Sequences

• A055151 → x ↦ x+1
  1 · 1 · 2, 1 · 4, 3 … Σ 1, 1, 3, 7, 21, 61, 191, 603, …
  (−∞, −1]nonnegative
• basis-map coefficients of A026729 → basis A013610
  1 · 1, 3 · 1, 6, 9 · 1, 9, 27, 27 … Σ 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  (−∞, 0]nonnegativerational roots
• A102547 → x ↦ (1/27)x → basis A013610
  1 · 3 · 9 · 28, 3 … Σ 1, 3, 9, 31, 105, 351, 1177, 3951, …
  (−∞, 0]nonnegative
• A111125 → Veronese 2 (j=1)
  1 · 5 · 14, 1 · 30, 9 … Σ 0, 1, 5, 15, 39, 100, 260, 681, …
  (−∞, 0]nonnegative
• A257615: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.
  1 · 5, 5 · 25, 70, 25 · 125, 715, 715, 125 … Σ 1, 10, 120, 1680, 26880, 483840, …
  (−∞, 0)nonnegative
• A104684 → x ↦ x/(x+1)
  1 · 2, 3 · 6, 18, 13 · 20, 90, 132, 63 … Σ 1, 5, 37, 305, 2641, 23525, 213445, …
  [−1, 0]nonnegative
• A131689 → x ↦ x/(x+1)
  1 · 0, 1 · 0, 1, 3 · 0, 1, 8, 13 … Σ 1, 1, 4, 22, 160, 1456, 15904, 202672, …
  [−1, 0]nonnegative
• A152830: Triangle read by rows, A007318 rows repeated four times .
  1 · 1 · 1 · 1 … Σ 1, 1, 1, 1, 2, 2, 2, 2, …
  (−∞, 0]nonnegativerational roots
• A249252 → basis A013610
  1 · 5, 12 · 17, 72, 72 · 61, 372, 720, 432 … Σ 1, 17, 161, 1585, 15553, 152657, …
  (−∞, 0]nonnegative
• A113214 → x ↦ x−5
  1 · -3, 1 · 10, -7, 1 · -35, 37, -11, 1 … Σ 1, -2, 4, -8, 16, -32, 64, -128, …
  (1, 5]
• A028421: theta_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers …
  1 · 1, 2 · 2, 6, 3 · 6, 22, 18, 4 … Σ 1, 3, 11, 50, 274, 1764, 13068, 109584, …
  (−∞, 0)nonnegative
• A078812 → D+1
  1 · 3, 1 · 7, 6, 1 · 14, 22, 9, 1 … Σ 1, 4, 14, 46, 145, 444, 1331, 3926, …
  (−∞, 0]nonnegative
• A146988: Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 4^(n-1) * binomial(n-2, k-1) otherwise.
  1 · 1, 1 · 1, 6, 1 · 1, 19, 19, 1 … Σ 1, 2, 8, 40, 272, 2080, 16448, 131200, …
  (−∞, 0)nonnegative
• A135838 → x ↦ x+1
  1 · 4, 2 · 8, 8, 2 · 32, 48, 24, 4 … Σ 1, 6, 18, 108, 324, 1944, 5832, 34992, …
  (−∞, −1]nonnegativerational roots
• A062196 → reverse
  1 · 3, 1 · 6, 8, 1 · 10, 30, 15, 1 … Σ 1, 4, 15, 56, 210, 792, 3003, 11440, …
  (−∞, 0]nonnegative
• basis-map coefficients of A021012 → Veronese 2 (j=1)
  -1 · 0, -6 · 0, 0, -120 · 0, 0, 0, -5040 … Σ 0, -1, 0, -6, 0, -120, 0, -5040, …
  (−∞, 0]rational roots
• A113214 → θ+1
  1 · 2, 2 · 0, 6, 3 · 0, 4, 12, 4 … Σ 1, 4, 9, 20, 40, 78, 147, 272, …
  (−∞, 0]nonnegative
• A257614 → Pₙ + Pₙ₊₁
  3, 2 · 6, 30, 4 · 12, 272, 248, 8 … Σ 5, 40, 540, 10080, 239400, 6894720, …
  (−∞, ∞)nonnegative
• A038220 → basis A007318
  1 · 5, 2 · 25, 20, 4 · 125, 150, 60, 8 … Σ 1, 7, 49, 343, 2401, 16807, 117649, …
  (−∞, −1]nonnegativerational roots
• A134991: Triangle of Ward numbers T(n,k) read by rows.
  1 · 1, 3 · 1, 10, 15 · 1, 25, 105, 105 … Σ 1, 4, 26, 236, 2752, 39208, 660032, …
  (−1, 0)nonnegative
• A334218: Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.
  1 · 1 · 0, 2 · 0, 3, 3 … Σ 1, 1, 2, 6, 24, 120, 720, 5040, …
  (−∞, 0]nonnegative
• basis-map coefficients of A026729, shifted → D+1
  1 · -1, -1 · 0, 2, 1 · 0, 0, -3, -1 … Σ 1, -2, 3, -4, 5, -6, 7, -8, …
  (−∞, 0]rational roots
• A169803 → basis A013611
  1 · 2, 4 · 3, 8 · 5, 20, 16 … Σ 1, 6, 11, 41, 96, 301, 781, 2286, …
  (−∞, 0]nonnegative
• A038243 → basis A013611
  1 · 6, 4 · 36, 48, 16 · 216, 432, 288, 64 … Σ 1, 10, 100, 1000, 10000, 100000, …
  (−∞, 0]nonnegativerational roots
• A035607 → x ↦ x+1
  1 · 1 · 3, 2 · 7, 8, 2 … Σ 1, 1, 5, 17, 61, 217, 773, 2753, …
  (−∞, −1]nonnegative
• A038207 → basis A008277
  1 · 3, 1 · 9, 7, 1 · 27, 37, 12, 1 … Σ 1, 4, 17, 77, 372, 1915, 10481, 60814, …
  (−∞, 0]nonnegative
• A038226 → basis A013610
  121, 528, 576 · 1331, 8712, 19008, 13824 … Σ 1225, 42875, 1500625, 52521875, …
  (−∞, 0]nonnegativerational roots
• A099040 → x ↦ x−2
  1 · -4, 2 · 12, -14, 4 · -32, 64, -40, 8 … Σ 1, -2, 2, 0, -4, 8, -8, 0, …
  (0, 2]
• A114656 → x ↦ x/(x+1)
  1 · 2, 3 · 4, 14, 11 · 8, 48, 84, 45 … Σ 1, 5, 29, 185, 1257, 8925, 65445, …
  [−1, 0]nonnegative
• A038281 → basis A013610
  121, 198, 81 · 1331, 3267, 2673, 729 … Σ 400, 8000, 160000, 3200000, 64000000, …
  (−∞, 0]nonnegativerational roots
• A152198: Triangle read by rows, A007318 rows repeated.
  1 · 1 · 1, 1 · 1, 1 … Σ 1, 1, 2, 2, 4, 4, 8, 8, …
  (−∞, 0]nonnegativerational roots+1
• A318772 → x ↦ (1/243)x → basis A013610
  1 · 3 · 9 · 27 … Σ 1, 3, 9, 27, 85, 267, 837, 2619, …
  (−∞, 0]nonnegative
• A136572 → basis A013610
  1 · 1, 3 · 2, 12, 18 · 6, 54, 162, 162 … Σ 1, 4, 32, 384, 6144, 122880, 2949120, …
  (−∞, 0]nonnegativerational roots
• A027465 → basis A008277
  1 · 4, 1 · 16, 9, 1 · 64, 61, 15, 1 … Σ 1, 5, 26, 141, 799, 4736, 29371, …
  (−∞, 0]nonnegative
• A257623: Triangle read by rows: T(n,k) = t(n-k, k), where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(n) = 5*n + 3.
  1 · 3, 3 · 9, 48, 9 · 27, 501, 501, 27 … Σ 1, 6, 66, 1056, 22176, 576576, …
  (−∞, 0)nonnegative
• A028326 → basis A007318
  2 · 4, 2 · 8, 8, 2 · 16, 24, 12, 2 … Σ 2, 6, 18, 54, 162, 486, 1458, 4374, …
  (−∞, −1]nonnegativerational roots
• A290319: Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.
  1 · 1, 1 · 5, 6, 1 · 45, 59, 15, 1 … Σ 1, 2, 12, 120, 1680, 30240, 665280, …
  (−∞, 0)nonnegativerational roots
• A257609: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.
  1 · 2, 2 · 4, 16, 4 · 8, 88, 88, 8 … Σ 1, 4, 24, 192, 1920, 23040, 322560, …
  (−∞, 0)nonnegative
• A106246 → θ+1
  1 · 2, 2 · 1, 8, 3 · 0, 6, 18, 4 … Σ 1, 4, 12, 28, 55, 96, 154, 232, …
  (−∞, 0]nonnegative
• A257180 → Pₙ + Pₙ₊₁
  36, 165, 150, 27 · 108, 867, 1551, 753, 81 … Σ 378, 3360, 33264, 362880, 4324320, …
  (−∞, ∞)nonnegative
• delannoy → Veronese 2 (j=1)
  1 · 3 · 5, 1 · 7, 7 … Σ 0, 1, 3, 6, 14, 35, 85, 204, …
  (−∞, 0]nonnegative
• A142075: Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.
  1 · 1, 2 · 1, 8, 4 · 1, 22, 44, 8 … Σ 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  (−∞, 0)nonnegative
• A266213 → basis A013610
  1 · 1 · 3, 6 · 7, 24, 18 … Σ 1, 1, 9, 49, 281, 1601, 9129, 52049, …
  (−∞, 0]nonnegative
• A118394 → basis A008277
  1 · 1 · 1 · 7, 6 … Σ 1, 1, 1, 13, 49, 121, 2041, 13021, …
  (−∞, 0]nonnegative
• A122542 → D
  1 · 2, 2 · 2, 8, 3 · 2, 16, 18, 4 … Σ 0, 1, 4, 13, 40, 117, 332, 921, …
  (−∞, 0]nonnegative
• A257615 → Pₙ + Pₙ₊₁
  6, 5 · 30, 75, 25 · 150, 785, 740, 125 … Σ 11, 130, 1800, 28560, 510720, 10160640, …
  (−∞, ∞)nonnegative
• A216973 → D
  2 · 6, 6 · 12, 24, 12 · 20, 60, 60, 20 … Σ 0, 0, 2, 12, 48, 160, 480, 1344, …
  (−∞, 0]nonnegativerational roots
• A034870 → (θ+1)²
  1 · 1, 8, 9 · 1, 16, 54, 64, 25 … Σ 1, 18, 160, 1120, 6912, 39424, 212992, …
  (−∞, 0]nonnegative
• A100862 → basis A013610
  1 · 2, 3 · 5, 15, 9 · 14, 63, 81, 27 … Σ 1, 5, 29, 185, 1273, 9325, 72085, …
  (−∞, 0]nonnegative
• A145597 → basis A007318
  1 · 6, 3 · 27, 27, 6 · 110, 165, 75, 10 … Σ 1, 9, 60, 360, 2061, 11529, 63728, …
  (−∞, −1]nonnegative