basis_map / A007318

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

219 sequences

• A099040 → basis A007318
  1 · 2, 2 · 6, 10, 4 · 16, 40, 32, 8 … row sums 1, 4, 20, 96, 464, 2240, 10816, 52224, …
  (−∞, −1] nonnegative
• A082985 → x ↦ x+1
  1 · 4, 3 · 11, 15, 5 · 29, 56, 35, 7 … row sums 1, 7, 31, 127, 511, 2047, 8191, 32767, …
  (−∞, −1] nonnegative
• A099093: Riordan array (1, 3+3x).
  1 · 0, 3 · 0, 3, 9 · 0, 0, 18, 27 … row sums 1, 3, 12, 45, 171, 648, 2457, 9315, …
  A099093 (−4/3, 0] nonnegative
• A038259 → basis A007318
  1 · 11, 5 · 121, 110, 25 · 1331, 1815, 825, 125 … row sums 1, 16, 256, 4096, 65536, 1048576, …
  (−∞, 0] nonnegative rational roots
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
  1 · 3, 2 · 9, 12, 4 · 27, 54, 36, 8 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A038220 (−∞, −1] nonnegative rational roots
• A085881 → basis A007318
  1 · 2, 1 · 12, 12, 3 · 120, 180, 90, 15 … row sums 1, 3, 27, 405, 8505, 229635, 7577955, …
  (−∞, 0] nonnegative rational roots
• A063007 → x ↦ x+1
  1 · 3, 2 · 13, 18, 6 · 63, 132, 90, 20 … row sums 1, 5, 37, 305, 2641, 23525, 213445, …
  (−2, −1) nonnegative
• A113953 → basis A007318
  1 · 1, 1 · 3, 4, 1 · 5, 11, 7, 1 … row sums 1, 2, 8, 24, 80, 256, 832, 2688, …
  (−∞, −1] nonnegative
• A013620 → basis A007318
  1 · 5, 3 · 25, 30, 9 · 125, 225, 135, 27 … row sums 1, 8, 64, 512, 4096, 32768, 262144, …
  (−∞, −1] nonnegative rational roots
• A033282 → x ↦ x+1
  1 · 3, 2 · 11, 15, 5 · 45, 93, 63, 14 … row sums 1, 5, 31, 215, 1597, 12425, 99955, …
  (−2, −1) 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
• 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
• 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
• 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
• A085841 → basis A007318
  1 · 7, 4 · 61, 72, 16 · 547, 1004, 528, 64 … row sums 1, 11, 149, 2143, 31273, 457907, …
  (−∞, 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
• 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
• 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
• A196347 → basis A007318
  1 · 2, 1 · 8, 8, 2 · 48, 72, 36, 6 … row sums 1, 3, 18, 162, 1944, 29160, 524880, …
  (−∞, 0] nonnegative rational roots
• A038337 → basis A007318
  529, 506, 121 · 12167, 17457, 8349, 1331 … row sums 1156, 39304, 1336336, 45435424, …
  (−∞, 0] nonnegative rational roots
• A274498 → x ↦ x+1
  1 · 3 · 9, 6 · 27, 18 … row sums 1, 3, 15, 45, 225, 675, 3375, 10125, …
  (−∞, −1] nonnegative rational roots
• A318772 → basis A007318
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 7, 13, 19, 25, …
  (−∞, −1] nonnegative
• A143491 → basis A007318
  12, 7, 1 · 60, 47, 12, 1 · 360, 342, 119, 18, 1 … row sums 20, 120, 840, 6720, 60480, 604800, …
  (−∞, 0] nonnegative rational roots
• A290319 → basis A007318
  1 · 2, 1 · 12, 8, 1 · 120, 92, 18, 1 … row sums 1, 3, 21, 231, 3465, 65835, 1514205, …
  (−∞, 0] nonnegative rational roots
• A099093 → basis A007318
  1 · 3, 3 · 12, 21, 9 · 45, 117, 99, 27 … row sums 1, 6, 42, 288, 1980, 13608, 93528, …
  (−∞, −1] nonnegative
• A299989 → basis A007318
  1, 1 · 8, 14, 7, 1 · 64, 160, 148, 64, 13, 1 … row sums 2, 30, 450, 6750, 101250, 1518750, …
  (−∞, 0] nonnegative rational roots
• A207824 → basis A007318
  1 · 6, 1 · 35, 12, 1 · 204, 106, 18, 1 … row sums 1, 7, 48, 329, 2255, 15456, 105937, …
  (−∞, 0] nonnegative
• A249252 → basis A007318
  1 · 5, 4 · 17, 24, 8 · 61, 124, 80, 16 … row sums 1, 9, 49, 281, 1601, 9129, 52049, …
  (−∞, −1] nonnegative
• A249250 → x ↦ x+1
  1 · 4, 3 · 10, 12, 3 · 28, 48, 24, 3 … row sums 1, 7, 25, 103, 409, 1639, 6553, 26215, …
  (−∞, −1] nonnegative
• A030528: Triangle read by rows: T(n,k) = binomial(k,n-k).
  1 · 1, 1 · 0, 2, 1 · 0, 1, 3, 1 … row sums 1, 2, 3, 5, 8, 13, 21, 34, …
  A030528 (−4, 0] nonnegative
• A159854 → x ↦ x+1
  1, 2, 1 · 2, 5, 4, 1 · 4, 12, 13, 6, 1 … row sums 4, 12, 36, 108, 324, 972, 2916, 8748, …
  (−∞, −1] nonnegative rational roots
• A038255: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).
  1 · 6, 1 · 36, 12, 1 · 216, 108, 18, 1 … row sums 1, 7, 49, 343, 2401, 16807, 117649, …
  A038255 (−∞, 0) nonnegative rational roots
• A286718: Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.
  1 · 1, 1 · 4, 5, 1 · 28, 39, 12, 1 … row sums 1, 2, 10, 80, 880, 12320, 209440, …
  A286718 (−∞, 0) nonnegative rational roots
• A109692 → basis A007318
  1 · 2, 1 · 8, 10, 3 · 48, 100, 68, 15 … row sums 1, 3, 21, 231, 3465, 65835, 1514205, …
  (−∞, 0] nonnegative rational roots
• A091042 → basis A007318
  1 · 4, 3 · 16, 20, 5 · 64, 112, 56, 7 … row sums 1, 7, 41, 239, 1393, 8119, 47321, …
  (−∞, −1] nonnegative
• delannoy → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 12, 18, 8, 1 … row sums 1, 3, 11, 39, 139, 495, 1763, 6279, …
  (−∞, −1] nonnegative
• A272866 → basis A007318
  1 · 5, 5, 1 · 25, 50, 35, 10, 1 … row sums 1, 11, 121, 1331, 14641, 161051, …
  (−∞, −1] nonnegative
• A204420 → basis A007318
  1 · 1, 1 · 9, 12, 3 · 225, 345, 135, 15 … row sums 1, 2, 24, 720, 40320, 3628800, …
  (−∞, 0] nonnegative rational roots
• A038338 → basis A007318
  576, 576, 144 · 13824, 20736, 10368, 1728 … row sums 1296, 46656, 1679616, 60466176, …
  (−∞, 0] nonnegative rational roots
• A038226 → basis A007318
  121, 176, 64 · 1331, 2904, 2112, 512 … row sums 361, 6859, 130321, 2476099, 47045881, …
  (−∞, 0] nonnegative rational roots
• A143685 → basis A007318
  1 · 2, 1 · 13, 13, 1 · 44, 66, 24, 1 … row sums 1, 3, 27, 135, 891, 5103, 31347, …
  (−∞, −1] nonnegative
• A207823: Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).
  1 · 4, 1 · 15, 8, 1 · 56, 46, 12, 1 … row sums 1, 5, 24, 115, 551, 2640, 12649, 60605, …
  A207823 (−6, −2) nonnegative
• A116414 → basis A007318
  1 · 5, 1 · 22, 10, 1 · 95, 69, 15, 1 … row sums 1, 6, 33, 180, 981, 5346, 29133, …
  (−9, −1) nonnegative
• A038303 → basis A007318
  1 · 11, 1 · 121, 22, 1 · 1331, 363, 33, 1 … row sums 1, 12, 144, 1728, 20736, 248832, …
  (−∞, 0] nonnegative rational roots
• A038321 → basis A007318
  324, 252, 49 · 5832, 6804, 2646, 343 … row sums 625, 15625, 390625, 9765625, 244140625, …
  (−∞, 0] nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.
  144, 264, 121 · 1728, 4752, 4356, 1331 … row sums 529, 12167, 279841, 6436343, 148035889, …
  (−∞, 0) nonnegative rational roots
• A154228 → basis A007318
  1 · 2, 1 · 18, 18, 1 · 96, 144, 50, 1 … row sums 1, 3, 37, 291, 4943, 67791, 1587413, …
  (−∞, 0] nonnegative
• A272867 → basis A007318
  1 · 6, 6, 1 · 36, 72, 48, 12, 1 … row sums 1, 13, 169, 2197, 28561, 371293, …
  (−∞, 0] nonnegative