basis_map

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

1056 sequences

• A038291 → basis A013611
  1 · 10, 4 · 100, 80, 16 · 1000, 1200, 480, 64 … row sums 1, 14, 196, 2744, 38416, 537824, …
  (−∞, 0] nonnegative rational roots
• A038328 → basis A013611
  196, 224, 64 · 2744, 4704, 2688, 512 … row sums 484, 10648, 234256, 5153632, 113379904, …
  (−∞, 0] nonnegative rational roots
• A114687 → basis A013611
  1 · 3, 8 · 11, 56, 64 · 45, 336, 768, 512 … row sums 1, 11, 131, 1661, 22101, 305151, …
  (−∞, 0] nonnegative
• A026374 → basis A013609
  1 · 2, 2 · 5, 10, 4 · 10, 30, 28, 8 … row sums 1, 4, 19, 76, 361, 1444, 6859, 27436, …
  (−∞, 0] 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
• A103327 → basis A013609
  1 · 4, 2 · 16, 24, 4 · 64, 160, 96, 8 … row sums 1, 6, 44, 328, 2448, 18272, 136384, …
  (−∞, 0] nonnegative
• A038333 → basis A008277
  361, 315, 49 · 6859, 10717, 3822, 343 … row sums 725, 21741, 700996, 24245693, …
  (−∞, 0] nonnegative
• 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
• A304236 → basis A013610
  1 · 1 · 4, 9 · 7, 18 … row sums 1, 1, 13, 25, 181, 481, 2653, 8425, …
  (−∞, 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
• A038221: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
  1 · 3, 3 · 9, 18, 9 · 27, 81, 81, 27 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …
  A038221 (−∞, 0) nonnegative rational roots
• A299989 → basis A008277
  1, 1 · 8, 22, 10, 1 · 64, 386, 446, 167, 23, 1 … row sums 2, 41, 1087, 36466, 1511407, 75738861, …
  (−∞, 0] nonnegative
• A178822 → basis A013610
  1 · 12, 18 · 84, 252, 189 · 448, 2016, 3024, 1512 … row sums 1, 30, 525, 7000, 78750, 787500, …
  (−∞, 0] nonnegative rational roots
• A135838 → basis A013611
  1 · 4, 8 · 8, 32, 32 · 32, 192, 384, 256 … row sums 1, 12, 72, 864, 5184, 62208, 373248, …
  (−∞, 0] nonnegative rational roots
• A103633 → basis A013610
  1 · 1, 3 · 2, 9, 9 · 2, 15, 36, 27 … row sums 1, 4, 20, 80, 400, 1600, 8000, 32000, …
  (−∞, 0] 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
• A249250 → basis A008277
  1 · 4, 3 · 10, 15, 3 · 28, 75, 33, 3 … row sums 1, 7, 28, 139, 721, 4036, 24031, …
  (−∞, 0] nonnegative
• A011973 → x ↦ x+2 (shift +1) → basis A013611
  1 · 4, 4 · 7, 8 · 19, 36, 16 … row sums 1, 8, 15, 71, 176, 673, 1905, 6616, …
  (−∞, 0] nonnegative
• A358735 → basis A013611
  1 · 2, 4 · 7, 32, 32 · 33, 240, 544, 384 … row sums 1, 6, 71, 1201, 26351, 710276, …
  (−∞, 0] nonnegative
• A078812 → basis A013609
  1 · 3, 2 · 8, 12, 4 · 21, 50, 36, 8 … row sums 1, 5, 24, 115, 551, 2640, 12649, 60605, …
  (−∞, 0] nonnegative
• A011973 → x ↦ x+2 (shift +1) → basis A013610
  1 · 4, 3 · 7, 6 · 19, 27, 9 … row sums 1, 7, 13, 55, 133, 463, 1261, 4039, …
  (−∞, 0] nonnegative
• A318772 → x ↦ (1/243)x → x ↦ x+1
  1 · 3 · 9 · 27 … row sums 1, 3, 9, 27, 83, 255, 783, 2403, …
  (−∞, −1] nonnegative
• A086645 → basis A008277
  1 · 2, 1 · 8, 9, 1 · 32, 67, 21, 1 … row sums 1, 3, 18, 121, 879, 6834, 56483, …
  (−∞, 0] nonnegative
• A118394 → basis A007318
  1 · 1 · 1 · 7, 6 … row sums 1, 1, 1, 13, 49, 121, 1681, 10501, …
  (−∞, −1] nonnegative
• A008955 → basis A013609
  1 · 2, 2 · 10, 26, 16 · 100, 440, 628, 288 … row sums 1, 4, 52, 1456, 71344, 5422144, …
  (−∞, 0] nonnegative rational roots
• A133080 → basis A013610
  1 · 2, 3 · 1, 6, 9 · 2, 15, 36, 27 … row sums 1, 5, 16, 80, 256, 1280, 4096, 20480, …
  (−∞, 0] nonnegative rational roots
• 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
• A104684 → basis A013610
  1 · 3, 3 · 13, 24, 9 · 63, 171, 135, 27 … row sums 1, 6, 46, 396, 3606, 33876, 324556, …
  (−∞, 0] nonnegative
• A286718 → basis A008277
  1 · 2, 1 · 10, 8, 1 · 80, 82, 18, 1 … row sums 1, 3, 19, 181, 2293, 36191, 683303, …
  (−∞, 0] nonnegative
• A038284 → basis A013610
  196, 504, 324 · 2744, 10584, 13608, 5832 … row sums 1024, 32768, 1048576, 33554432, …
  (−∞, 0] nonnegative rational roots
• A086645 → basis A013609
  1 · 2, 2 · 8, 16, 4 · 32, 96, 72, 8 … row sums 1, 4, 28, 208, 1552, 11584, 86464, …
  (−∞, 0] nonnegative
• A081582 → basis A013611
  1 · 2, 4 · 11, 44, 16 · 36, 216, 320, 64 … row sums 1, 6, 71, 636, 6301, 60066, 580931, …
  (−∞, 0] nonnegative
• A102547 → x ↦ (1/27)x → basis A008277
  1 · 3 · 9 · 28, 1 … row sums 1, 3, 9, 29, 93, 297, 950, 3042, …
  (−∞, 0] nonnegative
• A038319 → basis A013611
  256, 640, 400 · 4096, 15360, 19200, 8000 … row sums 1296, 46656, 1679616, 60466176, …
  (−∞, 0] 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
• A344565 → basis A013610
  3 · 30, 45 · 180, 540, 405 · 840, 3780, 5670, 2835 … row sums 3, 75, 1125, 13125, 131250, 1181250, …
  (−∞, 0] nonnegative rational roots
• A274498 → basis A008277
  1 · 3 · 9, 6 · 27, 18 … row sums 1, 3, 15, 45, 261, 783, 5211, 15633, …
  (−∞, 0] nonnegative
• basis-map coefficients of A030528, shifted → basis A013609
  1 · -1, -2 · 1, 4, 4 · -1, -6, -12, -8 … row sums 1, -3, 9, -27, 81, -243, 729, -2187, …
  (−∞, 0] rational roots
• A094638 → basis A008277
  1 · 2, 1 · 6, 9, 2 · 24, 81, 47, 6 … row sums 1, 3, 17, 158, 2194, 42439, 1085923, …
  (−∞, 0] nonnegative
• 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
• A081580 → basis A013610
  1 · 2, 3 · 9, 27, 9 · 28, 126, 144, 27 … row sums 1, 5, 45, 325, 2525, 19125, 146125, …
  (−∞, 0] nonnegative
• A327997 → basis A013611
  1 · 4, 4 · 17, 36, 16 · 78, 260, 240, 64 … row sums 1, 8, 69, 642, 6441, 69588, 808029, …
  (−∞, 0] nonnegative
• A021012: Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
  basis-map coefficients of A021012 → basis A130595
  1 · 1, -1 · 2, -4, 2 · 6, -18, 18, -6 … row sums 1, 0, 0, 0, 0, 0, 0, 0, …
  A021012 (−∞, ∞) rational roots
• A130123 → basis A008277
  1 · 2, 2 · 4, 12, 4 · 8, 56, 48, 8 … row sums 1, 4, 20, 120, 832, 6496, 56128, …
  (−∞, 0] nonnegative
• A046716 → basis A013609
  1 · 2, 2 · 5, 10, 4 · 16, 50, 44, 8 … row sums 1, 4, 19, 118, 1021, 12088, 183727, …
  (−∞, 0] nonnegative
• A178820 → basis A013610
  1 · 8, 12 · 40, 120, 90 · 160, 720, 1080, 540 … row sums 1, 20, 250, 2500, 21875, 175000, …
  (−∞, 0] nonnegative rational roots
• A060821: Triangle read by rows. T(n, k) are the coefficients of the Hermite polynomial of order n, for 0 <= k <= n.
  A023531 → basis A060821
  1 · 0, 2 · -2, 0, 4 · 0, -12, 0, 8 … row sums 1, 2, 2, -4, -20, -8, 184, 464, …
  A060821 (−∞, ∞)
• delannoy → basis A008277
  1 · 2, 1 · 5, 6, 1 · 12, 27, 11, 1 … row sums 1, 3, 12, 51, 237, 1190, 6415, 36899, …
  (−∞, 0] nonnegative