Sequences

• A133214 → basis A007318
  13, 16, 4 · 63, 114, 60, 8 · 321, 768, 624, 192, 16 … Σ 33, 245, 1921, 15525, 127905, 1067925, …
  (−∞, 0]nonnegative
• A038220 → Pₙ + Pₙ₊₁
  4, 2 · 12, 14, 4 · 36, 66, 40, 8 … Σ 6, 30, 150, 750, 3750, 18750, 93750, …
  (−∞, ∞)nonnegativerational roots
• A078812 → reverse
  1 · 1, 2 · 1, 4, 3 · 1, 6, 10, 4 … Σ 1, 3, 8, 21, 55, 144, 377, 987, …
  (−∞, 0]nonnegative
• A118933 → basis A013610
  1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 25, 121, 361, 841, …
  (−∞, 0]nonnegative
• A081579 → D
  1 · 6, 2 · 11, 22, 3 · 16, 92, 48, 4 … Σ 0, 1, 8, 36, 160, 640, 2496, 9408, …
  (−∞, 0]nonnegative
• A134082 → x ↦ x+1
  1 · 3, 1 · 5, 6, 1 · 7, 15, 9, 1 … Σ 1, 4, 12, 32, 80, 192, 448, 1024, …
  (−∞, −1]nonnegativerational roots
• A113413 → D
  1 · 4, 2 · 8, 12, 3 · 12, 36, 24, 4 … Σ 0, 1, 6, 23, 76, 233, 682, 1935, …
  (−∞, 0]nonnegative
• A172431 → D
  2 · 4, 6 · 6, 20, 12 · 8, 42, 60, 20 … Σ 0, 2, 10, 38, 130, 420, 1308, 3970, …
  (−∞, 0]nonnegative
• A116414 → Veronese 2 (j=1)
  1 · 8 · 42, 1 · 184, 16 … Σ 0, 1, 8, 43, 200, 880, 3800, 16333, …
  (−∞, 0]nonnegative
• A089231 → Pₙ + Pₙ₊₁
  2, 2 · 2, 8, 6 · 2, 18, 42, 24 · 2, 32, 156, 264, 120 … Σ 4, 16, 86, 574, 4552, 41684, 431986, …
  (−∞, ∞)nonnegative
• A249139 → Veronese 2
  1 · 3 · 5 · 11, 1 … Σ 1, 3, 5, 12, 24, 55, 119, 271, …
  (−∞, 0]nonnegative
• A008955 → basis A007318
  1 · 2, 1 · 10, 13, 4 · 100, 220, 157, 36 … Σ 1, 3, 27, 513, 16929, 863379, 63026667, …
  (−∞, 0]nonnegativerational roots
• A114655: Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a la…
  2 · 4, 2 · 8, 12, 2 · 16, 48, 24, 2 … Σ 2, 6, 22, 90, 394, 1806, 8558, 41586, …
  (−∞, 0)nonnegative
• A038255 → basis A013609
  1 · 7, 2 · 49, 28, 4 · 343, 294, 84, 8 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, 0]nonnegativerational roots
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
  1 · 1, 3 · 1, 6, 9 · 1, 9, 27, 27 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  (−∞, 0)[−1, 0]nonnegative+1
• A066325 → x ↦ x+1
  1 · 1, 1 · 0, 2, 1 · -2, 0, 3, 1 … Σ 1, 2, 3, 2, -5, -18, -11, 86, …
  (−∞, ∞)
• A100861 → basis A013609
  1 · 1 · 2, 2 · 4, 6 … Σ 1, 1, 4, 10, 46, 166, 856, 3844, …
  (−∞, 0]nonnegative
• veronese parent of Binomial coefficients C(n,k) with n-k even, read by rows.
  0, 1 · 1, 1 · 0, 1, 2, 1 · 1, 3, 3, 1 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, 0]nonnegativerational roots
• A257619: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 9*x + 2.
  1 · 2, 2 · 4, 44, 4 · 8, 564, 564, 8 … Σ 1, 4, 52, 1144, 35464, 1418560, …
  (−∞, 0)nonnegative
• A091042 → Veronese 2
  1 · 1 · 1, 5 · 1, 35 … Σ 1, 1, 6, 36, 136, 496, 2016, 8256, …
  (−∞, 0]nonnegative
• 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, 10, 1 · 1, 23, 23, 1 · 1, 50, 138, 50, 1 … Σ 12, 48, 240, 1440, 10080, 80640, …
  (−∞, 0)nonnegative
• A344565 → Veronese 2
  3 · 15 · 45, 45 · 105, 315 … Σ 3, 15, 90, 420, 1680, 6048, 20160, …
  (−∞, 0]nonnegative
• A232535 → basis A013610
  1 · 3, 6 · 12, 42, 27 · 48, 240, 333, 108 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, 0]nonnegative
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
  1 · 2, 1 · 4, 5, 1 · 8, 19, 9, 1 … Σ 1, 3, 10, 37, 151, 674, 3263, 17007, …
  (−∞, 0)nonnegative
• A049374 → Pₙ + Pₙ₊₁
  7, 1 · 48, 19, 1 · 378, 294, 37, 1 … Σ 8, 68, 710, 8894, 130460, 2191640, …
  (−∞, ∞)nonnegative
• A086645 → D
  1 · 6, 2 · 15, 30, 3 · 28, 140, 84, 4 … Σ 0, 1, 8, 48, 256, 1280, 6144, 28672, …
  (−∞, 0]nonnegative
• A091044 → ⊙ C(n+2,k+1) → x ↦ x/(x+1)
  1 · 1, 2 · 1, 7, 7 · 1, 17, 45, 30 … Σ 1, 3, 15, 93, 653, 4983, 40339, 341337, …
  [−1, 0]nonnegative
• A079638: Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.
  1 · 3, 1 · 14, 9, 1 · 90, 83, 18, 1 … Σ 1, 4, 24, 192, 1920, 23040, 322560, …
  (−∞, 0)nonnegative
• A081580: Pascal-(1,5,1) array.
  1 · 1, 1 · 1, 7, 1 · 1, 13, 13, 1 … Σ 1, 2, 9, 28, 101, 342, 1189, 4088, …
  (−∞, 0)nonnegative
• A114656 → basis A007318
  1 · 3, 1 · 11, 8, 1 · 45, 51, 15, 1 … Σ 1, 4, 20, 112, 672, 4224, 27456, …
  (−∞, −1]nonnegative
• A207606 → D+1
  11, 11, 2 · 21, 38, 17, 2 · 36, 102, 81, 23, 2 … Σ 24, 78, 244, 743, 2218, 6521, 18942, …
  (−∞, 0]nonnegative
• A008278 → x ↦ x+1 (shift +1) → D+1
  1 · 3, 1 · 10, 7, 1 · 38, 43, 13, 1 … Σ 1, 4, 18, 95, 579, 3966, 29944, 246141, …
  (−∞, 0]nonnegative
• A154227 → basis A013609
  1 · 2, 2 · 10, 20, 4 · 40, 120, 88, 8 … Σ 1, 4, 34, 256, 2554, 26344, 319912, …
  (−∞, 0]nonnegative
• A127648 → Veronese 2
  1 · 0, 3 · 0, 0, 5 · 0, 0, 0, 7 … Σ 0, 1, 0, 3, 0, 5, 0, 7, …
  (−∞, 0]nonnegativerational roots
• A152198 → basis A013610
  1 · 1 · 2, 3 · 2, 3 … Σ 1, 1, 5, 5, 25, 25, 125, 125, …
  (−∞, 0]nonnegativerational roots
• A127648 → θ+1
  1 · 0, 4 · 0, 0, 9 · 0, 0, 0, 16 … Σ 0, 1, 4, 9, 16, 25, 36, 49, …
  (−∞, 0]nonnegativerational roots
• A232535 → Veronese 2 (j=1)
  2 · 8 · 18, 4 · 32, 56 … Σ 0, 2, 8, 22, 88, 392, 1568, 6112, …
  (−∞, 0]nonnegative
• A124932 → D+1
  1 · 5, 3 · 12, 21, 6 · 22, 66, 54, 10 … Σ 1, 8, 39, 152, 520, 1632, 4816, 13568, …
  (−∞, 0]nonnegative
• A128813 → basis A007318
  1 · 2, 1 · 8, 10, 3 · 56, 118, 81, 18 … Σ 1, 3, 21, 273, 5733, 177723, 7642089, …
  (−∞, 0]nonnegativerational roots
• A099097 → x ↦ x/(x+1)
  1 · 0, 3 · 0, 1, 10 · 0, 0, 6, 33 … Σ 1, 3, 11, 39, 139, 495, 1763, 6279, …
  [−1, 0]nonnegative
• A028326 → x ↦ x/(x+1)
  2 · 2, 4 · 2, 8, 8 · 2, 12, 24, 16 … Σ 2, 6, 18, 54, 162, 486, 1458, 4374, …
  [−1, 0]nonnegativerational roots
• narayana → γ
  1 · 1 · 1, 1 · 1, 3 … Σ 1, 1, 2, 4, 9, 21, 51, 127, …
  (−∞, 0]nonnegative
• A100862 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … Σ 1, 3, 11, 45, 201, 963, 4899, 26253, …
  (−∞, −1]nonnegative
• A110441 → basis A007318
  1 · 4, 1 · 14, 8, 1 · 48, 44, 12, 1 … Σ 1, 5, 23, 105, 479, 2185, 9967, 45465, …
  (−7, −1)nonnegative
• A130595: Triangle read by rows: lower triangular matrix which is inverse to Pascal's triangle (A007318) regarded as a lower triangular matrix.
  1 · -1, 1 · 1, -2, 1 · -1, 3, -3, 1 … Σ 1, 0, 0, 0, 0, 0, 0, 0, …
  [0, ∞)rational roots
• A046716 → basis A013610
  1 · 2, 3 · 5, 15, 9 · 16, 75, 99, 27 … Σ 1, 5, 29, 217, 2297, 34349, 674965, …
  (−∞, 0]nonnegative
• A063967 → Veronese 2 (j=1)
  1 · 3 · 7, 1 · 15, 7 … Σ 0, 1, 3, 8, 22, 60, 164, 448, …
  (−∞, 0]nonnegative
• A344565 → Pₙ + Pₙ₊₁
  18, 15 · 60, 105, 45 · 150, 405, 360, 105 … Σ 33, 210, 1020, 4200, 15456, 52416, …
  (−∞, ∞)nonnegativerational roots
• A113413 → Veronese 2
  1 · 2 · 2, 1 · 2, 6 … Σ 1, 2, 3, 8, 21, 50, 119, 288, …
  (−∞, 0]nonnegative
• A082985 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 7, 11 · 1, 10, 31, 29 … Σ 1, 5, 19, 71, 265, 989, 3691, 13775, …
  [−1, 0]nonnegative