Sequences

• A249250 → Veronese 2 (j=1)
  3 · 6 · 9, 3 · 12, 24 … Σ 0, 3, 6, 12, 36, 108, 288, 768, …
  (−∞, 0]nonnegative
• A048993 → reverse
  1 · 1 · 1, 1 · 1, 3, 1 … Σ 1, 1, 2, 5, 15, 52, 203, 877, …
  (−∞, 0]nonnegative
• A118931 → basis A013611
  1 · 1 · 1 · 3, 8 … Σ 1, 1, 1, 11, 41, 101, 1201, 7351, …
  (−∞, 0]nonnegative
• A074246: Triangle of coefficients, read by rows, where the n-th row forms the polynomial P(n,x) = {Sum_{k=1..n} 1/(k+x)}*{Product_{k=1..n} (k+x)}.
  1 · 3, 2 · 11, 12, 3 · 50, 70, 30, 4 … Σ 1, 5, 26, 154, 1044, 8028, 69264, …
  (−∞, ∞)nonnegative
• A152830 → x ↦ x+1
  1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 3, 3, 3, 3, …
  (−∞, −1]nonnegativerational roots
• A153516: Triangle T(n,k,p,q,j) = T(n-1,k,p,q,j) + T(n-1,k-1,p,q,j) + (p*j+q)*prime(j)*T(n-2,k-1,p,q,j) with T(2,k,p,q,j) = prime(j) and (p,q,j) = (0,1,2), read…
  2 · 3, 3 · 2, 14, 2 · 2, 25, 25, 2 … Σ 2, 6, 18, 54, 162, 486, 1458, 4374, …
  (−∞, 0)nonnegative
• A038337 → basis A013609
  529, 1012, 484 · 12167, 34914, 33396, 10648 … Σ 2025, 91125, 4100625, 184528125, …
  (−∞, 0]nonnegativerational roots
• A133087 → D+1
  1 · 3, 1 · 3, 4, 1 · 7, 13, 7, 1 … Σ 1, 4, 8, 28, 48, 160, 256, 832, …
  (−∞, 0]nonnegative
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
  1 · 1, 2 · 1, 4, 4 · 1, 6, 12, 8 … Σ 1, 3, 9, 27, 81, 243, 729, 2187, …
  (−∞, 0)[−1, 0]nonnegative+1
• A152842: Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A…
  1 · 1, 3 · 1, 4, 3 · 1, 7, 15, 9 … Σ 1, 4, 8, 32, 64, 256, 512, 2048, …
  (−∞, 0]nonnegativerational roots
• A143494 → Pₙ + Pₙ₊₁
  3, 1 · 6, 6, 1 · 12, 24, 10, 1 · 24, 84, 64, 15, 1 … Σ 4, 13, 47, 188, 825, 3937, 20270, …
  (−∞, ∞)nonnegative
• A153653: Triangle T(n, k, j) = T(n-1, k, j) + T(n-1, k-1, j) + (2*j + 1)*prime(j)*T(n-2, k-1, j) with T(2, k, j) = prime(j) and j = 8, read by rows.
  2 · 19, 19 · 2, 718, 2 · 2, 6857, 6857, 2 … Σ 2, 38, 722, 13718, 260642, 4952198, …
  (−∞, 0)nonnegative
• A318772 → x ↦ (1/243)x → basis A013611
  1 · 3 · 9 · 27 … Σ 1, 3, 9, 27, 86, 273, 864, 2727, …
  (−∞, 0]nonnegative
• A144441: Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 14, 1 · 1, 83, 83, 1 … Σ 1, 2, 16, 168, 2416, 44160, 981184, …
  (−∞, 0)nonnegative
• A059364 → Pₙ + Pₙ₊₁
  3, 1 · 8, 8, 2 · 30, 53, 31, 6 … Σ 4, 18, 120, 1050, 11340, 145530, …
  (−∞, ∞)nonnegativerational roots
• A120910 → x ↦ x/(x+1)
  3 · 6, 9 · 12, 36, 27 · 24, 108, 162, 81 … Σ 3, 15, 75, 375, 1875, 9375, 46875, …
  [−1, 0]nonnegativerational roots
• A008277 → θ+1 → reverse
  1 · 2, 1 · 3, 6, 1 · 4, 18, 14, 1 … Σ 1, 3, 10, 37, 151, 674, 3263, 17007, …
  (−∞, 0]nonnegative
• A063967 → basis A013609
  1 · 2, 2 · 6, 10, 4 · 16, 40, 32, 8 … Σ 1, 4, 20, 96, 464, 2240, 10816, 52224, …
  (−∞, 0]nonnegative
• A178820 → basis A013609
  1 · 8, 8 · 40, 80, 40 · 160, 480, 480, 160 … Σ 1, 16, 160, 1280, 8960, 57344, 344064, …
  (−∞, 0]nonnegativerational roots
• A143683 → basis A008277
  1 · 2, 1 · 12, 13, 1 · 40, 83, 25, 1 … Σ 1, 3, 26, 149, 1119, 8162, 65271, …
  (−∞, 0]nonnegative
• A038284 → basis A007318
  196, 168, 36 · 2744, 3528, 1512, 216 … Σ 400, 8000, 160000, 3200000, 64000000, …
  (−∞, 0]nonnegativerational roots
• A008278 → θ+1
  1 · 1, 2 · 1, 6, 3 · 1, 12, 21, 4 … Σ 1, 3, 10, 38, 161, 747, 3753, 20253, …
  (−∞, 0]nonnegative
• A209599: Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the op…
  1 · 2 · 3, 1 · 5, 3 … Σ 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, −1]nonnegative
• A188440 → (θ+1)²
  1 · 1 · 1, 8 · 1, 8 … Σ 1, 1, 9, 9, 53, 53, 261, 261, …
  (−∞, 0]nonnegative
• A188440 → basis A008277
  1 · 1 · 3, 2 · 3, 2 … Σ 1, 1, 5, 5, 29, 29, 193, 193, …
  (−∞, 0]nonnegative
• chebyshev_u → D
  2 · 0, 8 · -4, 0, 24 · 0, -24, 0, 64 … Σ 0, 2, 8, 20, 40, 70, 112, 168, …
  (−∞, ∞)
• A133087 → x ↦ x/(x+1)
  1 · 2, 3 · 1, 4, 4 · 2, 11, 20, 12 … Σ 1, 5, 9, 45, 81, 405, 729, 3645, …
  [−1, 0]nonnegativerational roots
• A143683 → D
  1 · 10, 2 · 19, 38, 3 · 28, 236, 84, 4 … Σ 0, 1, 12, 60, 352, 1680, 8256, 38080, …
  (−∞, 0]nonnegative
• A334218 → (θ+1)²
  1 · 1 · 0, 8 · 0, 12, 27 … Σ 1, 1, 8, 39, 224, 1520, 11880, 105000, …
  (−∞, 0]nonnegative
• A094816 → basis A013611
  1 · 2, 4 · 5, 20, 16 · 16, 92, 144, 64 … Σ 1, 6, 41, 316, 2721, 25946, 271801, …
  (−∞, 0]nonnegative
• A397048: Triangle T(n, k) read by rows: T(n, k) = (Sum_{i=0..k} (-1)^i * binomial(k, i) * (n+i)^n) * (-1)^k / k! for 0 <= k <= n.
  1 · 1, 1 · 4, 5, 1 · 27, 37, 12, 1 … Σ 1, 2, 10, 77, 799, 10427, 163967, …
  (−∞, 0]nonnegative
• A266326: Triangle read by rows giving successive states of cellular automaton generated by "Rule 20" initiated with a single ON (black) cell.
  A133081 → reverse (shift +1)
  1 · 0, 1, 1 · 0, 0, 0, 0, 1 · 0, 0, 0, 0, 0, 1, 1 … Σ 1, 2, 1, 2, 1, 2, 1, 2, …
  (−∞, 0]nonnegativerational roots
• A188440 → Veronese 2 (j=1)
  2 · 2 · 4 · 4 … Σ 0, 0, 2, 2, 4, 4, 14, 14, …
  (−∞, 0]nonnegative
• A064861 → x ↦ x+1
  6, 7, 2 · 18, 33, 20, 4 · 36, 84, 73, 28, 4 … Σ 15, 75, 225, 1125, 3375, 16875, 50625, …
  (−∞, −1]nonnegativerational roots
• A100862 → D+1
  1 · 2, 1 · 4, 5, 1 · 7, 18, 9, 1 … Σ 1, 3, 10, 35, 129, 497, 1996, 8325, …
  (−∞, 0]nonnegative
• A052553 → basis A013611
  1 · 1 · 2, 4 · 3, 8 … Σ 1, 1, 6, 11, 41, 96, 301, 781, …
  (−∞, 0]nonnegative
• basis-map coefficients of A113214, shifted → x ↦ x/(x+1)
  1 · -1, -2 · 1, 4, 4 · -1, -6, -12, -8 … Σ 1, -3, 9, -27, 81, -243, 729, -2187, …
  [−1, 0]rational roots
• A347056 → D+1
  1 · 7, 4 · 22, 36, 10 · 50, 140, 110, 20 … Σ 1, 11, 68, 320, 1280, 4592, 15232, …
  (−∞, 0]nonnegative
• A347056 → x ↦ x+1
  1 · 7, 4 · 32, 36, 10 · 120, 200, 110, 20 … Σ 1, 11, 78, 450, 2295, 10773, 47628, …
  (−∞, −1]nonnegativerational roots
• A132014: T(n,j) for double application of an iterated mixed order Laguerre transform: Coefficients of Laguerre polynomial (-1)^n*n!*L(n,2-n,x).
  A023531 → D²−2D+1 (shift -1)
  1 · -2, 1 · 2, -4, 1 · 0, 6, -6, 1 … Σ 1, -1, -1, 1, 5, 11, 19, 29, …
  (−∞, ∞)
• A008292 → γ (scale 1/4) (shift +1) → (θ+1)²
  1 · 2 · 4, 8 · 8, 64 … Σ 1, 2, 12, 72, 512, 4144, 37632, 379008, …
  (−∞, 0]nonnegative
• A330609: T(n, k) = binomial(n-k-1, k-1)*(n-k)!/k! for n >= 0 and 0 <= k <= floor(n/2). Irregular triangle read by rows.
  1 · 0, 1 · 0, 2 · 0, 6, 1 … Σ 1, 0, 1, 2, 7, 30, 157, 972, …
  (−∞, 0]nonnegative
• A166973: Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.
  1 · 1, 1 · 1, 7, 1 · 1, 43, 18, 1 … Σ 1, 2, 9, 63, 536, 5307, 60389, 775988, …
  (−∞, 0)nonnegative
• A172431 → basis A007318
  1 · 3, 2 · 8, 10, 3 · 21, 38, 22, 4 … Σ 1, 5, 21, 85, 341, 1365, 5461, 21845, …
  (−∞, −1]nonnegative
• basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
  1 · -1, -1 · 1, 2, 1 · -1, -3, -3, -1 … Σ 1, -2, 4, -8, 16, -32, 64, -128, …
  (−∞, −1]rational roots
• A013620 → D
  3 · 12, 18 · 36, 108, 81 · 96, 432, 648, 324 … Σ 0, 3, 30, 225, 1500, 9375, 56250, …
  (−∞, 0]nonnegativerational roots
• A154233: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.
  1 · 1, 1 · 1, 7, 1 · 1, 19, 19, 1 … Σ 1, 2, 9, 40, 251, 1662, 13615, 118640, …
  (−∞, 0)nonnegative
• A035342 → Pₙ + Pₙ₊₁
  4, 1 · 18, 10, 1 · 120, 96, 19, 1 … Σ 5, 29, 236, 2447, 30707, 451370, …
  (−∞, ∞)nonnegative
• A094305 → θ+1
  1 · 3, 6 · 6, 24, 18 · 10, 60, 90, 40 … Σ 1, 9, 48, 200, 720, 2352, 7168, 20736, …
  (−∞, 0]nonnegativerational roots
• A347823 → x ↦ x+1
  1 · 5, 3 · 16, 18, 5 · 44, 72, 39, 7 … Σ 1, 8, 39, 162, 621, 2268, 8019, 27702, …
  (−∞, −1]nonnegativerational roots