Proof methods / preserver_image

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8,704 sequences

• A013612 → basis A013611 → (θ+1)²
  36, 960, 3600 · 216, 8640, 64800, 128000 … Σ 4596, 201656, 7920016, 289582176, …
  (−∞, 0]nonnegative
• A109692 → D+1
  1 · 2, 1 · 5, 10, 3 · 10, 55, 68, 15 … Σ 1, 3, 18, 148, 1520, 18656, 266112, …
  (−∞, 0]nonnegative
• A008276: Triangle of Stirling numbers of first kind, s(n, n-k+1), n >= 1, 1 <= k <= n. Also triangle T(n,k) giving coefficients in expansion of n!*binomial(x,n…
  1 · 1, -1 · 1, -3, 2 · 1, -6, 11, -6 … Σ 1, 0, 0, 0, 0, 0, 0, 0, …
  [0, ∞)rational roots
• delannoy → basis A013610 → x ↦ x+1
  1 · 5, 3 · 29, 33, 9 · 165, 279, 153, 27 … Σ 1, 8, 71, 624, 5489, 48280, 424663, …
  (−∞, −1]nonnegative
• A081581 → D+1
  1 · 2, 1 · 9, 10, 1 · 16, 45, 18, 1 … Σ 1, 3, 20, 80, 372, 1540, 6496, 26496, …
  (−∞, 0]nonnegative
• A128813 → basis A007318 → θ+1
  1 · 2, 2 · 8, 20, 9 · 56, 236, 243, 72 … Σ 1, 4, 37, 607, 15477, 565782, 28060809, …
  (−∞, 0]nonnegative
• A119900 → D
  2 · 1, 6 · 0, 8, 12 · 0, 2, 30, 20 … Σ 0, 2, 7, 20, 52, 128, 304, 704, …
  (−∞, 0]nonnegative
• A056241: Triangle T(n,k) = number of k-part order-consecutive partitions of n (1<=k<=n).
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … Σ 1, 2, 5, 14, 41, 122, 365, 1094, …
  (−∞, 0)nonnegative
• basis-map coefficients of A026729 → basis A007318
  1 · 1, 1 · 1, 2, 1 · 1, 3, 3, 1 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, −1]nonnegativerational roots
• A027467 → x ↦ x+1 → x ↦ x+1
  1 · 17, 1 · 289, 34, 1 · 4913, 867, 51, 1 … Σ 1, 18, 324, 5832, 104976, 1889568, …
  (−∞, −1]nonnegativerational roots
• A196347 → D+1
  1 · 2, 1 · 6, 8, 2 · 24, 54, 36, 6 … Σ 1, 3, 16, 120, 1152, 13440, 184320, …
  (−∞, 0]nonnegativerational roots
• A059473 → basis A013609 → D+1
  1 · 8, 4 · 54, 68, 16 · 320, 688, 416, 64 … Σ 1, 12, 138, 1488, 15444, 156048, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/2) (shift +1) → x ↦ x+1 → D+1
  1 · 1 · 3, 1 · 9, 4 … Σ 1, 1, 4, 13, 66, 351, 2292, 16425, …
  (−∞, 0]nonnegative
• A130123 → Veronese 2 (j=1) → x ↦ x+1
  2 · 8, 8 · 32, 64, 32 · 128, 384, 384, 128 … Σ 0, 2, 0, 16, 0, 128, 0, 1024, …
  (−∞, −1]nonnegativerational roots
• A099040 → basis A013611 → θ+1
  1 · 2, 16 · 6, 80, 192 · 16, 320, 1536, 2048 … Σ 1, 18, 278, 3920, 52460, 678200, …
  (−∞, 0]nonnegative
• A038284 → basis A013611 → x ↦ x/(x+1)
  196, 1064, 1444 · 2744, 22344, 60648, 54872 … Σ 2704, 140608, 7311616, 380204032, …
  [−1, 0]nonnegativerational roots
• A169803 → x ↦ x+1 → D+1
  1 · 3, 1 · 5, 2 · 10, 7, 1 … Σ 1, 4, 7, 18, 37, 84, 179, 390, …
  (−∞, 0]nonnegative
• A104978 → θ+1
  1 · 1, 2 · 2, 10, 9 · 5, 42, 84, 48 … Σ 1, 3, 21, 179, 1657, 16035, 159485, …
  (−∞, 0]nonnegative
• A169803 → Veronese 2 → x ↦ x+1
  1 · 1 · 1 · 2, 1 … Σ 1, 1, 1, 3, 7, 13, 21, 35, …
  (−∞, −1]nonnegative
• A038208 → Veronese 2 (j=1) → D
  8 · 64 · 320, 64 · 1280, 768 … Σ 0, 0, 0, 8, 64, 384, 2048, 10240, …
  (−∞, 0]nonnegative
• basis-map coefficients of A021012 → x ↦ x+1
  1 · -1, -1 · 2, 4, 2 · -6, -18, -18, -6 … Σ 1, -2, 8, -48, 384, -3840, 46080, …
  (−∞, −1]rational roots
• A038323 → basis A013609 → D
  18 · 720, 648 · 21600, 38880, 17496 … Σ 0, 18, 1368, 77976, 3950784, 187662240, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of A026729, shifted → D → basis A007318
  -1 · 2, 2 · -3, -6, -3 · 4, 12, 12, 4 … Σ 0, -1, 4, -12, 32, -80, 192, -448, …
  (−∞, −1]rational roots
• A178822 → basis A013611
  1 · 12, 24 · 84, 336, 336 · 448, 2688, 5376, 3584 … Σ 1, 36, 756, 12096, 163296, 1959552, …
  [−1, 0]nonnegativerational roots
• narayana → basis A007318 → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 9, 21, 14 … Σ 1, 3, 11, 45, 197, 903, 4279, 20793, …
  (−∞, 0]nonnegative
• A125662 → basis A013609 → x ↦ x/(x+1)
  1 · 4, 6 · 15, 46, 35 · 56, 260, 400, 204 … Σ 1, 10, 96, 920, 8816, 84480, 809536, …
  [−1, −1/2)nonnegative
• A038320 → x ↦ x+1 → θ+1
  289, 408, 108 · 4913, 10404, 5508, 864 … Σ 805, 21689, 571849, 14831573, …
  (−∞, 0]nonnegativerational roots
• A143685 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 13, 13 · 1, 24, 66, 44 … Σ 1, 3, 27, 135, 891, 5103, 31347, …
  [−1, 0]nonnegative
• A132393 → D+1 → θ+1
  1 · 1, 2 · 1, 6, 3 · 2, 16, 18, 4 … Σ 1, 3, 10, 40, 194, 1118, 7496, 57440, …
  (−∞, 0]nonnegative
• A144698 → D
  4 · 13, 32 · 32, 226, 192 · 71, 1062, 2463, 1024 … Σ 0, 4, 45, 450, 4620, 50400, 589680, …
  (−∞, 0]nonnegative
• A200536 → D+1
  1 · 4, 7, 2 · 7, 32, 49, 28, 4 … Σ 1, 13, 120, 972, 7344, 53136, 373248, …
  (−∞, 0]nonnegative
• A054142 → reverse → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 12, 15, 4 … Σ 1, 3, 10, 32, 99, 299, 887, 2595, …
  (−∞, 0]nonnegative
• A121574 → basis A013610 → θ+1
  1 · 3, 6 · 10, 42, 27 · 33, 210, 297, 108 … Σ 1, 9, 79, 648, 5116, 39312, 296128, …
  (−∞, 0]nonnegative
• A091044 → D → D
  6 · 56, 24 · 252, 360, 60 · 792, 2376, 1320, 120 … Σ 0, 0, 6, 80, 672, 4608, 28160, 159744, …
  (−∞, 0]nonnegative
• A122542 → basis A013609 → D
  2 · 8, 8 · 26, 56, 24 · 80, 256, 240, 64 … Σ 0, 2, 16, 106, 640, 3658, 20176, …
  (−∞, 0]nonnegative
• A225471 → basis A013609 → (θ+1)²
  1 · 4, 8 · 32, 96, 36 · 384, 1408, 864, 128 … Σ 1, 12, 164, 2784, 57424, 1401664, …
  (−∞, 0]nonnegative
• A075497: Stirling2 triangle with scaled diagonals (powers of 2).
  1 · 2, 1 · 4, 6, 1 · 8, 28, 12, 1 … Σ 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  (−∞, 0)nonnegative
• A127648 → Veronese 2 (j=1) → (θ+1)²
  2 · 0, 16 · 0, 0, 54 · 0, 0, 0, 128 … Σ 0, 0, 2, 0, 16, 0, 54, 0, …
  (−∞, 0]nonnegativerational roots
• A370258 → x ↦ x/(x+1)
  1, 12, 26 · 1, 24, 129, 190 · 1, 40, 384, 1312, 1462 … Σ 39, 344, 3199, 30645, 299220, 2960760, …
  [−1, 0]nonnegative
• A166973 → D
  1 · 7, 2 · 43, 36, 3 · 259, 482, 102, 4 … Σ 0, 1, 9, 82, 847, 9955, 131042, …
  (−∞, 0]nonnegative
• A100862 → x ↦ x+1 → θ+1
  1 · 2, 2 · 5, 10, 3 · 14, 42, 27, 4 … Σ 1, 4, 18, 87, 447, 2418, 13692, 80769, …
  (−∞, 0]nonnegative
• A249251 → D
  3 · 6, 12 · 9, 42, 36 · 12, 90, 180, 96 … Σ 0, 3, 18, 87, 378, 1548, 6102, 23421, …
  (−∞, 0]nonnegative
• A088996 → basis A007318 → θ+1
  1 · 1, 2 · 3, 10, 6 · 15, 68, 75, 24 … Σ 1, 3, 19, 182, 2322, 36908, 701356, …
  (−∞, 0]nonnegative
• A038335 → x ↦ x+1 → (θ+1)²
  1 · 21, 36 · 441, 1512, 729 … Σ 1, 57, 2682, 114480, 4600800, …
  (−∞, 0]nonnegative
• A128100 → Veronese 2 (j=1) → x ↦ x+1
  1 · 2 · 5 · 10 … Σ 0, 0, 1, 2, 5, 10, 22, 46, …
  (−∞, −1]nonnegative
• A008278 → x ↦ x+1 (shift +1) → x ↦ x+1 → x ↦ x/(x+1)
  1 · 3, 4 · 11, 29, 19 · 49, 193, 252, 109 … Σ 1, 7, 59, 603, 7291, 101107, 1571299, …
  [−1, 0]nonnegative
• A013611 → Veronese 2 → x ↦ x+1
  1 · 1 · 17, 16 · 49, 48 … Σ 1, 1, 33, 97, 1217, 5441, 48609, …
  (−∞, −1]nonnegative
• A232535 → reverse → x ↦ x+1
  1 · 3, 1 · 12, 10, 1 · 48, 64, 21, 1 … Σ 1, 4, 23, 134, 781, 4552, 26531, …
  (−∞, −1]nonnegative
• A178820: Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
  1 · 4, 4 · 10, 20, 10 · 20, 60, 60, 20 … Σ 1, 8, 40, 160, 560, 1792, 5376, 15360, …
  (−∞, 0)nonnegativerational roots
• A030528 → Veronese 2 → x ↦ x+1
  1 · 1 · 1, 1 · 3, 3 … Σ 1, 1, 2, 6, 10, 22, 48, 96, …
  (−∞, −1]nonnegative