basis_map / A007318

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

219 sequences

• A100862 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … row sums 1, 3, 11, 45, 201, 963, 4899, 26253, …
  (−∞, −1] nonnegative
• A110441 → basis A007318
  1 · 4, 1 · 14, 8, 1 · 48, 44, 12, 1 … row sums 1, 5, 23, 105, 479, 2185, 9967, 45465, …
  (−7, −1) nonnegative
• A038335 → basis A007318
  1 · 21, 9 · 441, 378, 81 · 9261, 11907, 5103, 729 … row sums 1, 30, 900, 27000, 810000, 24300000, …
  (−∞, 0] nonnegative rational roots
• A102547 → x ↦ x+1
  1 · 1 · 1 · 2, 1 … row sums 1, 1, 1, 3, 5, 7, 13, 23, …
  (−∞, −1] nonnegative
• A059364 → basis A007318
  1 · 3, 1 · 15, 11, 2 · 105, 122, 47, 6 … row sums 1, 4, 28, 280, 3640, 58240, 1106560, …
  (−∞, 0] nonnegative rational roots
• narayana → basis A007318
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  (−∞, −1] nonnegative
• A038320 → basis A007318
  289, 204, 36 · 4913, 5202, 1836, 216 … row sums 529, 12167, 279841, 6436343, 148035889, …
  (−∞, 0] nonnegative rational roots
• A152830 → x ↦ x+1
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 3, 3, 3, 3, …
  (−∞, −1] nonnegative rational roots
• A038284 → basis A007318
  196, 168, 36 · 2744, 3528, 1512, 216 … row sums 400, 8000, 160000, 3200000, 64000000, …
  (−∞, 0] nonnegative rational roots
• A064861 → x ↦ x+1
  6, 7, 2 · 18, 33, 20, 4 · 36, 84, 73, 28, 4 … row sums 15, 75, 225, 1125, 3375, 16875, 50625, …
  (−∞, −1] nonnegative rational roots
• A172431 → basis A007318
  1 · 3, 2 · 8, 10, 3 · 21, 38, 22, 4 … row sums 1, 5, 21, 85, 341, 1365, 5461, 21845, …
  (−∞, −1] nonnegative
• basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
  1 · -1, -1 · 1, 2, 1 · -1, -3, -3, -1 … row sums 1, -2, 4, -8, 16, -32, 64, -128, …
  (−∞, −1] rational roots
• A038277 → basis A007318
  1 · 18, 11 · 324, 396, 121 · 5832, 10692, 6534, 1331 … row sums 1, 29, 841, 24389, 707281, 20511149, …
  (−∞, 0] nonnegative rational roots
• A137320 → x ↦ x+1
  1 · 2, 2 · 6, 10, 4 · 24, 52, 36, 8 … row sums 1, 4, 20, 120, 840, 6720, 60480, …
  (−∞, −1] nonnegative rational roots
• A038327 → basis A007318
  1 · 13, 1 · 169, 26, 1 · 2197, 507, 39, 1 … row sums 1, 14, 196, 2744, 38416, 537824, …
  (−∞, 0] nonnegative rational roots
• A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
  1 · 5, 1 · 25, 10, 1 · 125, 75, 15, 1 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …
  A038243 (−∞, −1] nonnegative rational roots
• A038271 → basis A007318
  1 · 12, 5 · 144, 120, 25 · 1728, 2160, 900, 125 … row sums 1, 17, 289, 4913, 83521, 1419857, …
  (−∞, 0] nonnegative rational roots
• A128100: Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).
  A011973 → basis A007318
  1 · 1 · 2, 1 · 3, 2 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  A128100 (−∞, −1] nonnegative
• A038243 → basis A013610
  1 · 6, 3 · 36, 36, 9 · 216, 324, 162, 27 … row sums 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, 0] nonnegative rational roots
• A059364: Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.
  1 · 2, 1 · 6, 7, 2 · 24, 46, 29, 6 … row sums 1, 3, 15, 105, 945, 10395, 135135, …
  A059364 (−∞, −1] nonnegative rational roots
• A038281 → basis A007318
  121, 66, 9 · 1331, 1089, 297, 27 … row sums 196, 2744, 38416, 537824, 7529536, …
  (−∞, 0] nonnegative rational roots
• A100861 → basis A007318
  1 · 1 · 2, 1 · 4, 3 … row sums 1, 1, 3, 7, 25, 81, 331, 1303, …
  (−∞, −1] nonnegative
• A225470: Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
  1 · 2, 1 · 10, 7, 1 · 80, 66, 15, 1 … row sums 1, 3, 18, 162, 1944, 29160, 524880, …
  A225470 (−∞, 0) nonnegative rational roots
• A126216 → basis A007318
  1 · 3, 1 · 11, 7, 1 · 45, 42, 12, 1 … row sums 1, 4, 19, 100, 562, 3304, 20071, …
  (−∞, −1] nonnegative
• A079638 → basis A007318
  1 · 4, 1 · 24, 11, 1 · 192, 122, 21, 1 … row sums 1, 5, 36, 336, 3840, 51840, 806400, …
  (−∞, 0] nonnegative
• A038329 → basis A007318
  225, 90, 9 · 3375, 2025, 405, 27 … row sums 324, 5832, 104976, 1889568, 34012224, …
  (−∞, 0] nonnegative rational roots
• A008288 → basis A007318
  1 · 2, 1 · 5, 5, 1 · 12, 18, 8, 1 … row sums 1, 3, 11, 39, 139, 495, 1763, 6279, …
  (−∞, −1] nonnegative
• A103327 → basis A007318
  1 · 4, 1 · 16, 12, 1 · 64, 80, 24, 1 … row sums 1, 5, 29, 169, 985, 5741, 33461, …
  (−∞, −1] nonnegative
• A059344 → basis A007318
  1 · 1 · 3, 2 · 7, 6 … row sums 1, 1, 5, 13, 73, 281, 1741, 8485, …
  (−∞, −1] nonnegative
• A038326 → basis A007318
  1 · 23, 12 · 529, 552, 144 · 12167, 19044, 9936, 1728 … row sums 1, 35, 1225, 42875, 1500625, 52521875, …
  (−∞, 0] nonnegative rational roots
• A094816 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 16, 23, 9, 1 … row sums 1, 3, 11, 49, 261, 1631, 11743, 95901, …
  (−∞, −1] nonnegative
• A034870 → x ↦ x+1
  1 · 4, 4, 1 · 16, 32, 24, 8, 1 … row sums 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, −1] nonnegative rational roots
• A152198 → x ↦ x+1
  1 · 1 · 2, 1 · 2, 1 … row sums 1, 1, 3, 3, 9, 9, 27, 27, …
  (−∞, −1] nonnegative rational roots
• A085880 → x ↦ x+1
  1 · 2, 1 · 8, 8, 2 · 40, 60, 30, 5 … row sums 1, 3, 18, 135, 1134, 10206, 96228, …
  (−∞, −1] nonnegative rational roots
• A154233 → basis A007318
  1 · 2, 1 · 9, 9, 1 · 40, 60, 22, 1 … row sums 1, 3, 19, 123, 1091, 10407, 120683, …
  (−∞, 0] nonnegative
• A027465: Cube of lower triangular normalized binomial matrix.
  1 · 3, 1 · 9, 6, 1 · 27, 27, 9, 1 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A027465 (−∞, −1] nonnegative rational roots
• A038325 → basis A007318
  1 · 22, 11 · 484, 484, 121 · 10648, 15972, 7986, 1331 … row sums 1, 33, 1089, 35937, 1185921, 39135393, …
  (−∞, 0] nonnegative rational roots
• A318772 → x ↦ (1/243)x → x ↦ x+1
  1 · 3 · 9 · 27 … row sums 1, 3, 9, 27, 83, 255, 783, 2403, …
  (−∞, −1] nonnegative
• A118394 → basis A007318
  1 · 1 · 1 · 7, 6 … row sums 1, 1, 1, 13, 49, 121, 1681, 10501, …
  (−∞, −1] nonnegative
• A136124: Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).
  1 · 2, 1 · 6, 5, 1 · 24, 26, 9, 1 … row sums 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  A136124 (−∞, −1] nonnegative rational roots
• A013622 → basis A007318
  1 · 8, 5 · 64, 80, 25 · 512, 960, 600, 125 … row sums 1, 13, 169, 2197, 28561, 371293, …
  (−∞, 0] nonnegative rational roots
• basis-map coefficients of A026729, shifted → basis A007318
  1 · -1, -1 · 1, 2, 1 · -1, -3, -3, -1 … row sums 1, -2, 4, -8, 16, -32, 64, -128, …
  (−∞, −1] rational roots
• A132812: Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
  1 · 2, 2 · 3, 9, 3 · 4, 24, 24, 4 … row sums 1, 4, 15, 56, 210, 792, 3003, 11440, …
  A132812 (−∞, 0) nonnegative
• A391533 → Pₙ + Pₙ₊₁
  8, 12, 6, 1 · 8, 20, 18, 7, 1 · 8, 28, 38, 25, 8, 1 … row sums 27, 54, 108, 216, 432, 864, 1728, 3456, …
  (−∞, −1] nonnegative rational roots
• A035607 → basis A007318
  1 · 1 · 3, 2 · 7, 8, 2 … row sums 1, 1, 5, 17, 61, 217, 773, 2753, …
  (−∞, 0] nonnegative
• A081582 → x ↦ x+1
  1 · 2, 1 · 11, 11, 1 · 36, 54, 20, 1 … row sums 1, 3, 23, 111, 655, 3519, 19727, …
  (−∞, −1] nonnegative
• A038336 → basis A007318
  484, 440, 100 · 10648, 14520, 6600, 1000 … row sums 1024, 32768, 1048576, 33554432, …
  (−∞, 0] nonnegative rational roots
• A200536 → basis A007318
  1 · 6, 7, 2 · 36, 84, 73, 28, 4 … row sums 1, 15, 225, 3375, 50625, 759375, …
  (−∞, 0] nonnegative rational roots
• A120910 → basis A007318
  3 · 9, 3 · 27, 18, 3 · 81, 81, 27, 3 … row sums 3, 12, 48, 192, 768, 3072, 12288, …
  (−∞, −1] nonnegative rational roots
• A152828 → basis A007318
  1 · 1 · 1 · 2, 1 … row sums 1, 1, 1, 3, 3, 3, 9, 9, …
  (−∞, −1] nonnegative rational roots