Sequences

• A038221 → Pₙ + Pₙ₊₁
  4, 3 · 12, 21, 9 · 36, 99, 90, 27 … Σ 7, 42, 252, 1512, 9072, 54432, 326592, …
  (−∞, ∞)nonnegativerational roots
• A207607 → θ+1
  1 · 1, 4 · 1, 10, 6 · 1, 18, 27, 8 … Σ 1, 5, 17, 54, 166, 499, 1475, 4303, …
  (−∞, 0]nonnegative
• A249251 → (θ+1)²
  1 · 1, 12 · 1, 24, 54 · 1, 36, 189, 192 … Σ 1, 13, 79, 418, 2014, 9127, 39583, …
  (−∞, 0]nonnegative
• A133087 → Veronese 2
  1 · 2 · 1, 1 · 2, 4 … Σ 1, 2, 2, 6, 8, 24, 32, 96, …
  (−∞, 0]nonnegative
• A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
  1 · 2, 2 · 6, 12, 6 · 24, 72, 72, 24 … Σ 1, 4, 24, 192, 1920, 23040, 322560, …
  (−∞, 0)nonnegativerational 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 … Σ 1, 1, 2, 3, 5, 8, 13, 21, …
  (−∞, 0)nonnegative
• A060920 → Veronese 2
  1 · 2 · 5, 1 · 13, 9 … Σ 1, 2, 6, 22, 86, 342, 1366, 5462, …
  (−∞, 0]nonnegative
• A154233 → basis A013611
  1 · 2, 4 · 9, 36, 16 · 40, 240, 352, 64 … Σ 1, 6, 61, 696, 9971, 160746, 3008531, …
  (−∞, 0]nonnegative
• A188440 → basis A013610
  1 · 1 · 3, 6 · 3, 6 … Σ 1, 1, 9, 9, 81, 81, 729, 729, …
  (−∞, 0]nonnegativerational roots
• A038323 → basis A013609
  1 · 20, 18 · 400, 720, 324 · 8000, 21600, 19440, 5832 … Σ 1, 38, 1444, 54872, 2085136, 79235168, …
  (−∞, 0]nonnegativerational roots
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k…
  3 · 3, 6 · 3, 12, 12 · 3, 18, 36, 24 … Σ 3, 9, 27, 81, 243, 729, 2187, 6561, …
  (−∞, 0)nonnegativerational roots
• A099093 → x ↦ x−9
  1 · -27, 3 · 702, -159, 9 · -18225, 6237, -711, 27 … Σ 1, -24, 552, -12672, 290880, -6676992, …
  (23/3, 9]
• A026729 → θ+1
  1 · 0, 2 · 0, 2, 3 · 0, 0, 6, 4 … Σ 1, 2, 5, 10, 20, 38, 71, 130, …
  (−∞, 0]nonnegative
• A081582 → θ+1
  1 · 1, 2 · 1, 18, 3 · 1, 34, 51, 4 … Σ 1, 3, 22, 90, 447, 1925, 8572, 36612, …
  (−∞, 0]nonnegative
• A008292 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 24, 36, 14, 1 … Σ 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  (−∞, −1]nonnegative
• A125092 → (θ+1)²
  1 · 1, 16 · 1, 32, 81 · 1, 48, 243, 256 … Σ 1, 17, 114, 548, 2200, 7872, 25984, …
  (−∞, 0]nonnegative
• A163936 → Pₙ + Pₙ₊₁
  2 · 3, 1 · 8, 9, 1 · 30, 66, 23, 1 … Σ 2, 4, 18, 120, 1050, 11340, 145530, …
  (−∞, ∞)nonnegative
• Riordan array (1-x,x/(1-x)).
  0, 0, 1 · 0, 0, 1, 1 · 0, 0, 1, 2, 1 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, 0]nonnegativerational roots
• A204420 → basis A013610
  1 · 1, 3 · 9, 36, 27 · 225, 1035, 1215, 405 … Σ 1, 4, 72, 2880, 201600, 21772800, …
  (−∞, 0]nonnegativerational roots
• A124848: Triangle read by rows: T(n,k) = (k+1)*(k+2)*(k+3)*binomial(n,k)/6 (0 <= k <= n).
  1 · 1, 4 · 1, 8, 10 · 1, 12, 30, 20 … Σ 1, 5, 19, 63, 192, 552, 1520, 4048, …
  (−∞, 0)nonnegative
• A063007 → D+1
  1 · 3, 2 · 7, 18, 6 · 13, 72, 90, 20 … Σ 1, 5, 31, 195, 1221, 7593, 46915, …
  (−∞, 0]nonnegative
• A178820 → basis A013611
  1 · 8, 16 · 40, 160, 160 · 160, 960, 1920, 1280 … Σ 1, 24, 360, 4320, 45360, 435456, …
  (−∞, 0]nonnegativerational roots
• A053117 → D
  2 · 0, 8 · -4, 0, 24 · 0, -24, 0, 64 … Σ 0, 2, 8, 20, 40, 70, 112, 168, …
  (−∞, ∞)
• A038208 → θ+1
  1 · 2, 4 · 4, 16, 12 · 8, 48, 72, 32 … Σ 1, 6, 32, 160, 768, 3584, 16384, 73728, …
  (−∞, 0]nonnegativerational roots
• A038277 → basis A007318
  1 · 18, 11 · 324, 396, 121 · 5832, 10692, 6534, 1331 … Σ 1, 29, 841, 24389, 707281, 20511149, …
  (−∞, 0]nonnegativerational roots
• A001263 → basis A008277
  1 · 2, 1 · 5, 6, 1 · 14, 31, 12, 1 … Σ 1, 3, 12, 58, 323, 2014, 13793, 102351, …
  (−∞, 0]nonnegative
• A318772 → basis A013611
  1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 16, 31, 46, 61, …
  (−∞, 0]nonnegative
• A235706: (I + A132440)^3: Coefficients for normalized generalized Laguerre polynomials n!*Lag(n, 3-n, -x).
  A132440 → D+1 → (1/n)D²+(2/n)D+1/n
  1 · 3, 1 · 6, 6, 1 · 6, 18, 9, 1 … Σ 1, 4, 13, 34, 73, 136, 229, 358, …
  (−∞, 0]nonnegative
• A304236 → basis A013611
  1 · 1 · 4, 12 · 7, 24 … Σ 1, 1, 16, 31, 271, 736, 4801, 15841, …
  (−∞, 0]nonnegative
• A128099 → Veronese 2
  1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 5, 13, 25, 41, …
  (−∞, 0]nonnegative
• Stirling2 triangle with scaled diagonals (powers of 5).
  25, 15, 1 · 125, 175, 30, 1 · 625, 1875, 625, 50, 1 … Σ 41, 331, 3176, 35451, 447981, 6282416, …
  (−∞, 0)nonnegative
• A123519 → Veronese 2
  1, 4 · 1, 20 · 1, 60, 16 · 1, 140, 144 … Σ 5, 21, 77, 285, 1065, 3977, 14841, …
  (−∞, 0]nonnegative
• basis-map coefficients of A021012 → reverse
  1 · -1 · 2 · -6 … Σ 1, -1, 2, -6, 24, -120, 720, -5040, …
  (−∞, 0]rational roots
• A123519 → D
  6, 8 · 12, 40, 24 · 20, 120, 168, 64 … Σ 14, 76, 372, 1718, 7642, 33112, 140712, …
  (−∞, 0]nonnegative
• A124932: Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).
  1 · 2, 3 · 3, 9, 6 · 4, 18, 24, 10 … Σ 1, 5, 18, 56, 160, 432, 1120, 2816, …
  (−∞, 0)nonnegativerational roots
• A111125 → reverse
  1 · 1, 3 · 1, 5, 5 · 1, 7, 14, 7 … Σ 1, 4, 11, 29, 76, 199, 521, 1364, …
  (−∞, 0]nonnegative
• A272866 → Veronese 2
  1 · 1, 1 · 1, 11, 1 · 1, 30, 30, 1 … Σ 1, 2, 13, 62, 313, 1562, 7813, 39062, …
  (−∞, 0]nonnegative
• A011973 → basis A013609
  1 · 1 · 2, 2 · 3, 4 … Σ 1, 1, 4, 7, 19, 40, 97, 217, …
  (−∞, 0]nonnegative
• A104684: Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder p…
  1 · 2, 1 · 6, 6, 1 · 20, 30, 12, 1 … Σ 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  (−∞, 0)nonnegative
• A054142 → basis A008277
  1 · 2, 1 · 5, 6, 1 · 13, 30, 12, 1 … Σ 1, 3, 12, 56, 292, 1667, 10288, 68026, …
  (−∞, 0]nonnegative
• A152198 → D+1
  1 · 1 · 2, 1 · 2, 1 … Σ 1, 1, 3, 3, 8, 8, 20, 20, …
  (−∞, 0]nonnegativerational roots
• A133087 → Veronese 2 (j=1)
  1 · 2 · 5, 1 · 4, 4 … Σ 0, 1, 2, 6, 8, 24, 32, 96, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/4) (shift +1) → D+1
  1 · 2 · 6, 2 · 24, 16 … Σ 1, 2, 8, 40, 240, 1680, 13440, 120960, …
  (−∞, 0]nonnegative
• A126351 → x ↦ x/(x+1)
  1 · 1, 3 · 1, 7, 10 · 1, 12, 40, 37 … Σ 1, 4, 18, 90, 494, 2946, 18926, 130066, …
  [−1, 0]nonnegative
• A113953 → x ↦ x−2
  1 · -2, 1 · 0, -2, 1 · 8, -4, -2, 1 … Σ 1, -1, -1, 3, -1, -5, 7, 3, …
  (−6, 2]
• A008278 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 15, 23, 10, 1 … Σ 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  (−∞, −1]nonnegative
• A011973 → D+1
  1 · 1 · 2, 1 · 3, 2 … Σ 1, 1, 3, 5, 10, 18, 33, 59, …
  (−∞, 0]nonnegative
• A051338 → Pₙ + Pₙ₊₁
  -5, 1 · 36, -12, 1 · -294, 133, -20, 1 … Σ -4, 25, -180, 1470, -13440, 136080, …
  (−∞, ∞)rational roots
• A317496 → basis A013611
  1 · 1 · 1 · 4, 12 … Σ 1, 1, 1, 16, 31, 46, 286, 751, …
  (−∞, 0]nonnegative
• A178821: Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
  1 · 5, 5 · 15, 30, 15 · 35, 105, 105, 35 … Σ 1, 10, 60, 280, 1120, 4032, 13440, …
  (−∞, 0)nonnegativerational roots