basis_map / A007318

Image under a basis map u^k -> P_k. All operations

219 sequences

• A249251 → basis A007318
  1 · 4, 3 · 13, 18, 6 · 43, 87, 57, 12 … row sums 1, 7, 37, 199, 1069, 5743, 30853, …
  (−∞, −1] 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 … row sums 1, 9, 81, 729, 6561, 59049, 531441, …
  A038279 (−∞, 0) nonnegative rational roots
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A010054 (−∞, −1] [−1, 0] [−3−2√2, −3+2√2] nonnegative rational roots
• A026729: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.
  1 · 0, 1 · 0, 1, 1 · 0, 0, 2, 1 … row sums 1, 1, 2, 3, 5, 8, 13, 21, …
  A026729 (−4, 0] nonnegative
• A391533 → x ↦ x+1
  9, 6, 1 · 18, 21, 8, 1 · 36, 60, 37, 10, 1 … row sums 16, 48, 144, 432, 1296, 3888, 11664, …
  (−∞, −1] nonnegative rational roots
• A108838 → x ↦ x+1
  2 · 5, 2 · 14, 12, 2 · 42, 56, 21, 2 … row sums 2, 7, 28, 121, 550, 2591, 12536, 61921, …
  (−∞, −1] nonnegative
• A081577 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 16, 24, 10, 1 … row sums 1, 3, 13, 51, 205, 819, 3277, 13107, …
  (−∞, −1] nonnegative
• A113413 → basis A007318
  1 · 3, 1 · 7, 6, 1 · 17, 23, 9, 1 … row sums 1, 4, 14, 50, 178, 634, 2258, 8042, …
  (−∞, −1] nonnegative
• A081581 → basis A007318
  1 · 2, 1 · 10, 10, 1 · 32, 48, 18, 1 … row sums 1, 3, 21, 99, 549, 2835, 15093, 79299, …
  (−∞, −1] nonnegative
• A026374 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 10, 15, 7, 1 … row sums 1, 3, 11, 33, 121, 363, 1331, 3993, …
  (−∞, −1] nonnegative
• A154986 → basis A007318
  1 · 2, 1 · 6, 6, 1 · 24, 36, 14, 1 … row sums 1, 3, 13, 75, 537, 4611, 46053, 525483, …
  (−∞, −1] nonnegative
• A091044 → basis A007318
  1 · 4, 2 · 16, 16, 3 · 64, 96, 40, 4 … row sums 1, 6, 35, 204, 1189, 6930, 40391, …
  (−∞, −1] nonnegative
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
  1 · 4, 1 · 16, 8, 1 · 64, 48, 12, 1 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A038231 (−∞, −1] nonnegative rational roots
• A038291: Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.
  1 · 9, 1 · 81, 18, 1 · 729, 243, 27, 1 … row sums 1, 10, 100, 1000, 10000, 100000, …
  A038291 (−∞, 0) nonnegative rational roots
• A135838 → x ↦ x+1
  1 · 4, 2 · 8, 8, 2 · 32, 48, 24, 4 … row sums 1, 6, 18, 108, 324, 1944, 5832, 34992, …
  (−∞, −1] nonnegative rational roots
• A038220 → basis A007318
  1 · 5, 2 · 25, 20, 4 · 125, 150, 60, 8 … row sums 1, 7, 49, 343, 2401, 16807, 117649, …
  (−∞, −1] nonnegative rational roots
• A028326 → basis A007318
  2 · 4, 2 · 8, 8, 2 · 16, 24, 12, 2 … row sums 2, 6, 18, 54, 162, 486, 1458, 4374, …
  (−∞, −1] nonnegative rational roots
• A145597 → basis A007318
  1 · 6, 3 · 27, 27, 6 · 110, 165, 75, 10 … row sums 1, 9, 60, 360, 2061, 11529, 63728, …
  (−∞, −1] nonnegative
• A059473 → x ↦ x+1
  1 · 4, 2 · 18, 18, 4 · 80, 120, 56, 8 … row sums 1, 6, 40, 264, 1744, 11520, 76096, …
  (−∞, −1] nonnegative
• A013612 → basis A007318
  36, 60, 25 · 216, 540, 450, 125 … row sums 121, 1331, 14641, 161051, 1771561, …
  (−∞, 0] nonnegative rational roots
• A123519 → x ↦ x+1
  11, 14, 4 · 41, 76, 44, 8 · 153, 372, 324, 120, 16 … row sums 29, 169, 985, 5741, 33461, 195025, …
  (−3, −1) nonnegative
• A358735 → basis A007318
  1 · 2, 1 · 7, 8, 2 · 33, 60, 34, 6 … row sums 1, 3, 17, 133, 1313, 15623, 217409, …
  (−∞, 0] nonnegative
• A038331 → basis A007318
  289, 170, 25 · 4913, 4335, 1275, 125 … row sums 484, 10648, 234256, 5153632, 113379904, …
  (−∞, 0] nonnegative rational roots
• A038305 → basis A007318
  169, 78, 9 · 2197, 1521, 351, 27 … row sums 256, 4096, 65536, 1048576, 16777216, …
  (−∞, 0] nonnegative rational roots
• A139548 → basis A007318
  1 · 8, 8, 2 · 64, 128, 96, 32, 4 … row sums 1, 18, 324, 5832, 104976, 1889568, …
  (−∞, 0] nonnegative rational roots
• A304236 → basis A007318
  1 · 1 · 4, 3 · 7, 6 … row sums 1, 1, 7, 13, 55, 133, 463, 1261, …
  (−∞, −1] nonnegative
• A188440 → basis A007318
  1 · 1 · 3, 2 · 3, 2 … row sums 1, 1, 5, 5, 25, 25, 125, 125, …
  (−∞, −1] nonnegative rational roots
• A081580 → basis A007318
  1 · 2, 1 · 9, 9, 1 · 28, 42, 16, 1 … row sums 1, 3, 19, 87, 451, 2223, 11179, 55767, …
  (−∞, −1] nonnegative
• A030528 → basis A007318
  1 · 2, 1 · 3, 4, 1 · 5, 10, 6, 1 … row sums 1, 3, 8, 22, 60, 164, 448, 1224, …
  (−∞, −1] nonnegative
• A038333 → basis A007318
  361, 266, 49 · 6859, 7581, 2793, 343 … row sums 676, 17576, 456976, 11881376, …
  (−∞, 0] nonnegative rational roots
• A126216: Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A126216 (−∞, −1] nonnegative
• A008459 → x ↦ x+1
  1 · 2, 1 · 6, 6, 1 · 20, 30, 12, 1 … row sums 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  (−∞, −1] nonnegative
• A178822 → x ↦ x+1
  1 · 12, 6 · 84, 84, 21 · 448, 672, 336, 56 … row sums 1, 18, 189, 1512, 10206, 61236, 336798, …
  (−∞, −1] nonnegative rational roots
• A249248 → x ↦ x+1
  1 · 4, 3 · 11, 13, 3 · 38, 65, 31, 3 … row sums 1, 7, 27, 137, 681, 3687, 20595, …
  (−∞, −1] nonnegative
• A099097: Riordan array (1, 3+x).
  1 · 0, 3 · 0, 1, 9 · 0, 0, 6, 27 … row sums 1, 3, 10, 33, 109, 360, 1189, 3927, …
  A099097 (−4/9, 0] nonnegative
• A178820 → basis A007318
  1 · 8, 4 · 40, 40, 10 · 160, 240, 120, 20 … row sums 1, 12, 90, 540, 2835, 13608, 61236, …
  (−∞, −1] nonnegative rational roots
• A102662 → x ↦ x+1
  1 · 4, 3 · 9, 11, 3 · 22, 38, 20, 3 … row sums 1, 7, 23, 83, 295, 1051, 3743, 13331, …
  (−∞, −1] nonnegative
• A225471 → basis A007318
  1 · 4, 1 · 32, 12, 1 · 384, 176, 24, 1 … row sums 1, 5, 45, 585, 9945, 208845, 5221125, …
  (−∞, 0] nonnegative rational roots
• A052553 → x ↦ x+1
  1 · 1 · 2, 1 · 3, 2 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  (−∞, −1] nonnegative
• A136124 → basis A007318
  1 · 3, 1 · 12, 7, 1 · 60, 47, 12, 1 … row sums 1, 4, 20, 120, 840, 6720, 60480, …
  (−∞, 0] nonnegative rational roots
• A102547 → reverse → x ↦ x+1
  1, 1 · 1, 2, 1 · 1, 3, 3, 1 · 2, 7, 9, 5, 1 … row sums 2, 4, 8, 24, 64, 160, 416, 1088, …
  (−∞, −1] nonnegative
• A038334 → basis A007318
  400, 320, 64 · 8000, 9600, 3840, 512 … row sums 784, 21952, 614656, 17210368, …
  (−∞, 0] nonnegative rational roots
• A038324 → basis A007318
  1 · 21, 10 · 441, 420, 100 · 9261, 13230, 6300, 1000 … row sums 1, 31, 961, 29791, 923521, 28629151, …
  (−∞, 0] nonnegative rational roots
• A088996 → basis A007318
  1 · 1, 1 · 3, 5, 2 · 15, 34, 25, 6 … row sums 1, 2, 10, 80, 880, 12320, 209440, …
  (−∞, 0] nonnegative rational roots
• A111125 → x ↦ x+1
  1 · 4, 1 · 11, 7, 1 · 29, 31, 10, 1 … row sums 1, 5, 19, 71, 265, 989, 3691, 13775, …
  (−5, −1) nonnegative
• A152842 → x ↦ x+1
  1 · 4, 3 · 8, 10, 3 · 32, 64, 42, 9 … row sums 1, 7, 21, 147, 441, 3087, 9261, 64827, …
  (−∞, −1] nonnegative rational roots
• A133214 → basis A007318
  13, 16, 4 · 63, 114, 60, 8 · 321, 768, 624, 192, 16 … row sums 33, 245, 1921, 15525, 127905, 1067925, …
  (−∞, 0] nonnegative
• A008955 → basis A007318
  1 · 2, 1 · 10, 13, 4 · 100, 220, 157, 36 … row sums 1, 3, 27, 513, 16929, 863379, 63026667, …
  (−∞, 0] nonnegative rational roots
• A114656 → basis A007318
  1 · 3, 1 · 11, 8, 1 · 45, 51, 15, 1 … row sums 1, 4, 20, 112, 672, 4224, 27456, …
  (−∞, −1] nonnegative
• A128813 → basis A007318
  1 · 2, 1 · 8, 10, 3 · 56, 118, 81, 18 … row sums 1, 3, 21, 273, 5733, 177723, 7642089, …
  (−∞, 0] nonnegative rational roots