gamma_polynomial

Gamma polynomial of a palindromic parent. All operations

99 sequences

• A391403: Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).
  A011973 → ⊙ C(n,k)
  1 · 1 · 1, 2 · 1, 6 … row sums 1, 1, 3, 7, 19, 51, 141, 393, …
  A391403 (−∞, 0] nonnegative
• A257606 → γ
  1 · 4 · 16, 8 · 64, 104 … row sums 1, 4, 24, 168, 1368, 12456, 125496, …
  (−∞, 0] nonnegative
• A008292 → γ (scale 1/4)
  1 · 2 · 4, 2 · 8, 16 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  (−∞, 0] nonnegative
• A145597 → γ
  1 · 3 · 6, 3 · 10, 15 … row sums 1, 3, 9, 25, 69, 189, 518, 1422, …
  (−∞, 0] nonnegative
• A154228 → γ
  1 · 1 · 1, 14 · 1, 44 … row sums 1, 1, 15, 45, 870, 4965, 126765, …
  (−∞, 0] nonnegative
• A257622 → γ
  1 · 4 · 16, 24 · 64, 360 … row sums 1, 4, 40, 424, 6088, 98536, 1881640, …
  (−∞, 0] nonnegative
• A100861: Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
  1 · 1 · 1, 1 · 1, 3 … row sums 1, 1, 2, 4, 10, 26, 76, 232, …
  A100861 A144299 (−∞, 0] nonnegative
• A101280: Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).
  A008292 → γ (shift +1)
  1 · 1 · 1, 2 · 1, 8 … row sums 1, 1, 3, 9, 39, 189, 1107, 7281, …
  A101280 (−∞, 0] nonnegative
• A257608 → γ
  1 · 1 · 1, 18 · 1, 216 … row sums 1, 1, 19, 217, 6103, 175789, 7103971, …
  (−∞, 0] nonnegative
• A257180 → γ
  9, 6 · 27, 60 · 81, 402, 120 · 243, 2256, 2208 … row sums 15, 87, 603, 4707, 41175, 396063, …
  (−∞, 0] nonnegative
• A167884 → γ
  1 · 1 · 1, 16 · 1, 176 … row sums 1, 1, 17, 177, 4449, 114849, 4144113, …
  (−∞, 0] nonnegative
• A144443 → γ
  1 · 1 · 1, 18 · 1, 156 … row sums 1, 1, 19, 157, 3115, 61885, 1709707, …
  (−∞, 0] nonnegative
• A059473 → γ
  1 · 2 · 4, 2 · 8, 8 … row sums 1, 2, 6, 16, 44, 120, 328, 896, …
  (−∞, 0] nonnegative
• A257618 → γ
  1 · 2 · 4, 32 · 8, 448 … row sums 1, 2, 36, 456, 12048, 334368, 12561984, …
  (−∞, 0] nonnegative
• A257625 → γ
  1 · 3 · 9, 36 · 27, 540 … row sums 1, 3, 45, 567, 12393, 293787, 8955765, …
  (−∞, 0] nonnegative
• A132812 → γ
  1 · 2 · 3, 3 · 4, 12 … row sums 1, 2, 6, 16, 45, 126, 357, 1016, …
  (−∞, 0] nonnegative
• A081580 → γ
  1 · 1 · 1, 5 · 1, 10 … row sums 1, 1, 6, 11, 41, 96, 301, 781, …
  (−∞, 0] nonnegative
• A142459 → γ
  1 · 1 · 1, 8 · 1, 56 … row sums 1, 1, 9, 57, 753, 10449, 196281, …
  (−∞, 0] nonnegative
• A094503: Triangle read by rows: coefficients d(n,k) of André polynomials D(x,n) = Sum_{k>0} d(n,k)*x^k.
  A008292 → γ (scale 1/2) (shift +1)
  1 · 1 · 1, 1 · 1, 4 … row sums 1, 1, 2, 5, 16, 61, 272, 1385, …
  A094503 (−∞, 0] nonnegative
• A166344 → γ
  1, 4 · 1, 12 · 1, 30, 24 · 1, 68, 192 … row sums 5, 13, 55, 261, 1515, 9897, 73575, …
  (−∞, 0] nonnegative
• A143685 → γ
  1 · 1 · 1, 9 · 1, 18 … row sums 1, 1, 10, 19, 109, 280, 1261, 3781, …
  (−∞, 0] nonnegative
• A046739 → γ
  1, 5 · 1, 18 · 1, 47, 61 · 1, 108, 479 … row sums 6, 19, 109, 588, 4033, 29485, 246042, …
  (−∞, 0] nonnegative
• A142460 → γ
  1 · 1 · 1, 10 · 1, 80 … row sums 1, 1, 11, 81, 1311, 22101, 510651, …
  (−∞, 0] nonnegative
• A257626 → γ
  1 · 6 · 36, 36 · 216, 756 … row sums 1, 6, 72, 972, 16524, 317844, 7007148, …
  (−∞, 0] nonnegative
• A052553: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.
  1 · 1 · 1, 1 · 1, 2 … row sums 1, 1, 2, 3, 5, 8, 13, 21, …
  A052553 (−∞, 0] nonnegative
• A007318: Binomial
  1 · 1, 1 · 1, 2, 1 · 1, 3, 3, 1 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A007318 (−∞, −1] nonnegative rational roots
• A257624 → γ
  1 · 5 · 25, 30 · 125, 540 … row sums 1, 5, 55, 665, 10435, 185165, 3812095, …
  (−∞, 0] nonnegative
• A081579 → γ
  1 · 1 · 1, 4 · 1, 8 … row sums 1, 1, 5, 9, 29, 65, 181, 441, …
  (−∞, 0] nonnegative
• A257623 → γ
  1 · 3 · 9, 30 · 27, 420 … row sums 1, 3, 39, 447, 8451, 174843, 4577679, …
  (−∞, 0] nonnegative
• A069466 → γ
  1 · 2 · 6, 12 · 20, 120 … row sums 1, 2, 18, 140, 1330, 12852, 130284, …
  (−∞, 0] nonnegative
• A229032: Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).
  A011973 → x ↦ x−1/4
  1 · 2 · 3, 4 · 4, 16 … row sums 1, 2, 7, 20, 61, 182, 547, 1640, …
  A229032 (−∞, 0] nonnegative
• A038221 → γ
  1 · 3 · 9 · 27 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  (−∞, 0] nonnegative rational roots
• A257619 → γ
  1 · 2 · 4, 36 · 8, 540 … row sums 1, 2, 40, 548, 16108, 495068, 20737900, …
  (−∞, 0] nonnegative
• A154233 → γ
  1 · 1 · 1, 5 · 1, 16 … row sums 1, 1, 6, 17, 131, 624, 5995, 40315, …
  (−∞, 0] nonnegative
• A257612 → γ
  1 · 2 · 4, 16 · 8, 160 … row sums 1, 2, 20, 168, 2448, 38688, 786240, …
  (−∞, 0] nonnegative
• A007318 → D → γ
  1 · 2 · 3 · 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 0] nonnegative rational roots
• A356546 → γ
  1 · 2 · 6 · 20 … row sums 1, 2, 6, 20, 70, 252, 924, 3432, …
  (−∞, 0] nonnegative rational roots
• A256890 → γ
  4, 4 · 8, 28 · 16, 132, 56 · 32, 524, 752 … row sums 8, 36, 204, 1308, 9612, 78444, 708156, …
  (−∞, 0] nonnegative
• A038325 → γ
  1 · 11 · 121 · 1331 … row sums 1, 11, 121, 1331, 14641, 161051, …
  (−∞, 0] nonnegative rational roots
• A304236: Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.
  1 · 1 · 1, 3 · 1, 6 … row sums 1, 1, 4, 7, 19, 40, 97, 217, …
  A304236 (−∞, 0] nonnegative
• A142461 → γ
  1 · 1 · 1, 12 · 1, 108 … row sums 1, 1, 13, 109, 2089, 41449, 1136917, …
  (−∞, 0] nonnegative
• A128099: Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).
  1 · 1 · 1, 2 · 1, 4 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  A128099 (−∞, 0] nonnegative
• A143683 → γ
  1 · 1 · 1, 8 · 1, 16 … row sums 1, 1, 9, 17, 89, 225, 937, 2737, …
  (−∞, 0] nonnegative
• A178822 → γ
  1 · 6 · 21 · 56 … row sums 1, 6, 21, 56, 126, 252, 462, 792, …
  (−∞, 0] nonnegative rational roots
• A154986 → γ
  1 · 1 · 1, 2 · 1, 8 … row sums 1, 1, 3, 9, 45, 225, 1575, 11025, …
  (−∞, 0] nonnegative
• A166341 → γ
  1, 8 · 1, 20 · 1, 46, 40 · 1, 100, 304 … row sums 9, 21, 87, 405, 2331, 15129, 112023, …
  (−∞, 0] nonnegative
• A142462 → γ
  1 · 1 · 1, 14 · 1, 140 … row sums 1, 1, 15, 141, 3123, 71289, 2263815, …
  (−∞, 0] nonnegative
• A178820 → γ
  1 · 4 · 10 · 20 … row sums 1, 4, 10, 20, 35, 56, 84, 120, …
  (−∞, 0] nonnegative rational roots
• A257627 → γ
  1 · 3 · 9, 42 · 27, 672 … row sums 1, 3, 51, 699, 17343, 463167, 16112979, …
  (−∞, 0] nonnegative