Sequences

• A111125: Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
  1 · 3, 1 · 5, 5, 1 · 7, 14, 7, 1 … Σ 1, 4, 11, 29, 76, 199, 521, 1364, …
  (−4, 0)nonnegative
• delannoy → x ↦ x+1 → basis A013609
  1 · 3, 2 · 11, 14, 4 · 39, 74, 44, 8 … Σ 1, 5, 29, 165, 941, 5365, 30589, …
  (−∞, 0]nonnegative
• A132812 → basis A007318 → basis A013609
  1 · 6, 4 · 33, 42, 12 · 180, 336, 192, 32 … Σ 1, 10, 87, 740, 6285, 53550, 458115, …
  (−∞, 0]nonnegative
• A038207 → basis A008277 → (θ+1)²
  1 · 3, 4 · 9, 28, 9 · 27, 148, 108, 16 … Σ 1, 7, 46, 299, 1967, 13248, 91903, …
  (−∞, 0]nonnegative
• delannoy → basis A013609 → D+1
  1 · 4, 2 · 15, 18, 4 · 48, 100, 56, 8 … Σ 1, 6, 37, 212, 1173, 6322, 33425, …
  (−∞, 0]nonnegative
• A111125 → D → D+1
  1 · 7, 2 · 28, 20, 3 · 84, 108, 39, 4 … Σ 0, 1, 9, 51, 235, 960, 3624, 12941, …
  (−∞, 0]nonnegative
• reversed rows → D+1
  1 · 3, 2 · 4, 3 · 5, 8, 2 … Σ 1, 5, 7, 15, 26, 48, 85, 151, …
  (−∞, 0]nonnegative
• A352880 → reverse → θ+1
  8 · 18, 24 · 32, 128, 48 · 50, 400, 450, 80 … Σ 8, 42, 208, 980, 4464, 19866, 86944, …
  (−∞, 0]nonnegative
• A038319 → basis A013609 → (θ+1)²
  256, 1280, 900 · 4096, 30720, 43200, 16000 … Σ 2436, 94016, 3377296, 115579776, …
  (−∞, 0]nonnegative
• A046716 → basis A013609 → basis A013609
  1 · 4, 4 · 19, 36, 16 · 118, 324, 272, 64 … Σ 1, 8, 71, 778, 12125, 284012, 9241891, …
  (−∞, 0]nonnegative
• A114656 → reverse → x ↦ x/(x+1)
  1 · 1, 3 · 1, 8, 11 · 1, 15, 51, 45 … Σ 1, 4, 20, 112, 672, 4224, 27456, …
  [−1, 0]nonnegative
• A114656 → reverse → x ↦ x+1
  1 · 3, 2 · 11, 14, 4 · 45, 84, 48, 8 … Σ 1, 5, 29, 185, 1257, 8925, 65445, …
  (−∞, −1]nonnegative
• A249250 → D
  3 · 6, 6 · 9, 30, 9 · 12, 72, 72, 12 … Σ 0, 3, 12, 48, 168, 564, 1824, 5760, …
  (−∞, 0]nonnegative
• A091042 → basis A013610 → D+1
  1 · 13, 9 · 76, 150, 45 · 400, 1344, 1071, 189 … Σ 1, 22, 271, 3004, 31981, 332146, …
  (−∞, 0]nonnegative
• basis-map coefficients of A030528, shifted → D → basis A013609
  -1 · 2, 4 · -3, -12, -12 · 4, 24, 48, 32 … Σ 0, -1, 6, -27, 108, -405, 1458, -5103, …
  (−∞, 0]rational roots
• A013612 → basis A013610
  36, 180, 225 · 216, 1620, 4050, 3375 … Σ 441, 9261, 194481, 4084101, 85766121, …
  (−∞, 0]nonnegativerational roots
• A035342 → reverse
  1 · 1, 3 · 1, 9, 15 · 1, 18, 87, 105 … Σ 1, 4, 25, 211, 2236, 28471, 422899, …
  (−∞, 0]nonnegative
• A249251 → basis A007318 → D
  3 · 18, 12 · 87, 114, 36 · 378, 738, 468, 96 … Σ 0, 3, 30, 237, 1680, 11211, 71970, …
  (−∞, 0]nonnegative
• A038303 → basis A007318
  1 · 11, 1 · 121, 22, 1 · 1331, 363, 33, 1 … Σ 1, 12, 144, 1728, 20736, 248832, …
  (−∞, −1]nonnegativerational roots
• basis-map coefficients of A297191, shifted → basis A013610 → x ↦ x+1
  1 · 16, 24, 9 · 256, 768, 864, 432, 81 … Σ 1, 49, 2401, 117649, 5764801, …
  (−∞, −1]nonnegativerational roots
• A038324 → basis A013609 → reverse
  1 · 20, 21 · 400, 840, 441 · 8000, 25200, 26460, 9261 … Σ 1, 41, 1681, 68921, 2825761, 115856201, …
  (−∞, 0]nonnegativerational roots
• A130534 → basis A008277 → (θ+1)²
  1 · 2, 4 · 6, 24, 9 · 24, 144, 108, 16 … Σ 1, 6, 39, 292, 2505, 24306, 263431, …
  (−∞, 0]nonnegative
• A110441 → basis A013611 → basis A008277
  1 · 8, 4 · 62, 80, 16 · 480, 1200, 576, 64 … Σ 1, 12, 158, 2320, 37700, 671376, …
  (−∞, 0]nonnegative
• A038328 → basis A013609
  196, 112, 16 · 2744, 2352, 672, 64 … Σ 324, 5832, 104976, 1889568, 34012224, …
  (−∞, 0]nonnegativerational roots
• narayana → basis A013610 → θ+1
  1 · 2, 6 · 5, 30, 27 · 14, 126, 243, 108 … Σ 1, 8, 62, 491, 3975, 32730, 272940, …
  (−∞, 0]nonnegative
• A204420 → basis A013610 → θ+1
  1 · 1, 6 · 9, 72, 81 · 225, 2070, 3645, 1620 … Σ 1, 7, 162, 7560, 589680, 69128640, …
  (−∞, 0]nonnegative
• A139548 → basis A013609
  1 · 8, 16, 8 · 64, 256, 384, 256, 64 … Σ 1, 32, 1024, 32768, 1048576, 33554432, …
  (−∞, 0]nonnegativerational roots
• A131431 → θ+1 → basis A013610
  1 · 8, 24 · 21, 126, 189 · 40, 360, 1080, 1080 … Σ 1, 32, 336, 2560, 16640, 98304, 544768, …
  (−∞, 0]nonnegativerational roots
• A111125 → basis A007318 → (θ+1)²
  1 · 4, 4 · 11, 28, 9 · 29, 124, 90, 16 … Σ 1, 8, 48, 259, 1309, 6328, 29616, …
  (−∞, 0]nonnegative
• A038271 → basis A013611 → basis A008277
  1 · 32, 20 · 1024, 1680, 400 … Σ 1, 52, 3104, 211008, 16105216, …
  (−∞, 0]nonnegative
• A249139 → basis A007318 → basis A013611
  1 · 5, 4 · 9, 8 · 29, 44, 16 … Σ 1, 9, 17, 89, 225, 937, 2737, 10233, …
  (−∞, 0]nonnegative
• A207606 → x ↦ x+1 → θ+1
  13, 22, 6 · 34, 88, 51, 8 · 89, 310, 279, 92, 10 … Σ 41, 181, 780, 3301, 13775, 56841, …
  (−∞, 0]nonnegative
• A008288 → D
  1 · 3, 2 · 5, 10, 3 · 7, 26, 21, 4 … Σ 0, 1, 5, 18, 58, 175, 507, 1428, …
  (−∞, 0]nonnegative
• A060920 → Veronese 2 (j=1) → D
  1 · 14 · 105, 2 · 594, 54 … Σ 0, 0, 0, 1, 14, 107, 648, 3493, …
  (−∞, 0]nonnegative
• A249248 → reverse → basis A013610
  1 · 4, 3 · 11, 27, 9 · 38, 147, 135, 27 … Σ 1, 7, 47, 347, 2675, 21703, 183415, …
  (−∞, 0]nonnegative
• A269944 → basis A013609
  1 · 1, 2 · 2, 6, 4 · 10, 34, 32, 8 … Σ 1, 3, 12, 84, 1008, 19152, 536256, …
  (−∞, 0]nonnegativerational roots
• A038305 → basis A013609 → θ+1
  169, 312, 108 · 2197, 6084, 4212, 864 … Σ 589, 13357, 294937, 6385729, 136185445, …
  (−∞, 0]nonnegativerational roots
• A081579 → x ↦ x+1 → D
  1 · 8, 2 · 36, 28, 3 · 160, 200, 60, 4 … Σ 0, 1, 10, 67, 424, 2469, 13926, 76199, …
  (−∞, 0]nonnegative
• A272867 → basis A013610 → reverse
  1 · 9, 18, 6 · 81, 324, 432, 216, 36 … Σ 1, 33, 1089, 35937, 1185921, 39135393, …
  (−∞, 0]nonnegative
• A038327 → basis A013611 → (θ+1)²
  1 · 13, 16 · 169, 416, 144 · 2197, 8112, 5616, 1024 … Σ 1, 29, 729, 16949, 374833, 8003277, …
  (−∞, 0]nonnegative
• A027467 → basis A013609
  1 · 16, 2 · 256, 64, 4 · 4096, 1536, 192, 8 … Σ 1, 18, 324, 5832, 104976, 1889568, …
  (−∞, 0]nonnegativerational roots
• A094816: Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.
  1 · 1, 1 · 1, 3, 1 · 1, 8, 6, 1 … Σ 1, 2, 5, 16, 65, 326, 1957, 13700, …
  (−∞, 0)nonnegative
• A347056: Triangle read by rows: T(n,k) = (n+1)*(n+2)*(k+3)*binomial(n,k)/6, 0 <= k <= n.
  1 · 3, 4 · 6, 16, 10 · 10, 40, 50, 20 … Σ 1, 7, 32, 120, 400, 1232, 3584, 9984, …
  (−∞, 0)nonnegativerational roots
• A142461 → D+1
  1 · 2, 1 · 15, 16, 1 · 112, 333, 114, 1 … Σ 1, 3, 32, 560, 13440, 407680, 14909440, …
  (−∞, 0]nonnegative
• A257609 → D (shift +2) → x ↦ x/(x+1)
  11, 44, 36 · 26, 210, 420, 240 … Σ 91, 896, 9829, 120198, 1628911, …
  [−1, 0]nonnegative
• A344565 → x ↦ x/(x+1)
  3 · 15, 30 · 45, 180, 180 · 105, 630, 1260, 840 … Σ 3, 45, 405, 2835, 17010, 91854, 459270, …
  [−1, 0]nonnegativerational roots
• A078812 → x ↦ x+1 → (θ+1)²
  1 · 3, 4 · 8, 24, 9 · 21, 100, 81, 16 … Σ 1, 7, 41, 218, 1091, 5237, 24379, …
  (−∞, 0]nonnegative
• A113953 → basis A013611
  1 · 1, 4 · 3, 16, 16 · 5, 44, 112, 64 … Σ 1, 5, 35, 225, 1475, 9625, 62875, …
  (−∞, 0]nonnegative
• A125092 → θ+1 → x ↦ x/(x+1)
  1 · 1, 9 · 1, 18, 44 · 1, 27, 132, 170 … Σ 1, 10, 63, 330, 1557, 6858, 28755, …
  [−1, 0]nonnegative
• A038330 → basis A008277
  256, 144, 16 · 4096, 3904, 960, 64 … Σ 416, 9024, 204544, 4849664, 120303616, …
  (−∞, 0]nonnegative