basis_map

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

1056 sequences

• A099040 → basis A013610
  1 · 2, 6 · 6, 30, 36 · 16, 120, 288, 216 … row sums 1, 8, 72, 640, 5696, 50688, 451072, …
  (−∞, 0] nonnegative
• A133087 → basis A013610
  1 · 3, 3 · 4, 12, 9 · 12, 48, 63, 27 … row sums 1, 6, 25, 150, 625, 3750, 15625, 93750, …
  (−∞, 0] nonnegative rational roots
• A099093 → basis A008277
  1 · 3, 3 · 12, 30, 9 · 45, 243, 180, 27 … row sums 1, 6, 51, 495, 5472, 67392, 911655, …
  (−∞, 0] nonnegative
• A118933 → basis A013611
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 31, 151, 451, 1051, …
  (−∞, 0] nonnegative
• A272866 → basis A013611
  1 · 5, 20, 16 · 25, 200, 560, 640, 256 … row sums 1, 41, 1681, 68921, 2825761, 115856201, …
  (−∞, 0] nonnegative
• A034870: Even-numbered rows of Pascal's triangle.
  1 · 1, 2, 1 · 1, 4, 6, 4, 1 · 1, 6, 15, 20, 15, 6, 1 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A034870 (−∞, −1] nonnegative rational roots
• A091044 → basis A013610
  1 · 4, 6 · 16, 48, 27 · 64, 288, 360, 108 … row sums 1, 10, 91, 820, 7381, 66430, 597871, …
  (−∞, 0] nonnegative
• A013622 → basis A013609
  1 · 8, 10 · 64, 160, 100 · 512, 1920, 2400, 1000 … row sums 1, 18, 324, 5832, 104976, 1889568, …
  (−∞, 0] nonnegative rational roots
• A249250 → basis A013609
  1 · 4, 6 · 10, 24, 12 · 28, 96, 96, 24 … row sums 1, 10, 46, 244, 1252, 6472, 33400, …
  (−∞, 0] nonnegative
• A145597 → basis A013609
  1 · 6, 6 · 27, 54, 24 · 110, 330, 300, 80 … row sums 1, 12, 105, 820, 6081, 43932, 313012, …
  (−∞, 0] nonnegative
• A081578 → basis A013611
  1 · 2, 4 · 7, 28, 16 · 20, 120, 192, 64 … row sums 1, 6, 51, 396, 3141, 24786, 195831, …
  (−∞, 0] nonnegative
• A011973 → x ↦ x+2 (shift +1) → basis A008277
  1 · 4, 1 · 7, 2 · 19, 10, 1 … row sums 1, 5, 9, 30, 68, 200, 503, 1426, …
  (−∞, 0] nonnegative
• A038334 → basis A008277
  400, 384, 64 · 8000, 13952, 5376, 512 … row sums 848, 27840, 989440, 37927936, …
  (−∞, 0] nonnegative
• A356546 → basis A007318
  1 · 4, 2 · 24, 24, 6 · 160, 240, 120, 20 … row sums 1, 6, 54, 540, 5670, 61236, 673596, …
  (−∞, 0] nonnegative rational roots
• A249251 → basis A013611
  1 · 4, 12 · 13, 72, 96 · 43, 348, 912, 768 … row sums 1, 16, 181, 2071, 23686, 270901, …
  (−∞, 0] nonnegative
• A034870 → basis A013609
  1 · 4, 8, 4 · 16, 64, 96, 64, 16 … row sums 1, 16, 256, 4096, 65536, 1048576, …
  (−∞, 0] nonnegative rational roots
• A038284 → basis A013611
  196, 672, 576 · 2744, 14112, 24192, 13824 … row sums 1444, 54872, 2085136, 79235168, …
  (−∞, 0] nonnegative rational roots
• A081579 → basis A007318
  1 · 2, 1 · 8, 8, 1 · 24, 36, 14, 1 … row sums 1, 3, 17, 75, 361, 1683, 7937, 37275, …
  (−∞, −1] nonnegative
• A038208 → basis A013611
  1 · 4, 8 · 16, 64, 64 · 64, 384, 768, 512 … row sums 1, 12, 144, 1728, 20736, 248832, …
  (−∞, 0] nonnegative rational roots
• A274496 → basis A008277
  1 · 2 · 4, 2 · 8, 4 … row sums 1, 2, 6, 12, 40, 80, 296, 592, …
  (−∞, 0] nonnegative
• A034870 → basis A013611
  1 · 4, 16, 16 · 16, 128, 384, 512, 256 … row sums 1, 36, 1296, 46656, 1679616, 60466176, …
  (−∞, 0] nonnegative rational roots
• basis-map coefficients of A112227 → basis A112227
  -2, 0, 1 · 0, -6, 0, 1 · 12, 0, -12, 0, 1 … row sums -1, -5, 1, 41, 31, -461, -895, 6481, …
  (−∞, ∞)
• A143491 → basis A013610
  12, 21, 9 · 60, 141, 108, 27 … row sums 42, 336, 3024, 30240, 332640, 3991680, …
  (−∞, 0] nonnegative rational roots
• A137320 → basis A008277
  1 · 2, 2 · 6, 14, 4 · 24, 96, 60, 8 … row sums 1, 4, 24, 188, 1796, 20112, 257248, …
  (−∞, 0] nonnegative
• A038322 → basis A007318
  1 · 19, 8 · 361, 304, 64 · 6859, 8664, 3648, 512 … row sums 1, 27, 729, 19683, 531441, 14348907, …
  (−∞, 0] nonnegative rational roots
• A128099 → x ↦ x+1
  1 · 1 · 3, 2 · 5, 4 … row sums 1, 1, 5, 9, 29, 65, 181, 441, …
  (−∞, −1] nonnegative
• A064861 → basis A013610
  6, 21, 18 · 18, 99, 180, 108 · 36, 252, 657, 756, 324 … row sums 45, 405, 2025, 18225, 91125, 820125, …
  (−∞, 0] nonnegative rational roots
• A085841 → basis A007318
  1 · 7, 4 · 61, 72, 16 · 547, 1004, 528, 64 … row sums 1, 11, 149, 2143, 31273, 457907, …
  (−∞, 0] nonnegative
• A122431 → basis A008277
  8, 19, 9, 1 · 8, 46, 46, 13, 1 … row sums 37, 114, 409, 1657, 7432, 36401, …
  (−∞, 0] nonnegative
• A145596: basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x -> x + 1
  1 · 2, 2 · 3, 8, 3 · 4, 20, 20, 4 … row sums 1, 4, 14, 48, 165, 572, 2002, 7072, …
  A145596 (−∞, 0) nonnegative
• A111125 → basis A013611
  1 · 4, 4 · 11, 28, 16 · 29, 124, 160, 64 … row sums 1, 8, 55, 377, 2584, 17711, 121393, …
  (−∞, 0] nonnegative
• A143683 → basis A013610
  1 · 2, 3 · 12, 36, 9 · 40, 180, 198, 27 … row sums 1, 5, 57, 445, 4049, 34485, 301993, …
  (−∞, 0] nonnegative
• A128099 → basis A008277
  1 · 1 · 3, 2 · 5, 4 … row sums 1, 1, 5, 9, 33, 77, 261, 705, …
  (−∞, 0] nonnegative
• A318772 → basis A013610
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 13, 25, 37, 49, …
  (−∞, 0] nonnegative
• A028326 → basis A013610
  2 · 4, 6 · 8, 24, 18 · 16, 72, 108, 54 … row sums 2, 10, 50, 250, 1250, 6250, 31250, …
  (−∞, 0] nonnegative rational roots
• A110441 → basis A013609
  1 · 4, 2 · 14, 16, 4 · 48, 88, 48, 8 … row sums 1, 6, 34, 192, 1084, 6120, 34552, …
  (−∞, 0] nonnegative
• A038303 → basis A013609
  1 · 11, 2 · 121, 44, 4 · 1331, 726, 132, 8 … row sums 1, 13, 169, 2197, 28561, 371293, …
  (−∞, 0] nonnegative rational roots
• A008459 → basis A013610
  1 · 2, 3 · 6, 18, 9 · 20, 90, 108, 27 … row sums 1, 5, 33, 245, 1921, 15525, 127905, …
  (−∞, 0] nonnegative
• A104684 → basis A007318
  1 · 3, 1 · 13, 8, 1 · 63, 57, 15, 1 … row sums 1, 4, 22, 136, 886, 5944, 40636, …
  (−∞, −1] nonnegative
• A100861 → reverse → x ↦ x+1
  1 · 1, 1 · 2, 3, 1 · 4, 9, 6, 1 … row sums 1, 2, 6, 20, 76, 312, 1384, 6512, …
  (−∞, −1] nonnegative
• A085881 → basis A008277
  1 · 2, 1 · 12, 15, 3 · 120, 285, 135, 15 … row sums 1, 3, 30, 555, 15855, 636930, 33918885, …
  (−∞, 0] nonnegative
• A102547 → x ↦ (1/27)x → basis A013611
  1 · 3 · 9 · 28, 4 … row sums 1, 3, 9, 32, 111, 378, 1294, 4437, …
  (−∞, 0] nonnegative
• A152198 → basis A008277
  1 · 1 · 2, 1 · 2, 1 … row sums 1, 1, 3, 3, 10, 10, 37, 37, …
  (−∞, 0] nonnegative
• A154228 → basis A013609
  1 · 2, 2 · 18, 36, 4 · 96, 288, 200, 8 … row sums 1, 4, 58, 592, 11938, 209368, 5851432, …
  (−∞, 0] nonnegative
• A132393 → basis A013611
  1 · 1, 4 · 2, 12, 16 · 6, 44, 96, 64 … row sums 1, 5, 30, 210, 1680, 15120, 151200, …
  (−∞, 0] nonnegative rational roots
• A128813 → basis A013611
  1 · 2, 4 · 8, 40, 48 · 56, 472, 1296, 1152 … row sums 1, 6, 96, 2976, 151776, 11534976, …
  (−∞, 0] nonnegative rational roots
• A341287: Triangle read by rows: T(n,k) = Sum_{i=0..n-1} binomial(n-1, i)*T(n-1-i,k-1) - Sum_{i=1..n-1} binomial(n-1,i)*T(n-1-i,k) for 1 <= k <= n+1 with T(0,1) = 1 (and T(n,k) = 0 otherwise).
  A023531 → basis A341287
  1 · 0, 1 · -1, 1, 1 · -1, -2, 3, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A341287 (−∞, 1]
• A116412 → basis A007318
  1 · 4, 1 · 13, 8, 1 · 43, 42, 12, 1 … row sums 1, 5, 22, 98, 436, 1940, 8632, 38408, …
  (−∞, −1] nonnegative
• A114687 → basis A013609
  1 · 3, 4 · 11, 28, 16 · 45, 168, 192, 64 … row sums 1, 7, 55, 469, 4237, 39907, 387739, …
  (−∞, 0] nonnegative
• basis-map coefficients of A112227 → basis A013611
  1, 8, 16 · 1, 12, 48, 64 · 1, 16, 96, 256, 256 … row sums 25, 125, 625, 3125, 15625, 78125, …
  (−∞, 0] nonnegative rational roots