 · 1, 42, 252, 336 … Σ 1, 7, 61, 631, 7525, 101407, 1521661, …
  [−1, 0]nonnegative
• A134346: Triangle read by rows: T(n,k) = (2^(n+1)-1)*binomial(n,k).
  1 · 3, 3 · 7, 14, 7 · 15, 45, 45, 15 … Σ 1, 6, 28, 120, 496, 2016, 8128, 32640, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A113214, shifted → D → D+1
  -1 · 4, 2 · -9, -12, -3 · 16, 36, 24, 4 … Σ 0, -1, 6, -24, 80, -240, 672, -1792, …
  (−∞, 0]rational roots
• A075497 → x ↦ x+1 → reverse
  1 · 1, 3 · 1, 8, 11 · 1, 15, 55, 49 … Σ 1, 4, 20, 120, 832, 6496, 56128, …
  (−∞, 0]nonnegative
• A132774 → Veronese 2 (j=1) → D
  7 · 8 · 0, 22 · 0, 24 … Σ 0, 0, 0, 7, 8, 22, 24, 45, …
  (−∞, 0]nonnegativerational roots
• A172431 → Veronese 2 → D
  3 · 10 · 21, 10 · 36, 70 … Σ 0, 0, 3, 10, 31, 106, 328, 990, …
  (−∞, 0]nonnegative
• A038323 → x ↦ x+1 → D
  9 · 360, 162 · 10800, 9720, 2187 … Σ 0, 9, 522, 22707, 878004, 31827645, …
  (−∞, 0]nonnegativerational roots
• A085880 → x ↦ x/(x+1)
  1 · 1, 2 · 2, 8, 8 · 5, 30, 60, 40 … Σ 1, 3, 18, 135, 1134, 10206, 96228, …
  [−1, 0]nonnegativerational roots
• A113413 → basis A013610 → reverse
  1 · 3, 3 · 9, 18, 7 · 27, 81, 69, 17 … Σ 1, 6, 34, 194, 1106, 6306, 35954, …
  (−∞, 0]nonnegative
• A290319 → basis A013611 → (θ+1)²
  1 · 2, 16 · 12, 128, 144 · 120, 1472, 2592, 1024 … Σ 1, 18, 284, 5208, 112912, 2855968, …
  (−∞, 0]nonnegative