basis_map / A007318

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

219 sequences

• basis-map coefficients of A026729 → basis A007318
  1 · 1, 1 · 1, 2, 1 · 1, 3, 3, 1 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, −1] nonnegative rational roots
• basis-map coefficients of A021012 → x ↦ x+1
  1 · -1, -1 · 2, 4, 2 · -6, -18, -18, -6 … row sums 1, -2, 8, -48, 384, -3840, 46080, …
  (−∞, −1] rational roots
• A038303: Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.
  1 · 10, 1 · 100, 20, 1 · 1000, 300, 30, 1 … row sums 1, 11, 121, 1331, 14641, 161051, …
  A038303 (−∞, 0) nonnegative rational roots
• A099097 → basis A007318
  1 · 3, 3 · 10, 19, 9 · 33, 93, 87, 27 … row sums 1, 6, 38, 240, 1516, 9576, 60488, …
  (−∞, −1] nonnegative
• A154227 → basis A007318
  1 · 2, 1 · 10, 10, 1 · 40, 60, 22, 1 … row sums 1, 3, 21, 123, 999, 8163, 80433, …
  (−∞, 0] nonnegative
• A118933 → x ↦ x+1
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 13, 61, 181, 421, …
  (−∞, −1] nonnegative
• A169803 → x ↦ x+1
  1 · 2, 1 · 3, 2 · 5, 5, 1 … row sums 1, 3, 5, 11, 21, 43, 85, 171, …
  (−∞, −1] nonnegative
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
  1 · 2, 1 · 4, 4, 1 · 8, 12, 6, 1 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A038207 (−∞, −1] nonnegative rational roots
• A269947 → basis A007318
  1 · 1, 1 · 2, 3, 1 · 18, 29, 12, 1 … row sums 1, 2, 6, 60, 1740, 114840, 14584680, …
  (−∞, 0] nonnegative rational roots
• A118931 → basis A007318
  1 · 1 · 1 · 3, 2 … row sums 1, 1, 1, 5, 17, 41, 241, 1261, …
  (−∞, −1] nonnegative
• A317496 → x ↦ x+1
  1 · 1 · 1 · 4, 3 … row sums 1, 1, 1, 7, 13, 19, 61, 139, …
  (−∞, −1] nonnegative
• A103633 → basis A007318
  1 · 1, 1 · 2, 3, 1 · 2, 5, 4, 1 … row sums 1, 2, 6, 12, 36, 72, 216, 432, …
  (−∞, −1] nonnegative rational roots
• A038314 → basis A007318
  484, 528, 144 · 10648, 17424, 9504, 1728 … row sums 1156, 39304, 1336336, 45435424, …
  (−∞, 0] nonnegative rational roots
• A143683 → x ↦ x+1
  1 · 2, 1 · 12, 12, 1 · 40, 60, 22, 1 … row sums 1, 3, 25, 123, 769, 4275, 25129, …
  (−∞, −1] nonnegative
• A225470 → basis A007318
  1 · 3, 1 · 18, 9, 1 · 162, 99, 18, 1 … row sums 1, 4, 28, 280, 3640, 58240, 1106560, …
  (−∞, 0] nonnegative rational roots
• A114687 → basis A007318
  1 · 3, 2 · 11, 14, 4 · 45, 84, 48, 8 … row sums 1, 5, 29, 185, 1257, 8925, 65445, …
  (−∞, −1] nonnegative
• A145324 → basis A007318
  1 · 3, 2 · 12, 17, 6 · 60, 133, 98, 24 … row sums 1, 5, 35, 315, 3465, 45045, 675675, …
  (−∞, 0] nonnegative rational roots
• A121574 → basis A007318
  1 · 3, 1 · 10, 7, 1 · 33, 35, 11, 1 … row sums 1, 4, 18, 80, 356, 1584, 7048, 31360, …
  (−∞, −1] nonnegative
• A113953: A Jacobsthal triangle.
  1 · 0, 1 · 0, 2, 1 · 0, 0, 4, 1 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  A113953 (−8, 0] nonnegative
• A013613 → basis A007318
  1 · 7, 6 · 49, 84, 36 · 343, 882, 756, 216 … row sums 1, 13, 169, 2197, 28561, 371293, …
  (−∞, 0] nonnegative rational roots
• A161198 → basis A007318
  1 · 3, 2 · 15, 16, 4 · 105, 142, 60, 8 … row sums 1, 5, 35, 315, 3465, 45045, 675675, …
  (−∞, 0] nonnegative rational roots
• A081578 → x ↦ x+1
  1 · 2, 1 · 7, 7, 1 · 20, 30, 12, 1 … row sums 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  (−∞, −1] nonnegative
• A038301 → basis A007318
  400, 440, 121 · 8000, 13200, 7260, 1331 … row sums 961, 29791, 923521, 28629151, …
  (−∞, 0] nonnegative rational roots
• A086645 → x ↦ x+1
  1 · 2, 1 · 8, 8, 1 · 32, 48, 18, 1 … row sums 1, 3, 17, 99, 577, 3363, 19601, 114243, …
  (−∞, −1] nonnegative
• A027467 → basis A007318
  1 · 16, 1 · 256, 32, 1 · 4096, 768, 48, 1 … row sums 1, 17, 289, 4913, 83521, 1419857, …
  (−∞, 0] nonnegative rational roots
• A207607 → basis A007318
  1 · 3, 2 · 8, 9, 2 · 21, 33, 15, 2 … row sums 1, 5, 19, 71, 265, 989, 3691, 13775, …
  (−5, −1) nonnegative
• A113214: Riordan array (1+2x,x(1+x)).
  1 · 2, 1 · 0, 3, 1 · 0, 2, 4, 1 … row sums 1, 3, 4, 7, 11, 18, 29, 47, …
  A113214 (−4, 0] nonnegative
• A367270 → x ↦ x+1
  1 · 1 · 3, 2 · 10, 12, 3 … row sums 1, 1, 5, 25, 129, 681, 3653, 19825, …
  (−∞, −1] nonnegative
• A207824: Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).
  1 · 5, 1 · 24, 10, 1 · 115, 73, 15, 1 … row sums 1, 6, 35, 204, 1189, 6930, 40391, …
  A207824 (−7, −3) nonnegative
• A269944 → basis A007318
  1 · 1, 1 · 2, 3, 1 · 10, 17, 8, 1 … row sums 1, 2, 6, 36, 396, 7128, 192456, …
  (−∞, 0] nonnegative rational roots
• A063967 → basis A007318
  1 · 2, 1 · 6, 5, 1 · 16, 20, 8, 1 … row sums 1, 3, 12, 45, 171, 648, 2457, 9315, …
  (−∞, −1] nonnegative
• A063967: Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
  1 · 1, 1 · 2, 3, 1 · 3, 7, 5, 1 … row sums 1, 2, 6, 16, 44, 120, 328, 896, …
  A063967 (−5, −1] nonnegative
• A007318: Binomial
  1 · 1, 1 · 1, 2, 1 · 1, 3, 3, 1 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A007318 (−∞, −1] nonnegative rational roots
• A102547 → x ↦ (1/27)x → x ↦ x+1
  1 · 3 · 9 · 28, 1 … row sums 1, 3, 9, 29, 93, 297, 949, 3033, …
  (−∞, −1] nonnegative
• A196347: Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
  1 · 1, 1 · 2, 4, 2 · 6, 18, 18, 6 … row sums 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  A196347 (−∞, −1] nonnegative rational roots
• A232535 → x ↦ x+1
  1 · 3, 2 · 12, 14, 3 · 48, 80, 37, 4 … row sums 1, 5, 29, 169, 985, 5741, 33461, …
  (−∞, −1] nonnegative
• A013611 → basis A007318
  1 · 5, 4 · 25, 40, 16 · 125, 300, 240, 64 … row sums 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, −1] nonnegative rational roots
• A085478 → x ↦ x+1
  1 · 1 · 2, 1 · 5, 5, 1 … row sums 1, 1, 3, 11, 41, 153, 571, 2131, …
  (−5, −1) nonnegative
• A038208: Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
  1 · 2, 2 · 4, 8, 4 · 8, 24, 24, 8 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A038208 (−∞, −1] nonnegative rational roots
• A038323 → basis A007318
  1 · 20, 9 · 400, 360, 81 · 8000, 10800, 4860, 729 … row sums 1, 29, 841, 24389, 707281, 20511149, …
  (−∞, 0] nonnegative rational roots
• A274496 → x ↦ x+1
  1 · 2 · 4, 2 · 8, 4 … row sums 1, 2, 6, 12, 36, 72, 216, 432, …
  (−∞, −1] nonnegative rational roots
• A133087: A133080 * A007318.
  1 · 2, 1 · 1, 2, 1 · 2, 5, 4, 1 … row sums 1, 3, 4, 12, 16, 48, 64, 192, …
  A133087 (−∞, −1] nonnegative rational roots
• A038319 → basis A007318
  256, 160, 25 · 4096, 3840, 1200, 125 … row sums 441, 9261, 194481, 4084101, 85766121, …
  (−∞, 0] nonnegative rational roots
• basis-map coefficients of A112227 → x ↦ x+1
  1, 2, 1 · 1, 3, 3, 1 · 1, 4, 6, 4, 1 … row sums 4, 8, 16, 32, 64, 128, 256, 512, …
  (−∞, −1] nonnegative rational roots
• A207606 → basis A007318
  13, 11, 2 · 34, 44, 17, 2 · 89, 155, 93, 23, 2 … row sums 26, 97, 362, 1351, 5042, 18817, 70226, …
  (−5, −1) nonnegative
• A156992 → basis A007318
  1 · 4, 2 · 24, 24, 6 · 192, 288, 144, 24 … row sums 1, 6, 54, 648, 9720, 174960, 3674160, …
  (−∞, 0] nonnegative rational roots
• A327997 → basis A007318
  1 · 4, 1 · 17, 9, 1 · 78, 65, 15, 1 … row sums 1, 5, 27, 159, 1029, 7353, 58095, …
  (−∞, 0] nonnegative
• A133087 → x ↦ x+1
  1 · 3, 1 · 4, 4, 1 · 12, 16, 7, 1 … row sums 1, 4, 9, 36, 81, 324, 729, 2916, …
  (−∞, −1] nonnegative rational roots
• A011973 → x ↦ x+2 (shift +1) → basis A007318
  1 · 4, 1 · 7, 2 · 19, 9, 1 … row sums 1, 5, 9, 29, 65, 181, 441, 1165, …
  (−∞, −1] nonnegative
• A078812 → basis A007318
  1 · 3, 1 · 8, 6, 1 · 21, 25, 9, 1 … row sums 1, 4, 15, 56, 209, 780, 2911, 10864, …
  (−5, −1) nonnegative