gamma_polynomial

Gamma polynomial of a palindromic parent. All operations

99 sequences

• A257614 → γ
  1 · 2 · 4, 20 · 8, 220 … row sums 1, 2, 24, 228, 3996, 75132, 1850364, …
  (−∞, 0] nonnegative
• A008292 → γ
  1 · 1 · 1, 2 · 1, 8 … row sums 1, 1, 3, 9, 39, 189, 1107, 7281, …
  (−∞, 0] nonnegative
• A178821 → γ
  1 · 5 · 15 · 35 … row sums 1, 5, 15, 35, 70, 126, 210, 330, …
  (−∞, 0] nonnegative rational roots
• A038338 → γ
  144 · 1728 · 20736 · 248832 … row sums 144, 1728, 20736, 248832, 2985984, …
  (−∞, 0] nonnegative rational roots
• A135838 → γ
  1 · 2 · 2 · 4 … row sums 1, 2, 2, 4, 4, 8, 8, 16, …
  (−∞, 0] nonnegative rational roots
• A257621 → γ
  1 · 3 · 9, 24 · 27, 312 … row sums 1, 3, 33, 339, 5409, 95427, 2087937, …
  (−∞, 0] nonnegative
• A055151: Triangular array of Motzkin polynomial coefficients.
  narayana → γ (shift +1)
  1 · 1 · 1, 1 · 1, 3 … row sums 1, 1, 2, 4, 9, 21, 51, 127, …
  A055151 (−∞, 0] nonnegative
• A257617 → γ
  1 · 2 · 4, 28 · 8, 364 … row sums 1, 2, 32, 372, 8724, 216108, 7186068, …
  (−∞, 0] nonnegative
• A166343 → γ
  1, 10 · 1, 24 · 1, 54, 48 · 1, 116, 360 … row sums 11, 25, 103, 477, 2739, 17745, 131247, …
  (−∞, 0] nonnegative
• A144440 → γ
  1 · 1 · 1, 9 · 1, 51 … row sums 1, 1, 10, 52, 559, 6112, 89494, …
  (−∞, 0] nonnegative
• A257615 → γ
  1 · 5 · 25, 20 · 125, 340 … row sums 1, 5, 45, 465, 5865, 83565, 1346805, …
  (−∞, 0] nonnegative
• A257609 → γ
  1 · 2 · 4, 8 · 8, 64 … row sums 1, 2, 12, 72, 624, 6048, 70848, 931968, …
  (−∞, 0] nonnegative
• A257610 → γ
  1 · 2 · 4, 12 · 8, 108 … row sums 1, 2, 16, 116, 1348, 17228, 276004, …
  (−∞, 0] nonnegative
• A144441 → γ
  1 · 1 · 1, 12 · 1, 80 … row sums 1, 1, 13, 81, 1117, 15537, 294733, …
  (−∞, 0] nonnegative
• A144445 → γ
  1 · 1 · 1, 21 · 1, 203 … row sums 1, 1, 22, 204, 4663, 106548, 3404962, …
  (−∞, 0] nonnegative
• A257607 → γ
  1 · 5 · 25, 10 · 125, 160 … row sums 1, 5, 35, 285, 2655, 27465, 311715, …
  (−∞, 0] nonnegative
• A085881 → γ
  1 · 1 · 3 · 15 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, 0] nonnegative rational roots
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A010054 (−∞, −1] [−1, 0] [−3−2√2, −3+2√2] nonnegative rational roots
• A008303: Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= ceiling(n/2)-1) = number of permutations of [n] with k peaks.
  A008292 → γ (scale 1/4) (shift +1)
  1 · 2 · 4, 2 · 8, 16 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A008303 (−∞, 0] nonnegative
• A152198: Triangle read by rows, A007318 rows repeated.
  1 · 1 · 1, 1 · 1, 1 … row sums 1, 1, 2, 2, 4, 4, 8, 8, …
  A152198 A152815 (−∞, 0] nonnegative rational roots
• A086645 → γ
  1 · 1 · 1, 4 · 1, 12 … row sums 1, 1, 5, 13, 41, 121, 365, 1093, …
  (−∞, 0] nonnegative
• A257613 → γ
  1 · 4 · 16, 16 · 64, 224 … row sums 1, 4, 32, 288, 3264, 41856, 615168, …
  (−∞, 0] nonnegative
• A142458 → γ
  1 · 1 · 1, 6 · 1, 36 … row sums 1, 1, 7, 37, 379, 4129, 59743, 972973, …
  (−∞, 0] nonnegative
• A156992 → γ
  1 · 2 · 6 · 24 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, 0] nonnegative rational roots
• A344565 → γ
  3 · 15 · 45 · 105 … row sums 3, 15, 45, 105, 210, 378, 630, 990, …
  (−∞, 0] nonnegative rational roots
• A154227 → γ
  1 · 1 · 1, 6 · 1, 16 … row sums 1, 1, 7, 17, 122, 479, 3895, 21139, …
  (−∞, 0] nonnegative
• A144439 → γ
  1 · 1 · 1, 6 · 1, 28 … row sums 1, 1, 7, 29, 223, 1773, 18287, 210509, …
  (−∞, 0] nonnegative
• A010054 → reverse → reverse
  1 · 3 · 6 · 10 … row sums 1, 3, 6, 10, 15, 21, 28, 36, …
  (−∞, 0] nonnegative rational roots
• A081581 → γ
  1 · 1 · 1, 6 · 1, 12 … row sums 1, 1, 7, 13, 55, 133, 463, 1261, …
  (−∞, 0] nonnegative
• A144436 → γ
  1 · 1 · 1, 6 · 1, 20 … row sums 1, 1, 7, 21, 115, 585, 3727, 25549, …
  (−∞, 0] nonnegative
• A130123 → reverse
  1 · 2 · 4 · 8 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, 0] nonnegative rational roots
• A060187 → γ
  1 · 1 · 1, 4 · 1, 20 … row sums 1, 1, 5, 21, 153, 1209, 12285, 140589, …
  (−∞, 0] nonnegative
• A008971: Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2.
  type_b_eulerian → γ (scale 1/4)
  1 · 1 · 1, 1 · 1, 5 … row sums 1, 1, 2, 6, 24, 120, 720, 5040, …
  A008971 (−∞, 0] nonnegative
• A196347 → γ
  1 · 1 · 2 · 6 … row sums 1, 1, 2, 6, 24, 120, 720, 5040, …
  (−∞, 0] nonnegative rational roots
• A054142: Triangular array binomial(2*n-k, k), k=0..n, n >= 0.
  1 · 1, 1 · 1, 3, 1 · 1, 5, 6, 1 … row sums 1, 2, 5, 13, 34, 89, 233, 610, …
  A054142 (−∞, 0) nonnegative
• basis-map coefficients of A030528, shifted → reverse
  1 · -1 · 1 · -1 … row sums 1, -1, 1, -1, 1, -1, 1, -1, …
  (−∞, 0] rational roots
• A081582 → γ
  1 · 1 · 1, 7 · 1, 14 … row sums 1, 1, 8, 15, 71, 176, 673, 1905, …
  (−∞, 0] nonnegative
• narayana → γ
  1 · 1 · 1, 1 · 1, 3 … row sums 1, 1, 2, 4, 9, 21, 51, 127, …
  (−∞, 0] nonnegative
• A091044 → ⊙ C(n+2,k+1) → γ
  1 · 1 · 1, 3 · 1, 11 … row sums 1, 1, 4, 12, 53, 221, 1055, 5055, …
  (−∞, 0] nonnegative
• A144442 → γ
  1 · 1 · 1, 15 · 1, 115 … row sums 1, 1, 16, 116, 1951, 32946, 767576, …
  (−∞, 0] nonnegative
• A257611 → γ
  1 · 3 · 9, 12 · 27, 132 … row sums 1, 3, 21, 159, 1593, 18027, 238509, …
  (−∞, 0] nonnegative
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
  1 · 1, 2 · 1, 4, 4 · 1, 6, 12, 8 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A013609 (−∞, 0) [−1, 0] nonnegative rational roots
• A011973: Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.
  1 · 1 · 1, 1 · 1, 2 … row sums 1, 1, 2, 3, 5, 8, 13, 21, …
  A011973 (−∞, 0] nonnegative
• type_b_eulerian → γ
  1 · 1 · 1, 4 · 1, 20 … row sums 1, 1, 5, 21, 153, 1209, 12285, 140589, …
  (−∞, 0] nonnegative
• A028326 → γ
  2 · 2 · 2 · 2 … row sums 2, 2, 2, 2, 2, 2, 2, 2, …
  (−∞, 0] nonnegative rational roots
• A257620 → γ
  1 · 3 · 9, 18 · 27, 216 … row sums 1, 3, 27, 243, 3159, 45927, 807003, …
  (−∞, 0] nonnegative
• A166340 → γ
  1, 6 · 1, 16 · 1, 38, 32 · 1, 84, 248 … row sums 7, 17, 71, 333, 1923, 12513, 92799, …
  (−∞, 0] nonnegative
• A108838 → ⊙ C(n+1,k+1) (shift +1) → γ
  1 · 1 · 1, 2 · 1, 7 … row sums 1, 1, 3, 8, 29, 106, 436, 1849, …
  (−∞, 0] nonnegative
• A257616 → γ
  1 · 2 · 4, 24 · 8, 288 … row sums 1, 2, 28, 296, 6064, 132128, 3823552, …
  (−∞, 0] nonnegative
• A144438 → γ
  1 · 1 · 1, 3 · 1, 11 … row sums 1, 1, 4, 12, 55, 270, 1612, 10708, …
  (−∞, 0] nonnegative