Sequences

• A257621 → Pₙ + Pₙ₊₁
  4, 3 · 12, 45, 9 · 36, 435, 402, 27 … Σ 7, 66, 900, 15960, 347760, 8981280, …
  (−∞, ∞)nonnegative
• A008292 → γ (scale 1/4) → x ↦ x+1
  1 · 2 · 6, 2 · 24, 16 … Σ 1, 2, 8, 40, 256, 1952, 17408, 177280, …
  (−∞, −1]nonnegative
• A019538 → Pₙ + Pₙ₊₁
  2, 2 · 2, 8, 6 · 2, 20, 42, 24 · 2, 44, 186, 264, 120 … Σ 4, 16, 88, 616, 5224, 51976, 593128, …
  (−∞, ∞)nonnegative
• A180013: Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have exactly k cycles; n>=1, 1<=k<=n.
  A028338 → x ↦ 2x−1
  1 · 0, 2 · 0, 3, 3 · 0, 8, 12, 4 … Σ 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, 1/2]nonnegativerational roots
• A152842 → θ+1
  1 · 1, 6 · 1, 8, 9 · 1, 14, 45, 36 … Σ 1, 7, 18, 96, 224, 1088, 2432, 11264, …
  (−∞, 0]nonnegative
• A137597 → x ↦ x/(x+1)
  1 · 2, 3 · 5, 15, 11 · 15, 67, 98, 47 … Σ 1, 5, 31, 227, 1897, 17693, 181351, …
  [−1, 0]nonnegative
• A172431 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 6, 8 · 1, 9, 25, 21 … Σ 1, 4, 15, 56, 209, 780, 2911, 10864, …
  [−1, 0]nonnegative
• A272866 → D+1
  1 · 4, 5, 1 · 7, 28, 29, 10, 1 … Σ 1, 10, 75, 500, 3125, 18750, 109375, …
  (−∞, 0]nonnegative
• A142458 → γ
  1 · 1 · 1, 6 · 1, 36 … Σ 1, 1, 7, 37, 379, 4129, 59743, 972973, …
  (−∞, 0]nonnegative
• A120910 → basis A008277
  3 · 9, 3 · 27, 21, 3 · 81, 111, 36, 3 … Σ 3, 12, 51, 231, 1116, 5745, 31443, …
  (−∞, 0]nonnegative
• A156992 → basis A013611
  1 · 4, 8 · 24, 96, 96 · 192, 1152, 2304, 1536 … Σ 1, 12, 216, 5184, 155520, 5598720, …
  (−∞, 0]nonnegativerational roots
• A156992 → γ
  1 · 2 · 6 · 24 … Σ 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, 0]nonnegativerational roots
• basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0
  1 · 0, -1 · 0, 0, 1 · 0, 0, 0, -1 … Σ 1, -1, 1, -1, 1, -1, 1, -1, …
  [−1, 0]rational roots
• A099097 → Veronese 2
  1 · 0, 9 · 0, 6 · 0, 1, 81 … Σ 1, 0, 9, 6, 82, 108, 783, 1470, …
  (−∞, 0]nonnegative
• A094305 → D
  3 · 12, 12 · 30, 60, 30 · 60, 180, 180, 60 … Σ 0, 3, 24, 120, 480, 1680, 5376, 16128, …
  (−∞, 0]nonnegativerational roots
• A123519 → x ↦ x+1
  11, 14, 4 · 41, 76, 44, 8 · 153, 372, 324, 120, 16 … Σ 29, 169, 985, 5741, 33461, 195025, …
  (−3, −1)nonnegative
• A145597 → basis A013611
  1 · 6, 12 · 27, 108, 96 · 110, 660, 1200, 640 … Σ 1, 18, 231, 2610, 27765, 286146, …
  (−∞, 0]nonnegative
• A173018 → reverse (shift +1) → D+1
  1 · 1, 1 · 1, 3, 1 · 1, 9, 7, 1 … Σ 1, 2, 5, 18, 84, 480, 3240, 25200, …
  (−∞, 0]nonnegative
• A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).
  0, 3, 1 · 0, 3, 6, 1 · 0, 0, 15, 10, 1 … Σ 4, 10, 26, 76, 232, 764, 2620, 9496, …
  (−∞, 0]nonnegative
• A038255 → basis A008277
  1 · 7, 1 · 49, 15, 1 · 343, 169, 24, 1 … Σ 1, 8, 65, 537, 4516, 38699, 338233, …
  (−∞, 0]nonnegative
• A144084: T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.
  1 · 1, 1 · 1, 4, 2 · 1, 9, 18, 6 … Σ 1, 2, 7, 34, 209, 1546, 13327, 130922, …
  (−∞, 0)nonnegative
• A156365: T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) = 1.
  1, 8, 4 · 1, 22, 44, 8 · 1, 52, 264, 208, 16 … Σ 13, 75, 541, 4683, 47293, 545835, …
  (−∞, 0)nonnegative
• A113953 → Veronese 2
  1 · 0, 1 · 0, 4 · 0, 4, 1 … Σ 1, 0, 1, 4, 5, 8, 25, 44, …
  (−∞, 0]nonnegative
• A121574 → Veronese 2 (j=1)
  1 · 5 · 16, 1 · 44, 11 … Σ 0, 1, 5, 17, 55, 180, 594, 1963, …
  (−∞, 0]nonnegative
• A094638 → Veronese 2 (j=1)
  1 · 3 · 6, 6 · 10, 50 … Σ 0, 1, 3, 12, 60, 360, 2520, 20160, …
  (−∞, 0]nonnegative
• A391045 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 12, 31, 21 … Σ 1, 3, 13, 65, 353, 2023, 12041, 73713, …
  [−1, 0]nonnegative
• basis-map coefficients of A113214, shifted → basis A013610
  1 · -2, -3 · 4, 12, 9 · -8, -36, -54, -27 … Σ 1, -5, 25, -125, 625, -3125, 15625, …
  (−∞, 0]rational roots
• A038325 → basis A013610
  1 · 22, 33 · 484, 1452, 1089 … Σ 1, 55, 3025, 166375, 9150625, …
  (−∞, 0]nonnegativerational roots
• A094638 → (θ+1)²
  1 · 1, 4 · 1, 12, 18 · 1, 24, 99, 96 … Σ 1, 5, 31, 220, 1756, 15596, 152804, …
  (−∞, 0]nonnegative
• A122431 → basis A013609
  8, 24, 24, 8 · 8, 40, 72, 56, 16 … Σ 64, 192, 576, 1728, 5184, 15552, 46656, …
  (−∞, 0]nonnegativerational roots
• A179900: Triangle T(n,k) read by rows: coefficient of [x^k] of the polynomial p_n(x)=(5-x)*p_{n-1}(x)-p_{n-2}(x), p_0=1, p_1=5-x.
  1 · 5, -1 · 24, -10, 1 · 115, -73, 15, -1 … Σ 1, 4, 15, 56, 209, 780, 2911, 10864, …
  (3, 7)
• A008277 → θ+2 (shift +1) → θ+1
  2 · 2, 6 · 2, 18, 12 · 2, 42, 72, 20 … Σ 2, 8, 32, 136, 622, 3060, 16130, 90688, …
  (−∞, 0]nonnegative
• A075502 → Pₙ + Pₙ₊₁
  392, 364, 43, 1 · 2744, 5488, 1267, 71, 1 … Σ 800, 9571, 134241, 2151648, 38484259, …
  (−∞, ∞)nonnegative
• row_factor parent of Triangle read by rows: T(n,k), for k=-n..n-1, is the scaled (by 2^n n!) probability that the sum of n uniform [-1, 1] variables is between…
  1 · 1, 1 · 1, 4, 1 · 1, 11, 11, 1 … Σ 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, 0)nonnegative
• A356546 → basis A013610
  1 · 4, 6 · 24, 72, 54 · 160, 720, 1080, 540 … Σ 1, 10, 150, 2500, 43750, 787500, …
  (−∞, 0]nonnegativerational roots
• A122848 → basis A008277
  1 · 1, 1 · 2, 4, 1 · 4, 16, 9, 1 … Σ 1, 2, 7, 30, 157, 948, 6487, 49332, …
  (−∞, 0]nonnegative
• A201701: Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 17, 41, 99, 239, …
  (−∞, 0)nonnegative
• A166343 → Pₙ + Pₙ₊₁
  2, 39, 28, 1 · 2, 85, 189, 59, 1 … Σ 70, 336, 1960, 13440, 105840, 940800, …
  (−∞, ∞)nonnegative
• A136394: Triangle read by rows: T(n,k) is the number of permutations of an n-set having k cycles of size > 1 (0<=k<=floor(n/2)).
  1 · 1 · 1, 1 · 1, 5 … Σ 1, 1, 2, 6, 24, 120, 720, 5040, …
  (−∞, 0)nonnegative
• A344565 → γ
  3 · 15 · 45 · 105 … Σ 3, 15, 45, 105, 210, 378, 630, 990, …
  (−∞, 0]nonnegativerational roots
• A038208 → Veronese 2 (j=1)
  2 · 8 · 24, 8 · 64, 64 … Σ 0, 2, 8, 32, 128, 512, 2048, 8192, …
  (−∞, 0]nonnegative
• A103327 → θ+1
  1 · 3, 2 · 5, 20, 3 · 7, 70, 63, 4 … Σ 1, 5, 28, 144, 704, 3328, 15360, 69632, …
  (−∞, 0]nonnegative
• A154227 → γ
  1 · 1 · 1, 6 · 1, 16 … Σ 1, 1, 7, 17, 122, 479, 3895, 21139, …
  (−∞, 0]nonnegative
• A317496 → basis A013609
  1 · 1 · 1 · 4, 6 … Σ 1, 1, 1, 10, 19, 28, 118, 289, …
  (−∞, 0]nonnegative
• A011973 → Veronese 2
  1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 2, 4, 7, 11, …
  (−∞, 0]nonnegative
• A105278: Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.
  1 · 2, 1 · 6, 6, 1 · 24, 36, 12, 1 … Σ 1, 3, 13, 73, 501, 4051, 37633, 394353, …
  (−∞, 0)nonnegative
• A358735 → basis A007318
  1 · 2, 1 · 7, 8, 2 · 33, 60, 34, 6 … Σ 1, 3, 17, 133, 1313, 15623, 217409, …
  (−∞, 0]nonnegative
• A114655 → x ↦ x/(x+1)
  2 · 4, 6 · 8, 28, 22 · 16, 96, 168, 90 … Σ 2, 10, 58, 370, 2514, 17850, 130890, …
  [−1, 0]nonnegative
• A122542 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 2, 3 · 0, 2, 8, 7 … Σ 1, 1, 5, 17, 61, 217, 773, 2753, …
  [−1, 0]nonnegative
• A120910 → basis A013611
  3 · 9, 12 · 27, 72, 48 · 81, 324, 432, 192 … Σ 3, 21, 147, 1029, 7203, 50421, 352947, …
  (−∞, 0]nonnegativerational roots