Sequences

3,043 sequences

• A013611 → θ+1
  1 · 1, 8 · 1, 16, 48 · 1, 24, 144, 256 … row sums 1, 9, 65, 425, 2625, 15625, 90625, …
  (−∞, 0] nonnegative rational roots
• A039991 → D+1
  1 · 1 · 2, -2, -1 · 4, -6, -3 … row sums 1, 1, -1, -5, -11, -19, -29, -41, …
  (−∞, ∞)
• A008277 → θ+1 → D
  2 · 6, 6 · 14, 36, 12 · 30, 150, 120, 20 … row sums 0, 2, 12, 62, 320, 1712, 9604, 56674, …
  (−∞, 0] nonnegative
• A267482: Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=n.
  1 · 1, 1 · -1, 1, 1 · -1, -2, 1, 1 … row sums 1, 2, 1, -1, -2, -1, 1, 2, …
  A267482 (−2, 2)
• A356546 → basis A007318
  1 · 4, 2 · 24, 24, 6 · 160, 240, 120, 20 … row sums 1, 6, 54, 540, 5670, 61236, 673596, …
  (−∞, 0] nonnegative rational roots
• A249251 → basis A013611
  1 · 4, 12 · 13, 72, 96 · 43, 348, 912, 768 … row sums 1, 16, 181, 2071, 23686, 270901, …
  (−∞, 0] nonnegative
• A034870 → basis A013609
  1 · 4, 8, 4 · 16, 64, 96, 64, 16 … row sums 1, 16, 256, 4096, 65536, 1048576, …
  (−∞, 0] nonnegative rational roots
• A038284 → basis A013611
  196, 672, 576 · 2744, 14112, 24192, 13824 … row sums 1444, 54872, 2085136, 79235168, …
  (−∞, 0] nonnegative rational roots
• A132393 → D
  1 · 1, 2 · 2, 6, 3 · 6, 22, 18, 4 … row sums 0, 1, 3, 11, 50, 274, 1764, 13068, …
  (−∞, 0] nonnegative
• A053123: Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order).
  1 · 1, -2 · 1, -4, 3 · 1, -6, 10, -4 … row sums 1, -1, 0, 1, -1, 0, 1, -1, …
  A053123 (−∞, ∞)
• A081579 → basis A007318
  1 · 2, 1 · 8, 8, 1 · 24, 36, 14, 1 … row sums 1, 3, 17, 75, 361, 1683, 7937, 37275, …
  (−∞, −1] nonnegative
• A038208 → basis A013611
  1 · 4, 8 · 16, 64, 64 · 64, 384, 768, 512 … row sums 1, 12, 144, 1728, 20736, 248832, …
  (−∞, 0] nonnegative rational roots
• delannoy → Veronese 2
  1 · 1 · 1, 1 · 1, 5 … row sums 1, 1, 2, 6, 15, 35, 84, 204, …
  (−∞, 0] nonnegative
• A257611: Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 2*n + 3.
  1 · 3, 3 · 9, 30, 9 · 27, 213, 213, 27 … row sums 1, 6, 48, 480, 5760, 80640, 1290240, …
  A257611 (−∞, 0) nonnegative
• A274496 → basis A008277
  1 · 2 · 4, 2 · 8, 4 … row sums 1, 2, 6, 12, 40, 80, 296, 592, …
  (−∞, 0] nonnegative
• A099093 → Veronese 2
  1 · 0, 9 · 0, 18 · 0, 9, 81 … row sums 1, 0, 9, 18, 90, 324, 1215, 4698, …
  (−∞, 0] nonnegative
• A039756 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 24, 38, 16, 1 … row sums 1, 3, 13, 79, 601, 5339, 53861, 607527, …
  (−∞, −1] nonnegative
• A172431 → ⊙ C(n,k+1) (shift -1) → D
  1 · 4, 2 · 9, 20, 3 · 16, 84, 60, 4 … row sums 0, 1, 6, 32, 164, 830, 4184, 21077, …
  (−∞, 0] nonnegative
• A008278 → x ↦ x+1 (shift +1) → D
  1 · 5, 2 · 23, 20, 3 · 109, 152, 57, 4 … row sums 0, 1, 7, 46, 322, 2427, 19645, 170236, …
  (−∞, 0] nonnegative
• A034870 → basis A013611
  1 · 4, 16, 16 · 16, 128, 384, 512, 256 … row sums 1, 36, 1296, 46656, 1679616, 60466176, …
  (−∞, 0] nonnegative rational roots
• basis-map coefficients of A112227 → basis A112227
  -2, 0, 1 · 0, -6, 0, 1 · 12, 0, -12, 0, 1 … row sums -1, -5, 1, 41, 31, -461, -895, 6481, …
  (−∞, ∞)
• A069138: Triangle formed by multiplying Stirling numbers of 2nd kind S2(n,m) (A008277) by m+1.
  A008277 → θ+2 (shift +1)
  2 · 2, 3 · 2, 9, 4 · 2, 21, 24, 5 … row sums 2, 5, 15, 52, 203, 877, 4140, 21147, …
  A069138 (−∞, 0] nonnegative
• A257627 → Pₙ + Pₙ₊₁
  4, 3 · 12, 63, 9 · 36, 813, 762, 27 … row sums 7, 84, 1638, 43680, 1474200, 60147360, …
  (−∞, ∞) nonnegative
• A143491 → basis A013610
  12, 21, 9 · 60, 141, 108, 27 … row sums 42, 336, 3024, 30240, 332640, 3991680, …
  (−∞, 0] nonnegative rational roots
• A099097 → x ↦ x−8
  1 · -24, 3 · 568, -143, 9 · -13440, 5088, -642, 27 … row sums 1, -21, 434, -8967, 185269, -3827880, …
  (68/9, 8]
• A137320 → basis A008277
  1 · 2, 2 · 6, 14, 4 · 24, 96, 60, 8 … row sums 1, 4, 24, 188, 1796, 20112, 257248, …
  (−∞, 0] nonnegative
• A102547: Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318).
  1 · 1 · 1 · 1, 1 … row sums 1, 1, 1, 2, 3, 4, 6, 9, …
  A102547 (−∞, 0] nonnegative
• A114656: 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 semilength n and having k peaks (1 <= k <=…
  1 · 2, 1 · 4, 6, 1 · 8, 24, 12, 1 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A114656 (−∞, 0) nonnegative
• type_b_eulerian → γ (scale 1/4) → θ+1
  1 · 1 · 1, 2 · 1, 10 … row sums 1, 1, 3, 11, 52, 300, 2040, 15960, …
  (−∞, 0] nonnegative
• A038322 → basis A007318
  1 · 19, 8 · 361, 304, 64 · 6859, 8664, 3648, 512 … row sums 1, 27, 729, 19683, 531441, 14348907, …
  (−∞, 0] nonnegative rational roots
• A128099 → x ↦ x+1
  1 · 1 · 3, 2 · 5, 4 … row sums 1, 1, 5, 9, 29, 65, 181, 441, …
  (−∞, −1] nonnegative
• A178821 → γ
  1 · 5 · 15 · 35 … row sums 1, 5, 15, 35, 70, 126, 210, 330, …
  (−∞, 0] nonnegative rational roots
• A064861 → basis A013610
  6, 21, 18 · 18, 99, 180, 108 · 36, 252, 657, 756, 324 … row sums 45, 405, 2025, 18225, 91125, 820125, …
  (−∞, 0] nonnegative rational roots
• A051379: Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 7.
  A028338 → x ↦ −2x+15
  1 · -8, 1 · 72, -17, 1 · -720, 242, -27, 1 … row sums 1, -7, 56, -504, 5040, -55440, 665280, …
  A051379 [15/2, ∞) rational roots
• A085841 → basis A007318
  1 · 7, 4 · 61, 72, 16 · 547, 1004, 528, 64 … row sums 1, 11, 149, 2143, 31273, 457907, …
  (−∞, 0] nonnegative
• A038330 → Pₙ + Pₙ₊₁
  1872, 1824, 592, 64 · 22464, 29376, 14400, 3136, 256 … row sums 4352, 69632, 1114112, 17825792, …
  (−∞, ∞) nonnegative rational roots
• A122431 → basis A008277
  8, 19, 9, 1 · 8, 46, 46, 13, 1 … row sums 37, 114, 409, 1657, 7432, 36401, …
  (−∞, 0] nonnegative
• A053123 → D+1
  1 · -1, -2 · -3, 2, 3 · -5, 14, -2, -4 … row sums 1, -3, 2, 3, -7, 4, 5, -11, …
  (−∞, ∞)
• A055356 → D+1
  1 · 1 · 2, 1 · 5, 8, 2 … row sums 1, 1, 3, 15, 101, 851, 8603, 101423, …
  (−∞, 0] nonnegative
• A133080 → Veronese 2
  1 · 1 · 0, 1 · 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  (−∞, 0] nonnegative rational roots
• A173018 → Pₙ + Pₙ₊₁
  2 · 2, 1 · 2, 5, 1 · 2, 15, 12, 1 … row sums 2, 3, 8, 30, 144, 840, 5760, 45360, …
  (−∞, ∞) nonnegative
• A113214 → x ↦ x−9
  1 · -7, 1 · 54, -15, 1 · -423, 173, -23, 1 … row sums 1, -6, 40, -272, 1856, -12672, 86528, …
  (5, 9]
• A113214 → x ↦ x−10
  1 · -8, 1 · 70, -17, 1 · -620, 222, -26, 1 … row sums 1, -7, 54, -423, 3321, -26082, 204849, …
  (6, 10]
• A142460 → Pₙ + Pₙ₊₁
  2, 1 · 2, 13, 1 · 2, 95, 84, 1 · 2, 597, 1909, 515, 1 … row sums 3, 16, 182, 3024, 65688, 1759296, …
  (−∞, ∞) nonnegative
• A145596: basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x -> x + 1
  1 · 2, 2 · 3, 8, 3 · 4, 20, 20, 4 … row sums 1, 4, 14, 48, 165, 572, 2002, 7072, …
  A145596 (−∞, 0) nonnegative
• A232535 → θ+1
  1 · 1, 4 · 1, 16, 9 · 1, 36, 75, 16 … row sums 1, 5, 26, 128, 608, 2816, 12800, 57344, …
  (−∞, 0] nonnegative
• A111125 → basis A013611
  1 · 4, 4 · 11, 28, 16 · 29, 124, 160, 64 … row sums 1, 8, 55, 377, 2584, 17711, 121393, …
  (−∞, 0] nonnegative
• A249251 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 8, 13 · 1, 12, 42, 43 … row sums 1, 5, 22, 98, 436, 1940, 8632, 38408, …
  [−1, 0] nonnegative
• A007318 → ×(−(1/2)n+(1/2)n²) → x ↦ x/(x+1)
  1, 4, 4 · 3, 18, 36, 24 · 6, 48, 144, 192, 96 … row sums 0, 0, 9, 81, 486, 2430, 10935, 45927, …
  [−1, 0] nonnegative rational roots
• A143683 → basis A013610
  1 · 2, 3 · 12, 36, 9 · 40, 180, 198, 27 … row sums 1, 5, 57, 445, 4049, 34485, 301993, …
  (−∞, 0] nonnegative