Sequences

• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.
  144, 264, 121 · 1728, 4752, 4356, 1331 … Σ 529, 12167, 279841, 6436343, 148035889, …
  (−∞, 0)nonnegativerational roots
• A099040 → Veronese 2
  1 · 0, 4 · 0, 8 · 0, 4, 16 … Σ 1, 0, 4, 8, 20, 64, 160, 448, …
  (−∞, 0]nonnegative
• A178820 → basis A008277
  1 · 8, 4 · 40, 50, 10 · 160, 380, 180, 20 … Σ 1, 12, 100, 740, 5285, 37744, 274092, …
  (−∞, 0]nonnegative
• A028262: Elements in 3-Pascal triangle (by row).
  1 · 1, 1 · 1, 3, 1 · 1, 4, 4, 1 … Σ 1, 2, 5, 10, 20, 40, 80, 160, …
  (−∞, 0)nonnegative
• A120909 → (θ+1)²
  3 · 3, 24 · 3, 48, 108 · 3, 72, 324, 384 … Σ 3, 27, 159, 783, 3483, 14499, 57591, …
  (−∞, 0]nonnegative
• A132774 → reverse
  1 · 3, 2 · 5, 4 · 7, 6 … Σ 1, 5, 9, 13, 17, 21, 25, 29, …
  (−∞, 0]nonnegativerational roots
• A154228 → basis A007318
  1 · 2, 1 · 18, 18, 1 · 96, 144, 50, 1 … Σ 1, 3, 37, 291, 4943, 67791, 1587413, …
  (−∞, 0]nonnegative
• A121574 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 13, 10 · 8, 40, 64, 33 … Σ 1, 5, 27, 145, 779, 4185, 22483, …
  [−1, 0]nonnegative
• A272867 → basis A007318
  1 · 6, 6, 1 · 36, 72, 48, 12, 1 … Σ 1, 13, 169, 2197, 28561, 371293, …
  (−∞, 0]nonnegative
• A081582 → basis A013609
  1 · 2, 2 · 11, 22, 4 · 36, 108, 80, 8 … Σ 1, 4, 37, 232, 1705, 11692, 82573, …
  (−∞, 0]nonnegative
• A063967 → x ↦ x−8
  1 · -7, 1 · 42, -13, 1 · -245, 119, -19, 1 … Σ 1, -6, 30, -144, 684, -3240, 15336, …
  (3, 7]
• A110441 → basis A013611
  1 · 4, 4 · 14, 32, 16 · 48, 176, 192, 64 … Σ 1, 8, 62, 480, 3716, 28768, 222712, …
  (−∞, 0]nonnegative
• A207607: Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.
  1 · 1, 2 · 1, 5, 2 · 1, 9, 9, 2 … Σ 1, 3, 8, 21, 55, 144, 377, 987, …
  (−4, 0)nonnegative
• A136521: Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.
  1 · 0, 2 · 0, 0, 2 · 0, 0, 0, 2 … Σ 1, 2, 2, 2, 2, 2, 2, 2, …
  (−∞, 0]nonnegativerational roots
• A046716 → basis A008277
  1 · 2, 1 · 5, 6, 1 · 16, 37, 14, 1 … Σ 1, 3, 12, 68, 578, 7412, 136468, …
  (−∞, 0]nonnegative
• delannoy → basis A013610
  1 · 2, 3 · 5, 15, 9 · 12, 54, 72, 27 … Σ 1, 5, 29, 165, 941, 5365, 30589, …
  (−∞, 0]nonnegative
• A030528 → x ↦ x/(x+1)
  1 · 1, 2 · 0, 2, 3 · 0, 1, 5, 5 … Σ 1, 3, 5, 11, 21, 43, 85, 171, …
  [−1, 0]nonnegative
• A249251 → basis A007318
  1 · 4, 3 · 13, 18, 6 · 43, 87, 57, 12 … Σ 1, 7, 37, 199, 1069, 5743, 30853, …
  (−∞, −1]nonnegative
• A123519 → (θ+1)²
  1, 24, 36 · 1, 48, 180, 128 · 1, 80, 540, 896, 400 … Σ 61, 357, 1917, 9717, 47249, 222633, …
  (−∞, 0]nonnegative
• A103633: Triangle read by rows: triangle of repeated stepped binomial coefficients.
  1 · 0, 1 · 0, 1, 1 · 0, 0, 1, 1 … Σ 1, 1, 2, 2, 4, 4, 8, 8, …
  (−∞, 0]nonnegativerational roots
• A207824 → basis A008277
  1 · 6, 1 · 35, 13, 1 · 204, 125, 21, 1 … Σ 1, 7, 49, 351, 2578, 19439, 150631, …
  (−∞, 0]nonnegative
• A267482 → x ↦ x+1
  1 · 2, 1 · 1, 3, 1 · -1, 3, 4, 1 … Σ 1, 3, 5, 7, 9, 11, 13, 15, …
  (−3, 1)
• A144699: Triangle of 5-Eulerian numbers.
  1 · 1, 5 · 1, 16, 25 · 1, 39, 171, 125 … Σ 1, 6, 42, 336, 3024, 30240, 332640, …
  (−∞, 0)nonnegative
• A085881 → basis A013610
  1 · 2, 3 · 12, 36, 27 · 120, 540, 810, 405 … Σ 1, 5, 75, 1875, 65625, 2953125, …
  (−∞, 0]nonnegativerational roots
• delannoy → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 10, 15, 4 … Σ 1, 3, 10, 30, 87, 245, 676, 1836, …
  (−∞, 0]nonnegative
• A280580: Triangle read by rows: T(n,k) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=n.
  1 · 1, 1 · 2, 6, 1 · 5, 30, 15, 1 … Σ 1, 2, 9, 51, 323, 2188, 15511, 113634, …
  (−∞, 0)nonnegative
• A063967 → θ+1
  1 · 1, 2 · 2, 6, 3 · 3, 14, 15, 4 … Σ 1, 3, 11, 36, 116, 364, 1124, 3424, …
  (−∞, 0]nonnegative
• A038207 → Pₙ + Pₙ₊₁
  3, 1 · 6, 5, 1 · 12, 16, 7, 1 · 24, 44, 30, 9, 1 … Σ 4, 12, 36, 108, 324, 972, 2916, 8748, …
  (−∞, ∞)nonnegativerational roots
• A249248 → x ↦ x/(x+1)
  1 · 1, 4 · 1, 9, 11 · 1, 15, 49, 38 … Σ 1, 5, 21, 103, 519, 2793, 15645, 91623, …
  [−1, 0]nonnegative
• A113214 → basis A008277
  1 · 3, 1 · 4, 6, 1 · 7, 21, 10, 1 … Σ 1, 4, 11, 39, 152, 660, 3131, 16078, …
  (−∞, 0]nonnegative
• A038279: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.
  1 · 8, 1 · 64, 16, 1 · 512, 192, 24, 1 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, 0)nonnegativerational roots
• A124038 → x ↦ x+1
  1 · -1, 1 · -2, 0, 1 · -1, -3, 1, 1 … Σ 1, 0, -1, -2, -3, -4, -5, -6, …
  (−3, 2)
• A049029 → Pₙ + Pₙ₊₁
  6, 1 · 50, 16, 1 · 630, 270, 31, 1 … Σ 7, 67, 932, 16867, 374887, 9868532, …
  (−∞, ∞)nonnegative
• A143494 → x ↦ x+1
  1 · 3, 1 · 10, 7, 1 · 37, 40, 12, 1 … Σ 1, 4, 18, 90, 494, 2946, 18926, 130066, …
  (−∞, −1]nonnegative
• A144445 → γ
  1 · 1 · 1, 21 · 1, 203 … Σ 1, 1, 22, 204, 4663, 106548, 3404962, …
  (−∞, 0]nonnegative
• A110441 → reverse
  1 · 1, 3 · 1, 6, 7 · 1, 9, 23, 15 … Σ 1, 4, 14, 48, 164, 560, 1912, 6528, …
  (−∞, 0]nonnegative
• A113214 → Veronese 2 (j=1)
  1 · 3 · 2, 1 · 0, 5 … Σ 0, 1, 3, 3, 5, 10, 14, 23, …
  (−∞, 0]nonnegative
• A266213 → Veronese 2 (j=1)
  2 · 4 · 6, 2 · 8, 12 … Σ 0, 0, 2, 4, 8, 20, 50, 120, …
  (−∞, 0]nonnegative
• A249248 → reverse
  1 · 3, 1 · 3, 7, 1 · 3, 22, 12, 1 … Σ 1, 4, 11, 38, 131, 490, 1897, 7714, …
  (−∞, 0]nonnegative
• A334218 → θ+1
  1 · 1 · 0, 4 · 0, 6, 9 … Σ 1, 1, 4, 15, 72, 420, 2880, 22680, …
  (−∞, 0]nonnegative
• A038231 → Veronese 2 (j=1)
  1 · 8 · 48, 1 · 256, 16 … Σ 0, 1, 8, 49, 272, 1441, 7448, 37969, …
  (−∞, 0]nonnegative
• A135065: A127733 * A007318 as infinite lower triangular matrices.
  1 · 4, 4 · 9, 18, 9 · 16, 48, 48, 16 … Σ 1, 8, 36, 128, 400, 1152, 3136, 8192, …
  (−∞, −1]nonnegativerational roots
• A100861 → basis A013611
  1 · 1 · 2, 4 · 4, 12 … Σ 1, 1, 6, 16, 106, 426, 3076, 15856, …
  (−∞, 0]nonnegative
• A097805: Number of compositions of n with k parts, T(n, k) = binomial(n-1, k-1) for n, k >= 1 and T(n, 0) = 0^n, triangle read by rows for n >= 0 and 0 <= k <=…
  1 · 0, 1 · 0, 1, 1 · 0, 1, 2, 1 … Σ 1, 1, 2, 4, 8, 16, 32, 64, …
  (−∞, 0]nonnegativerational roots
• A257607 → γ
  1 · 5 · 25, 10 · 125, 160 … Σ 1, 5, 35, 285, 2655, 27465, 311715, …
  (−∞, 0]nonnegative
• A008292 → γ (scale 1/4) (shift +1) → θ+1
  1 · 2 · 4, 4 · 8, 32 … Σ 1, 2, 8, 40, 240, 1680, 13440, 120960, …
  (−∞, 0]nonnegative
• A038303 → Pₙ + Pₙ₊₁
  11, 1 · 110, 21, 1 · 1100, 320, 31, 1 … Σ 12, 132, 1452, 15972, 175692, 1932612, …
  (−∞, ∞)nonnegativerational roots
• A081577 → basis A013609
  1 · 2, 2 · 6, 12, 4 · 16, 48, 40, 8 … Σ 1, 4, 22, 112, 580, 2992, 15448, 79744, …
  (−∞, 0]nonnegative
• A159854 → Veronese 2
  0, 1 · 0, 1 · 0, 1, 1 · 0, 1, 3 … Σ 1, 1, 2, 4, 8, 16, 32, 64, …
  (−∞, 0]nonnegative
• Triangle read by rows: T(n, m) = S1(n, m)*6^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).
  72, -18, 1 · -1296, 396, -36, 1 … Σ 55, -935, 21505, -623645, 21827575, …
  [0, ∞)rational roots