Sequences

• A154929: shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(k,n-k).
  1 · 2, 1 · 3, 4, 1 · 5, 10, 6, 1 … Σ 1, 3, 8, 22, 60, 164, 448, 1224, …
  (−∞, −1]nonnegative
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical conse…
  3 · 6, 3 · 12, 12, 3 · 24, 36, 18, 3 … Σ 3, 9, 27, 81, 243, 729, 2187, 6561, …
  (−∞, 0)nonnegativerational roots
• A038323 → basis A013611
  1 · 20, 36 · 400, 1440, 1296 … Σ 1, 56, 3136, 175616, 9834496, …
  (−∞, 0]nonnegativerational roots
• A094638 → D+1
  1 · 2, 1 · 4, 7, 2 · 7, 28, 29, 6 … Σ 1, 3, 13, 70, 446, 3276, 27252, 253296, …
  (−∞, 0]nonnegative
• A081578 → Veronese 2 (j=1)
  1 · 5 · 9, 1 · 13, 13 … Σ 0, 1, 5, 10, 26, 91, 287, 820, …
  (−∞, 0]nonnegative
• A094816 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 5, 5 · 1, 11, 25, 16 … Σ 1, 3, 11, 53, 345, 2947, 31411, 400437, …
  [−1, 0]nonnegative
• A063967 → D
  1 · 3, 2 · 7, 10, 3 · 15, 32, 21, 4 … Σ 0, 1, 5, 20, 72, 244, 796, 2528, …
  (−∞, 0]nonnegative
• A063967 → basis A013610
  1 · 2, 3 · 6, 15, 9 · 16, 60, 72, 27 … Σ 1, 5, 30, 175, 1025, 6000, 35125, …
  (−∞, 0]nonnegative
• A209414: Triangle of coefficients of polynomials u(n,x) jointly generated with A112351; see the Formula section.
  1 · 1, 1 · 1, 4, 1 · 1, 7, 9, 1 … Σ 1, 2, 6, 18, 54, 162, 486, 1458, …
  (−∞, 0)nonnegative
• A099093 → D+1
  1 · 3, 3 · 3, 21, 9 · 0, 36, 99, 27 … Σ 1, 6, 33, 162, 756, 3402, 14931, 64314, …
  (−∞, 0]nonnegative
• A127773 → reverse
  1 · 3 · 6 · 10 … Σ 1, 3, 6, 10, 15, 21, 28, 36, …
  (−∞, 0]nonnegativerational roots
• Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(…
  1, 12, 1 · 1, 27, 27, 1 · 1, 58, 162, 58, 1 … Σ 14, 56, 280, 1680, 11760, 94080, …
  (−∞, 0)nonnegative
• A142963 → Pₙ + Pₙ₊₁
  2, 2 · 2, 12, 4 · 2, 40, 76, 8 · 2, 104, 588, 472, 16 … Σ 4, 18, 126, 1182, 13734, 189918, …
  (−∞, ∞)nonnegative
• A038284 → basis A008277
  196, 204, 36 · 2744, 5256, 2160, 216 … Σ 436, 10376, 268864, 7547936, 228241216, …
  (−∞, 0]nonnegative
• Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.
  -2, 0, 1 · 0, -8, 0, 1 · 24, 0, -20, 0, 1 … Σ -1, -7, 5, 145, -5, -6095, -5815, …
  (−∞, ∞)
• A188440 → θ+1
  1 · 1 · 1, 4 · 1, 4 … Σ 1, 1, 5, 5, 21, 21, 81, 81, …
  (−∞, 0]nonnegativerational roots
• A178822 → D
  6 · 42, 42 · 168, 336, 168 · 504, 1512, 1512, 504 … Σ 0, 6, 84, 672, 4032, 20160, 88704, …
  (−∞, 0]nonnegativerational roots
• A249251 → θ+1
  1 · 1, 6 · 1, 12, 18 · 1, 18, 63, 48 … Σ 1, 7, 31, 130, 520, 2017, 7651, 28537, …
  (−∞, 0]nonnegative
• A130534 → basis A013611
  1 · 2, 4 · 6, 20, 16 · 24, 104, 144, 64 … Σ 1, 6, 42, 336, 3024, 30240, 332640, …
  (−∞, 0]nonnegativerational roots
• A344565 → basis A013611
  3 · 30, 60 · 180, 720, 720 · 840, 5040, 10080, 6720 … Σ 3, 90, 1620, 22680, 272160, 2939328, …
  (−∞, 0]nonnegativerational roots
• A125092 → θ+1
  1 · 1, 8 · 1, 16, 27 · 1, 24, 81, 64 … Σ 1, 9, 44, 170, 576, 1792, 5248, 14688, …
  (−∞, 0]nonnegative
• A038335 → basis A013611
  1 · 21, 36 · 441, 1512, 1296 … Σ 1, 57, 3249, 185193, 10556001, …
  (−∞, 0]nonnegativerational roots
• A081577 → x ↦ x/(x+1)
  1 · 1, 2 · 1, 6, 6 · 1, 10, 24, 16 … Σ 1, 3, 13, 51, 205, 819, 3277, 13107, …
  [−1, 0]nonnegative
• A330510: Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <=…
  2 · 3, 3 · 4, 10, 4 · 5, 22, 22, 5 … Σ 2, 6, 18, 54, 162, 486, 1458, 4374, …
  (−∞, 0)nonnegative
• A013609 → basis A013610
  1 · 3, 6 · 9, 36, 36 · 27, 162, 324, 216 … Σ 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, 0]nonnegativerational roots
• A152828 → D
  1 · 1 · 1 · 2, 2 … Σ 0, 0, 0, 1, 1, 1, 4, 4, …
  (−∞, 0]nonnegativerational roots
• A038321 → basis A013609
  324, 504, 196 · 5832, 13608, 10584, 2744 … Σ 1024, 32768, 1048576, 33554432, …
  (−∞, 0]nonnegativerational roots
• A178821 → basis A013610
  1 · 10, 15 · 60, 180, 135 · 280, 1260, 1890, 945 … Σ 1, 25, 375, 4375, 43750, 393750, …
  (−∞, 0]nonnegativerational roots
• A193730: Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (2x+1)^n and q(n,x) = (2x+1)^n.
  1 · 2, 1 · 4, 8, 3 · 8, 28, 30, 9 … Σ 1, 3, 15, 75, 375, 1875, 9375, 46875, …
  (−∞, 0)nonnegativerational roots
• A236076: A skewed version of triangular array A122075.
  1 · 0, 2 · 0, 1, 3 · 0, 0, 3, 5 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  [−1, 0]nonnegative
• A152842 → reverse
  1 · 3, 1 · 3, 4, 1 · 9, 15, 7, 1 … Σ 1, 4, 8, 32, 64, 256, 512, 2048, …
  (−∞, 0]nonnegativerational roots
• A269944: Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1,…
  1 · 0, 1 · 0, 1, 1 · 0, 4, 5, 1 … Σ 1, 1, 2, 10, 100, 1700, 44200, 1635400, …
  (−∞, 0]nonnegativerational roots
• A027466 → basis A008277
  1 · 8, 1 · 64, 17, 1 · 512, 217, 27, 1 … Σ 1, 9, 82, 757, 7087, 67340, 649931, …
  (−∞, 0]nonnegative
• A318772 → x ↦ (1/243)x → basis A008277
  1 · 3 · 9 · 27 … Σ 1, 3, 9, 27, 83, 255, 783, 2403, …
  (−∞, 0]nonnegative
• A135838 → basis A013610
  1 · 4, 6 · 8, 24, 18 · 32, 144, 216, 108 … Σ 1, 10, 50, 500, 2500, 25000, 125000, …
  (−∞, 0]nonnegativerational roots
• A113953 → basis A013609
  1 · 1, 2 · 3, 8, 4 · 5, 22, 28, 8 … Σ 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  (−∞, 0]nonnegative
• A120909 → basis A013611
  3 · 9, 24 · 27, 144, 192 · 81, 648, 1728, 1536 … Σ 3, 33, 363, 3993, 43923, 483153, …
  (−∞, 0]nonnegativerational roots
• A395972 → Pₙ + Pₙ₊₁
  12, 88, 152, 48 · 24, 328, 1296, 1584, 384 … Σ 300, 3616, 52816, 902400, 17610944, …
  (−∞, ∞)nonnegative
• A113953 → x ↦ x−5
  1 · -5, 1 · 15, -8, 1 · -25, 35, -11, 1 … Σ 1, -4, 8, 0, -64, 256, -512, 0, …
  (−3, 5]
• A049374: A triangle of numbers related to triangle A030527.
  1 · 6, 1 · 42, 18, 1 · 336, 276, 36, 1 … Σ 1, 7, 61, 649, 8245, 122215, 2069425, …
  (−∞, 0)nonnegative
• A185410: A decomposition of the double factorials A001147.
  1 · 1 · 1, 2 · 1, 10, 4 … Σ 1, 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, 0)nonnegative
• A225467 → Pₙ + Pₙ₊₁
  4, 4 · 12, 44, 16 · 36, 356, 352, 64 … Σ 8, 72, 808, 10664, 160072, 2676744, …
  (−∞, ∞)nonnegative
• A078812 → Veronese 2 (j=1)
  1 · 4 · 10, 1 · 20, 8 … Σ 0, 1, 4, 11, 28, 72, 188, 493, …
  (−∞, 0]nonnegative
• A232535 → reverse
  1 · 2, 1 · 3, 8, 1 · 4, 25, 18, 1 … Σ 1, 3, 12, 48, 192, 768, 3072, 12288, …
  (−∞, 0]nonnegative
• A038333 → basis A007318
  361, 266, 49 · 6859, 7581, 2793, 343 … Σ 676, 17576, 456976, 11881376, …
  (−∞, 0]nonnegativerational roots
• A124038 → D
  1 · -2, 2 · -2, -4, 3 · 4, -6, -6, 4 … Σ 0, 1, 0, -3, -4, 0, 6, 7, …
  (−∞, ∞)
• A207824 → basis A013610
  1 · 6, 3 · 35, 36, 9 · 204, 318, 162, 27 … Σ 1, 9, 80, 711, 6319, 56160, 499121, …
  (−∞, 0]nonnegative
• A133087 → basis A013611
  1 · 3, 4 · 4, 16, 16 · 12, 64, 112, 64 … Σ 1, 7, 36, 252, 1296, 9072, 46656, …
  (−∞, 0]nonnegativerational roots
• Triangle read by rows: T(n, m) = S1(n, m)*4^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n).
  32, -12, 1 · -384, 176, -24, 1 … Σ 21, -231, 3465, -65835, 1514205, …
  [2, ∞)rational roots
• A091044 → basis A013611
  1 · 4, 8 · 16, 64, 48 · 64, 384, 640, 256 … Σ 1, 12, 128, 1344, 14080, 147456, …
  (−∞, 0]nonnegative