basis_map_theorem

Basis map theorem for the node's own rule. All operations

152 sequences

• Stirling2 triangle with scaled diagonals (powers of 9).
  81, 27, 1 · 729, 567, 54, 1 … row sums 109, 1351, 19612, 333451, 6493069, …
  (−∞, 0) nonnegative
• 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
• A165253: Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
  1 · 1 · 1, 1 · 1, 3, 1 … row sums 1, 1, 2, 5, 13, 34, 89, 233, …
  A165253 (−4, 0) nonnegative
• A028326: Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).
  2 · 2, 2 · 2, 4, 2 · 2, 6, 6, 2 … row sums 2, 4, 8, 16, 32, 64, 128, 256, …
  A028326 (−∞, 0) nonnegative rational roots
• A267482: Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=n.
  1 · 1, 1 · -1, 1, 1 · -1, -2, 1, 1 … row sums 1, 2, 1, -1, -2, -1, 1, 2, …
  A267482 (−2, 2)
• A085478: Triangle read by rows: T(n, k) = binomial(n + k, 2*k).
  1 · 1, 1 · 1, 3, 1 · 1, 6, 5, 1 … row sums 1, 2, 5, 13, 34, 89, 233, 610, …
  A085478 (−4, 0) nonnegative
• A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
  64, 48, 9 · 512, 576, 216, 27 … row sums 121, 1331, 14641, 161051, 1771561, …
  A038281 (−∞, 0) nonnegative rational roots
• A102662: Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).
  1 · 1, 3 · 1, 5, 3 · 1, 7, 11, 3 … row sums 1, 4, 9, 22, 53, 128, 309, 746, …
  A102662 (−∞, 0) nonnegative
• A225472: Triangle read by rows, k!*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  1 · 2, 3 · 4, 21, 18 · 8, 117, 270, 162 … row sums 1, 5, 43, 557, 9643, 208685, 5419243, …
  A225472 (−1, 0) nonnegative
• A075498: Stirling2 triangle with scaled diagonals (powers of 3).
  1 · 3, 1 · 9, 9, 1 · 27, 63, 18, 1 … row sums 1, 4, 19, 109, 742, 5815, 51193, …
  A075498 (−∞, 0) nonnegative
• A059473: Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...
  1 · 2, 2 · 4, 10, 4 · 8, 32, 32, 8 … row sums 1, 4, 18, 80, 356, 1584, 7048, 31360, …
  A059473 (−∞, 0) nonnegative
• A143683: Pascal-(1,8,1) array.
  1 · 1, 1 · 1, 10, 1 · 1, 19, 19, 1 … row sums 1, 2, 12, 40, 176, 672, 2752, 10880, …
  A143683 (−∞, 0) nonnegative
• A081582: Pascal-(1,7,1) array.
  1 · 1, 1 · 1, 9, 1 · 1, 17, 17, 1 … row sums 1, 2, 11, 36, 149, 550, 2143, 8136, …
  A081582 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
  121, 132, 36 · 1331, 2178, 1188, 216 … row sums 289, 4913, 83521, 1419857, 24137569, …
  (−∞, 0) nonnegative rational roots
• A081581: Pascal-(1,6,1) array.
  1 · 1, 1 · 1, 8, 1 · 1, 15, 15, 1 … row sums 1, 2, 10, 32, 124, 440, 1624, 5888, …
  A081581 (−∞, 0) nonnegative
• 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
• Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
  9, 48, 64 · 27, 216, 576, 512 … row sums 121, 1331, 14641, 161051, 1771561, …
  (−∞, 0) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
  100, 240, 144 · 1000, 3600, 4320, 1728 … row sums 484, 10648, 234256, 5153632, 113379904, …
  (−∞, 0) nonnegative rational roots
• Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m = 0.
  1, 11, 1 · 1, 111, 30, 1 · 1, 1111, 681, 58, 1 … row sums 13, 143, 1852, 27563, 473725, 9290396, …
  (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
  121, 110, 25 · 1331, 1815, 825, 125 … row sums 256, 4096, 65536, 1048576, 16777216, …
  (−∞, 0) nonnegative rational roots
• A131689: Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.
  1 · 0, 1 · 0, 1, 2 · 0, 1, 6, 6 … row sums 1, 1, 3, 13, 75, 541, 4683, 47293, …
  A131689 (−1, 0] nonnegative
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
  1 · 1, 4 · 1, 8, 16 · 1, 12, 48, 64 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A013611 (−∞, 0) [−1, 0] nonnegative rational roots
• A038322: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.
  1 · 11, 8 · 121, 176, 64 · 1331, 2904, 2112, 512 … row sums 1, 19, 361, 6859, 130321, 2476099, …
  A038322 (−∞, 0) nonnegative rational roots
• 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
• A282629: Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].
  1 · 1, 3 · 1, 15, 9 · 1, 63, 108, 27 … row sums 1, 4, 25, 199, 1876, 20257, 245017, …
  A282629 (−∞, 0) nonnegative
• A111125: Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
  1 · 3, 1 · 5, 5, 1 · 7, 14, 7, 1 … row sums 1, 4, 11, 29, 76, 199, 521, 1364, …
  A111125 (−4, 0) nonnegative
• 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
• A207607: Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.
  1 · 1, 2 · 1, 5, 2 · 1, 9, 9, 2 … row sums 1, 3, 8, 21, 55, 144, 377, 987, …
  A207607 (−4, 0) nonnegative
• A038279: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.
  1 · 8, 1 · 64, 16, 1 · 512, 192, 24, 1 … row sums 1, 9, 81, 729, 6561, 59049, 531441, …
  A038279 (−∞, 0) nonnegative rational roots
• A010054: a(n) = 1 if n is a triangular number, otherwise 0.
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A010054 (−∞, −1] [−1, 0] [−3−2√2, −3+2√2] nonnegative rational roots
• A053124: Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in increasing order).
  1 · -2, 4 · 3, -16, 16 · -4, 40, -96, 64 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  A053124 (0, 1)
• A008277: Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
  1 · 1, 1 · 1, 3, 1 · 1, 7, 6, 1 … row sums 1, 2, 5, 15, 52, 203, 877, 4140, …
  A008277 (−∞, 0) nonnegative
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
  1 · 4, 1 · 16, 8, 1 · 64, 48, 12, 1 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A038231 (−∞, −1] nonnegative rational roots
• A038291: Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.
  1 · 9, 1 · 81, 18, 1 · 729, 243, 27, 1 … row sums 1, 10, 100, 1000, 10000, 100000, …
  A038291 (−∞, 0) nonnegative rational roots
• A152198: Triangle read by rows, A007318 rows repeated.
  1 · 1 · 1, 1 · 1, 1 … row sums 1, 1, 2, 2, 4, 4, 8, 8, …
  A152198 A152815 (−∞, 0] nonnegative rational roots
• A290319: Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.
  1 · 1, 1 · 5, 6, 1 · 45, 59, 15, 1 … row sums 1, 2, 12, 120, 1680, 30240, 665280, …
  A290319 (−∞, 0) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
  100, 60, 9 · 1000, 900, 270, 27 … row sums 169, 2197, 28561, 371293, 4826809, …
  (−∞, 0) nonnegative rational roots
• A285061: Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).
  1 · 1, 4 · 1, 24, 16 · 1, 124, 240, 64 … row sums 1, 5, 41, 429, 5329, 75989, 1215481, …
  A285061 (−∞, 0) nonnegative
• A124038: Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.
  1 · -2, 1 · -1, -2, 1 · 2, -2, -2, 1 … row sums 1, -1, -2, -1, 1, 2, 1, -1, …
  A124038 (−2, 3)
• A038323: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.
  1 · 11, 9 · 121, 198, 81 · 1331, 3267, 2673, 729 … row sums 1, 20, 400, 8000, 160000, 3200000, …
  A038323 (−∞, 0) nonnegative rational roots
• A028338: Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
  1 · 1, 1 · 3, 4, 1 · 15, 23, 9, 1 … row sums 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  A028338 (−∞, 0) nonnegative rational roots
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.
  3 · 6, 3 · 12, 12, 3 · 24, 36, 18, 3 … row sums 3, 9, 27, 81, 243, 729, 2187, 6561, …
  A120910 (−∞, 0) nonnegative rational roots
• A049374: A triangle of numbers related to triangle A030527.
  1 · 6, 1 · 42, 18, 1 · 336, 276, 36, 1 … row sums 1, 7, 61, 649, 8245, 122215, 2069425, …
  A049374 (−∞, 0) nonnegative
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
  1 · 2, 3 · 4, 12, 9 · 8, 36, 54, 27 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A013620 (−∞, 0) [−1, 0] nonnegative rational roots
• A225468: Triangle read by rows, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  1 · 2, 1 · 4, 7, 1 · 8, 39, 15, 1 … row sums 1, 3, 12, 63, 405, 3024, 25515, 239355, …
  A225468 (−∞, 0) nonnegative
• A049029: Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.
  1 · 5, 1 · 45, 15, 1 · 585, 255, 30, 1 … row sums 1, 6, 61, 871, 15996, 358891, 9509641, …
  A049029 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
  144, 168, 49 · 1728, 3024, 1764, 343 … row sums 361, 6859, 130321, 2476099, 47045881, …
  (−∞, 0) nonnegative rational roots
• A137320: Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows…
  1 · 0, 2 · 0, 2, 4 · 0, 4, 12, 8 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A137320 (−∞, 0] nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
  144, 72, 9 · 1728, 1296, 324, 27 … row sums 225, 3375, 50625, 759375, 11390625, …
  (−∞, 0) nonnegative rational roots