Proof methods / preserver_image

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8,704 sequences

• A038329 → basis A013610 → D
  270, 162 · 6075, 7290, 2187 … Σ 432, 15552, 497664, 14929920, …
  (−∞, 0]nonnegativerational roots
• A038231 → D
  1 · 8, 2 · 48, 24, 3 · 256, 192, 48, 4 … Σ 0, 1, 10, 75, 500, 3125, 18750, 109375, …
  (−∞, −1]nonnegativerational roots
• A147716 → basis A013611 → x ↦ x/(x+1)
  1 · 15, 19 · 225, 570, 361 · 3375, 12825, 16245, 6859 … Σ 1, 34, 1156, 39304, 1336336, 45435424, …
  [−1, 0]nonnegativerational roots
• A049029 → reverse
  1 · 1, 5 · 1, 15, 45 · 1, 30, 255, 585 … Σ 1, 6, 61, 871, 15996, 358891, 9509641, …
  (−∞, 0]nonnegative
• A026374 → D+1 → θ+1
  1 · 2, 2 · 4, 10, 3 · 5, 24, 21, 4 … Σ 1, 4, 17, 54, 195, 540, 1775, 4550, …
  (−∞, 0]nonnegative
• A038305 → basis A013610 → D
  234, 162 · 4563, 6318, 2187 … Σ 396, 13068, 383328, 10541520, …
  (−∞, 0]nonnegativerational roots
• A125092 → θ+1 → D
  8 · 16, 54 · 24, 162, 192 · 32, 324, 768, 500 … Σ 0, 8, 70, 378, 1624, 6080, 20736, …
  (−∞, 0]nonnegative
• A038279 → basis A013610 → (θ+1)²
  1 · 9, 12 · 81, 216, 81 · 729, 2916, 2187, 432 … Σ 1, 21, 378, 6264, 98496, 1492992, …
  (−∞, 0]nonnegative
• A154227 → D+1
  1 · 2, 1 · 9, 10, 1 · 20, 57, 22, 1 … Σ 1, 3, 20, 100, 690, 4550, 36160, …
  (−∞, 0]nonnegative
• A145597 → D+1
  1 · 6, 3 · 21, 27, 6 · 55, 135, 75, 10 … Σ 1, 9, 54, 275, 1287, 5733, 24752, …
  (−∞, 0]nonnegative
• A113214 → basis A013610 → θ+1
  1 · 3, 6 · 4, 30, 27 · 7, 78, 189, 108 … Σ 1, 9, 61, 382, 2264, 12960, 72368, …
  (−∞, 0]nonnegative
• A123519 → basis A013610 → D
  42, 72 · 228, 792, 648 · 1116, 5832, 9720, 5184 … Σ 114, 1668, 21852, 269178, 3187902, …
  (−∞, 0]nonnegative
• A038226 → basis A013610 → x ↦ x/(x+1)
  121, 770, 1225 · 1331, 12705, 40425, 42875 … Σ 2116, 97336, 4477456, 205962976, …
  [−1, 0]nonnegativerational roots
• A062145 → D+1
  1 · 5, 4 · 11, 30, 10 · 19, 108, 105, 20 … Σ 1, 9, 51, 252, 1170, 5247, 23023, …
  (−∞, 0]nonnegative
• A013622 → x ↦ x/(x+1) → D
  8 · 48, 128 · 216, 1152, 1536 … Σ 0, 8, 176, 2904, 42592, 585640, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A112227 → (θ+1)² → D+1
  0, 18, 9 · 0, 0, 48, 16 · 0, 0, 0, 100, 25 … Σ 27, 64, 125, 216, 343, 512, 729, 1000, …
  (−∞, 0]nonnegativerational roots
• A272866 → Veronese 2 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 13, 13 · 1, 33, 93, 62 … Σ 1, 3, 27, 189, 1377, 9963, 72171, …
  [−1, 0]nonnegative
• basis-map coefficients of A112227 → (θ+1)²
  0, 0, 9 · 0, 0, 0, 16 · 0, 0, 0, 0, 25 … Σ 9, 16, 25, 36, 49, 64, 81, 100, …
  [−1, 0]nonnegativerational roots
• A001263 → basis A013610 → x ↦ x/(x+1)
  1 · 2, 5 · 5, 25, 29 · 14, 105, 249, 185 … Σ 1, 7, 59, 553, 5541, 58107, 629799, …
  [−1, 0]nonnegative
• A081580 → basis A013611 → D+1
  1 · 6, 4 · 45, 68, 16 · 196, 680, 448, 64 … Σ 1, 10, 129, 1388, 14837, 152526, …
  (−∞, 0]nonnegative
• A131689 → x ↦ x+1 → reverse
  1 · 1, 1 · 2, 5, 3 · 6, 24, 31, 13 … Σ 1, 2, 10, 74, 730, 9002, 133210, …
  [−1, 0]nonnegative
• A013613: Triangle of coefficients in expansion of (1+6x)^n.
  1 · 1, 6 · 1, 12, 36 · 1, 18, 108, 216 … Σ 1, 7, 49, 343, 2401, 16807, 117649, …
  (−∞, 0)nonnegativerational roots
• A046716 → x ↦ x+1 → D
  1 · 5, 2 · 25, 22, 3 · 144, 214, 84, 4 … Σ 0, 1, 7, 50, 446, 4983, 66641, 1029172, …
  (−∞, 0]nonnegative
• A049029 → θ+1
  1 · 5, 2 · 45, 30, 3 · 585, 510, 90, 4 … Σ 1, 7, 78, 1189, 22975, 537366, …
  (−∞, 0]nonnegative
• A103406 → Veronese 2 (j=1) → D+1
  2 · 4 · 10, 4 · 24, 16 … Σ 0, 2, 4, 14, 40, 108, 280, 704, …
  (−∞, 0]nonnegative
• A038333 → x ↦ x/(x+1)
  144, 456, 361 · 1728, 8208, 12996, 6859 … Σ 961, 29791, 923521, 28629151, …
  [−1, 0]nonnegativerational roots
• A269944 → x ↦ x+1 → D
  1 · 3, 2 · 17, 16, 3 · 180, 194, 54, 4 … Σ 0, 1, 5, 36, 432, 8172, 227772, …
  (−∞, 0]nonnegative
• A130534: Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is al…
  1 · 1, 1 · 2, 3, 1 · 6, 11, 6, 1 … Σ 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, −1]nonnegativerational roots
• A103327 → D
  1 · 10, 2 · 35, 42, 3 · 84, 252, 108, 4 … Σ 0, 1, 12, 80, 448, 2304, 11264, 53248, …
  (−∞, 0]nonnegative
• A131431 → x ↦ x+1
  1 · 4, 4 · 7, 14, 7 · 10, 30, 30, 10 … Σ 1, 8, 28, 80, 208, 512, 1216, 2816, …
  (−∞, −1]nonnegativerational roots
• A144698 → reverse
  1 · 4, 1 · 16, 13, 1 · 64, 113, 32, 1 … Σ 1, 5, 30, 210, 1680, 15120, 151200, …
  (−∞, 0]nonnegative
• A134991 → (θ+1)²
  1 · 1, 12 · 1, 40, 135 · 1, 100, 945, 1680 … Σ 1, 13, 176, 2726, 48420, 976284, …
  (−∞, 0]nonnegative
• A122848 → basis A013610 → D
  3 · 9, 18 · 27, 108, 81 · 84, 486, 810, 324 … Σ 0, 3, 27, 216, 1704, 13680, 112752, …
  (−∞, 0]nonnegative
• A028338 → x ↦ x+1 → reverse
  1 · 1, 2 · 1, 6, 8 · 1, 12, 44, 48 … Σ 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, 0]nonnegativerational roots
• A243662: Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.
  1 · 3, 1 · 12, 8, 1 · 55, 55, 15, 1 … Σ 1, 4, 21, 126, 818, 5594, 39693, …
  (−∞, −1]nonnegative
• A347056 → (θ+1)²
  1 · 3, 16 · 6, 64, 90 · 10, 160, 450, 320 … Σ 1, 19, 160, 940, 4480, 18592, 69888, …
  (−∞, 0]nonnegative
• A038291 → basis A013611 → basis A007318
  1 · 14, 4 · 196, 112, 16 · 2744, 2352, 672, 64 … Σ 1, 18, 324, 5832, 104976, 1889568, …
  (−∞, −1]nonnegativerational roots
• A227450 → reverse
  2 · 4, 2 · 8, 12, 2 · 16, 48, 28, 2 … Σ 2, 6, 22, 94, 454, 2430, 14214, 89918, …
  (−∞, 0]nonnegative
• A103327 → basis A013609 → x ↦ x/(x+1)
  1 · 4, 6 · 16, 56, 44 · 64, 352, 608, 328 … Σ 1, 10, 116, 1352, 15760, 183712, …
  [−1, 0]nonnegative
• A200536 → basis A013610 → x ↦ x/(x+1)
  1 · 6, 33, 45 · 36, 396, 1629, 2970, 2025 … Σ 1, 84, 7056, 592704, 49787136, …
  [−1, 0]nonnegativerational roots
• A124932 → θ+1
  1 · 2, 6 · 3, 18, 18 · 4, 36, 72, 40 … Σ 1, 8, 39, 152, 520, 1632, 4816, 13568, …
  (−∞, 0]nonnegative
• A330381 → θ+1
  1 · 1, 4 · 1, 10, 9 · 1, 18, 39, 16 … Σ 1, 5, 20, 74, 263, 911, 3098, 10388, …
  (−∞, 0]nonnegative
• A038243 → basis A013611 → θ+1
  1 · 6, 8 · 36, 96, 48 · 216, 864, 864, 256 … Σ 1, 14, 180, 2200, 26000, 300000, …
  (−∞, 0]nonnegativerational roots
• A304236 → basis A013609 → basis A007318
  1 · 1 · 10, 6 · 19, 12 … Σ 1, 1, 16, 31, 271, 736, 4801, 15841, …
  (−∞, −1]nonnegative
• A051380: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 8.
  1 · -9, 1 · 90, -19, 1 · -990, 299, -30, 1 … Σ 1, -8, 72, -720, 7920, -95040, 1235520, …
  [17/2, ∞)rational roots
• A038326 → basis A013609 → D+1
  1 · 47, 24 · 1633, 2256, 576 … Σ 1, 71, 4465, 262871, 14846689, …
  (−∞, 0]nonnegativerational roots
• A116666 → Veronese 2 → D
  5 · 15 · 30, 18 · 50, 90 … Σ 0, 0, 5, 15, 48, 140, 384, 1008, …
  (−∞, 0]nonnegative
• A123519 → basis A007318 → D
  14, 8 · 76, 88, 24 · 372, 648, 360, 64 … Σ 22, 188, 1444, 10446, 72714, 492760, …
  (−∞, 0]nonnegative
• A136572 → Veronese 2 → D+1
  1 · 2, 2 · 0, 48, 24 · 0, 0, 2160, 720 … Σ 1, 0, 4, 0, 72, 0, 2880, 0, …
  (−∞, 0]nonnegativerational roots
• A039755 → reverse
  1 · 1, 1 · 1, 4, 1 · 1, 9, 13, 1 … Σ 1, 2, 6, 24, 116, 648, 4088, 28640, …
  (−∞, 0]nonnegative