basis_map_theorem

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

152 sequences

• A038277: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.
  1 · 7, 11 · 49, 154, 121 · 343, 1617, 2541, 1331 … row sums 1, 18, 324, 5832, 104976, 1889568, …
  A038277 (−∞, 0) nonnegative rational roots
• A132393: Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
  1 · 0, 1 · 0, 1, 1 · 0, 2, 3, 1 … row sums 1, 1, 2, 6, 24, 120, 720, 5040, …
  A132393 (−∞, 0] nonnegative rational roots
• A188440: Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.
  1 · 1 · 1, 2 · 1, 2 … row sums 1, 1, 3, 3, 9, 9, 27, 27, …
  A188440 (−∞, 0] nonnegative rational roots
• A193685: 5-Stirling numbers of the second kind.
  1 · 5, 1 · 25, 11, 1 · 125, 91, 18, 1 … row sums 1, 6, 37, 235, 1540, 10427, 73013, …
  A193685 (−∞, 0) nonnegative
• Triangle of coefficients in expansion of (1+5x)^n.
  1, 10, 25 · 1, 15, 75, 125 · 1, 20, 150, 500, 625 … row sums 36, 216, 1296, 7776, 46656, 279936, …
  (−∞, 0) nonnegative rational roots
• A225466: Triangle read by rows, 3^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, 9 · 8, 117, 135, 27 … row sums 1, 5, 34, 287, 2839, 31898, 399079, …
  A225466 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
  64, 96, 36 · 512, 1152, 864, 216 … row sums 196, 2744, 38416, 537824, 7529536, …
  (−∞, 0) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
  144, 288, 144 · 1728, 5184, 5184, 1728 … row sums 576, 13824, 331776, 7962624, 191102976, …
  (−∞, 0) nonnegative rational roots
• Unsigned 2-Stirling numbers of the first kind.
  6, 5, 1 · 24, 26, 9, 1 · 120, 154, 71, 14, 1 … row sums 12, 60, 360, 2520, 20160, 181440, …
  (−∞, 0) nonnegative rational roots
• 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
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
  1 · 3, 5 · 9, 30, 25 · 27, 135, 225, 125 … row sums 1, 8, 64, 512, 4096, 32768, 262144, …
  A013622 (−∞, 0) [−1, 0] nonnegative rational roots
• A052553: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.
  1 · 1 · 1, 1 · 1, 2 … row sums 1, 1, 2, 3, 5, 8, 13, 21, …
  A052553 (−∞, 0] 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
• A035342: The convolution matrix of the double factorial of odd numbers (A001147).
  1 · 3, 1 · 15, 9, 1 · 105, 87, 18, 1 … row sums 1, 4, 25, 211, 2236, 28471, 422899, …
  A035342 (−∞, 0) nonnegative
• A111577: Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.
  1 · 1, 1 · 1, 5, 1 · 1, 21, 12, 1 … row sums 1, 2, 7, 35, 214, 1523, 12349, 112052, …
  A111577 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*11^j.
  81, 198, 121 · 729, 2673, 3267, 1331 … row sums 400, 8000, 160000, 3200000, 64000000, …
  (−∞, 0) nonnegative rational roots
• 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
• A039755: Triangle of B-analogs of Stirling numbers of the second kind.
  1 · 1, 1 · 1, 4, 1 · 1, 13, 9, 1 … row sums 1, 2, 6, 24, 116, 648, 4088, 28640, …
  A039755 (−∞, 0) nonnegative
• A121574: Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
  1 · 2, 1 · 4, 5, 1 · 8, 16, 8, 1 … row sums 1, 3, 10, 33, 109, 360, 1189, 3927, …
  A121574 (−∞, 0) nonnegative
• A227450: Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.
  2 · 2, 4 · 2, 12, 8 · 2, 28, 48, 16 … row sums 2, 6, 22, 94, 454, 2430, 14214, 89918, …
  A227450 (−∞, 0) nonnegative
• A116412: Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
  1 · 3, 1 · 6, 6, 1 · 12, 21, 9, 1 … row sums 1, 4, 13, 43, 142, 469, 1549, 5116, …
  A116412 (−∞, 0) nonnegative
• A081578: Pascal-(1,3,1) array.
  1 · 1, 1 · 1, 5, 1 · 1, 9, 9, 1 … row sums 1, 2, 7, 20, 61, 182, 547, 1640, …
  A081578 (−∞, 0) nonnegative
• A019538: Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).
  1 · 1, 2 · 1, 6, 6 · 1, 14, 36, 24 … row sums 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  A019538 (−1, 0) nonnegative
• A225467: Triangle read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  1 · 3, 4 · 9, 40, 16 · 27, 316, 336, 64 … row sums 1, 7, 65, 743, 9921, 150151, 2526593, …
  A225467 (−∞, 0) nonnegative
• A035469: Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.
  1 · 4, 1 · 28, 12, 1 · 280, 160, 24, 1 … row sums 1, 5, 41, 465, 6721, 117941, 2433145, …
  A035469 (−∞, 0) nonnegative
• delannoy
  1 · 1, 1 · 1, 3, 1 · 1, 5, 5, 1 … row sums 1, 2, 5, 12, 29, 70, 169, 408, …
  [−3−2√2, −3+2√2] nonnegative
• A035607: Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).
  1 · 1, 2 · 1, 4, 2 · 1, 6, 8, 2 … row sums 1, 3, 7, 17, 41, 99, 239, 577, …
  A035607 (−∞, 0) nonnegative
• A304236: Triangle T(n,k) = T(n-1,k) + 3*T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.
  1 · 1 · 1, 3 · 1, 6 … row sums 1, 1, 4, 7, 19, 40, 97, 217, …
  A304236 (−∞, 0] nonnegative
• A038324: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*10^j.
  1 · 11, 10 · 121, 220, 100 · 1331, 3630, 3300, 1000 … row sums 1, 21, 441, 9261, 194481, 4084101, …
  A038324 (−∞, 0) nonnegative rational roots
• Riordan array ((1+x)^3,x).
  1, 3, 3, 1 · 0, 1, 3, 3, 1 · 0, 0, 1, 3, 3, 1 … row sums 8, 8, 8, 8, 8, 8, 8, 8, …
  (−∞, 0] nonnegative rational roots
• A128099: Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).
  1 · 1 · 1, 2 · 1, 4 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  A128099 (−∞, 0] nonnegative
• A249252: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 4 · 1, 8, 8 · 1, 12, 32, 16 … row sums 1, 5, 17, 61, 217, 773, 2753, 9805, …
  A249252 (−∞, 0) nonnegative
• A038259: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j)*5^j.
  1 · 6, 5 · 36, 60, 25 · 216, 540, 450, 125 … row sums 1, 11, 121, 1331, 14641, 161051, …
  A038259 (−∞, 0) nonnegative rational roots
• A249251: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 3 · 1, 6, 6 · 1, 9, 21, 12 … row sums 1, 4, 13, 43, 142, 469, 1549, 5116, …
  A249251 (−∞, 0) nonnegative
• A322944: Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.
  1 · 1, 1 · 2, 6, 1 · 6, 38, 15, 1 … row sums 1, 2, 9, 60, 513, 5286, 63417, 865824, …
  A322944 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*2^j.
  144, 48, 4 · 1728, 864, 144, 8 … row sums 196, 2744, 38416, 537824, 7529536, …
  (−∞, 0) nonnegative rational roots
• A249250: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 3 · 1, 6, 3 · 1, 9, 15, 3 … row sums 1, 4, 10, 28, 76, 208, 568, 1552, …
  A249250 (−∞, 0) nonnegative
• A113413: A Riordan array of coordination sequences.
  1 · 2, 1 · 2, 4, 1 · 2, 8, 6, 1 … row sums 1, 3, 7, 17, 41, 99, 239, 577, …
  A113413 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*10^j.
  144, 240, 100 · 1728, 4320, 3600, 1000 … row sums 484, 10648, 234256, 5153632, 113379904, …
  (−∞, 0) nonnegative rational roots
• A081579: Pascal-(1,4,1) array.
  1 · 1, 1 · 1, 6, 1 · 1, 11, 11, 1 … row sums 1, 2, 8, 24, 80, 256, 832, 2688, …
  A081579 (−∞, 0) nonnegative
• chebyshev_u
  1 · 0, 2 · -1, 0, 4 · 0, -4, 0, 8 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  (−1, 1)
• A013613: Triangle of coefficients in expansion of (1+6x)^n.
  1 · 1, 6 · 1, 12, 36 · 1, 18, 108, 216 … row sums 1, 7, 49, 343, 2401, 16807, 117649, …
  A013613 (−∞, 0) nonnegative rational roots
• A130534: Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, …
  1 · 1, 1 · 2, 3, 1 · 6, 11, 6, 1 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A130534 (−∞, −1] nonnegative rational roots
• A027466: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).
  1 · 7, 1 · 49, 14, 1 · 343, 147, 21, 1 … row sums 1, 8, 64, 512, 4096, 32768, 262144, …
  A027466 (−∞, 0) nonnegative rational roots
• A081577: Pascal-(1,2,1) array read by antidiagonals.
  1 · 1, 1 · 1, 4, 1 · 1, 7, 7, 1 … row sums 1, 2, 6, 16, 44, 120, 328, 896, …
  A081577 (−∞, 0) nonnegative
• A125662: A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
  1 · 3, 1 · 8, 6, 1 · 21, 25, 9, 1 … row sums 1, 4, 15, 56, 209, 780, 2911, 10864, …
  A125662 (−5, −1) nonnegative
• A225473: Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  1 · 3, 4 · 9, 40, 32 · 27, 316, 672, 384 … row sums 1, 7, 81, 1399, 32289, 931687, …
  A225473 (−1, 0) nonnegative
• A038335: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*9^j.
  1 · 12, 9 · 144, 216, 81 · 1728, 3888, 2916, 729 … row sums 1, 21, 441, 9261, 194481, 4084101, …
  A038335 (−∞, 0) nonnegative rational roots
• A391533: The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
  4, 4, 1 · 4, 8, 5, 1 · 4, 12, 13, 6, 1 … row sums 9, 18, 36, 72, 144, 288, 576, 1152, …
  A391533 (−∞, 0) nonnegative rational roots
• A053122: Triangle of coefficients of Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in increasing order).
  1 · -2, 1 · 3, -4, 1 · -4, 10, -6, 1 … row sums 1, -1, 0, 1, -1, 0, 1, -1, …
  A053122 (0, 4)