basis_map_theorem

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

152 sequences

• A116414: Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).
  1 · 4, 1 · 13, 8, 1 · 40, 42, 12, 1 … row sums 1, 5, 22, 95, 409, 1760, 7573, 32585, …
  A116414 (−8, 0) nonnegative
• Stirling2 triangle with scaled diagonals (powers of 10).
  100, 30, 1 · 1000, 700, 60, 1 … row sums 131, 1761, 27601, 506651, 10674211, …
  (−∞, 0) nonnegative
• A028246: Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.
  1 · 1, 1 · 1, 3, 2 · 1, 7, 12, 6 … row sums 1, 2, 6, 26, 150, 1082, 9366, 94586, …
  A028246 [−1, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
  144, 192, 64 · 1728, 3456, 2304, 512 … row sums 400, 8000, 160000, 3200000, 64000000, …
  (−∞, 0) nonnegative rational roots
• Stirling2 triangle with scaled diagonals (powers of 8).
  64, 24, 1 · 512, 448, 48, 1 · 4096, 7680, 1600, 80, 1 … row sums 89, 1009, 13457, 210105, 3747753, …
  (−∞, 0) nonnegative
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
  1 · 1, 3 · 1, 6, 9 · 1, 9, 27, 27 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A013610 (−∞, 0) [−1, 0] nonnegative rational roots
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
  1 · 2, 1 · 4, 5, 1 · 8, 19, 9, 1 … row sums 1, 3, 10, 37, 151, 674, 3263, 17007, …
  A143494 (−∞, 0) nonnegative
• A081580: Pascal-(1,5,1) array.
  1 · 1, 1 · 1, 7, 1 · 1, 13, 13, 1 … row sums 1, 2, 9, 28, 101, 342, 1189, 4088, …
  A081580 (−∞, 0) nonnegative
• A023531: a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.
  1 · 0, 1 · 0, 0, 1 · 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A023531 [−1, 0] [0, ∞) nonnegative rational roots
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
  1 · 1, 2 · 1, 4, 4 · 1, 6, 12, 8 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A013609 (−∞, 0) [−1, 0] nonnegative rational roots
• A166973: Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.
  1 · 1, 1 · 1, 7, 1 · 1, 43, 18, 1 … row sums 1, 2, 9, 63, 536, 5307, 60389, 775988, …
  A166973 (−∞, 0) nonnegative
• A011973: Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci polynomials.
  1 · 1 · 1, 1 · 1, 2 … row sums 1, 1, 2, 3, 5, 8, 13, 21, …
  A011973 (−∞, 0] nonnegative
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).
  3 · 3, 6 · 3, 12, 12 · 3, 18, 36, 24 … row sums 3, 9, 27, 81, 243, 729, 2187, 6561, …
  A120909 (−∞, 0) nonnegative rational roots
• Riordan array (1-x,x/(1-x)).
  0, 0, 1 · 0, 0, 1, 1 · 0, 0, 1, 2, 1 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, 0] nonnegative rational roots
• Stirling2 triangle with scaled diagonals (powers of 5).
  25, 15, 1 · 125, 175, 30, 1 · 625, 1875, 625, 50, 1 … row sums 41, 331, 3176, 35451, 447981, 6282416, …
  (−∞, 0) nonnegative
• A143685: Pascal-(1,9,1) array.
  1 · 1, 1 · 1, 11, 1 · 1, 21, 21, 1 … row sums 1, 2, 13, 44, 205, 806, 3457, 14168, …
  A143685 (−∞, 0) nonnegative
• 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
• A274498: Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).
  1 · 3 · 3, 6 · 9, 18 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A274498 (−∞, 0] nonnegative rational roots
• A161198: Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).
  1 · 1, 2 · 3, 8, 4 · 15, 46, 36, 8 … row sums 1, 3, 15, 105, 945, 10395, 135135, …
  A161198 (−∞, 0) nonnegative rational roots
• A225469: Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  1 · 3, 1 · 9, 10, 1 · 27, 79, 21, 1 … row sums 1, 4, 20, 128, 1008, 9280, 96704, …
  A225469 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
  121, 154, 49 · 1331, 2541, 1617, 343 … row sums 324, 5832, 104976, 1889568, 34012224, …
  (−∞, 0) nonnegative rational roots
• A038325: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.
  1 · 11, 11 · 121, 242, 121 · 1331, 3993, 3993, 1331 … row sums 1, 22, 484, 10648, 234256, 5153632, …
  A038325 (−∞, 0) nonnegative rational roots
• A008288: Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
  1 · 1, 1 · 1, 3, 1 · 1, 5, 5, 1 … row sums 1, 2, 5, 12, 29, 70, 169, 408, …
  A008288 [−3−2√2, −3+2√2] nonnegative
• A038327: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.
  1 · 12, 1 · 144, 24, 1 · 1728, 432, 36, 1 … row sums 1, 13, 169, 2197, 28561, 371293, …
  A038327 (−∞, 0) nonnegative rational roots
• 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
• A225476: Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  1 · 1, 1 · 1, 4, 2 · 1, 13, 18, 6 … row sums 1, 2, 7, 38, 277, 2522, 27547, 351038, …
  A225476 (−2, 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
• 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
• A078812: Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
  1 · 2, 1 · 3, 4, 1 · 4, 10, 6, 1 … row sums 1, 3, 8, 21, 55, 144, 377, 987, …
  A078812 (−4, 0) nonnegative
• chebyshev_t
  1 · 0, 1 · -1, 0, 2 · 0, -3, 0, 4 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  (−1, 1)
• A110441: Triangular array formed by the Mersenne numbers.
  1 · 3, 1 · 7, 6, 1 · 15, 23, 9, 1 … row sums 1, 4, 14, 48, 164, 560, 1912, 6528, …
  A110441 (−6, 0) nonnegative
• A225471: Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
  1 · 3, 1 · 21, 10, 1 · 231, 131, 21, 1 … row sums 1, 4, 32, 384, 6144, 122880, 2949120, …
  A225471 (−∞, 0) nonnegative rational roots
• A038326: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.
  1 · 11, 12 · 121, 264, 144 · 1331, 4356, 4752, 1728 … row sums 1, 23, 529, 12167, 279841, 6436343, …
  A038326 (−∞, 0) nonnegative rational roots
• A266213: Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*bi…
  1 · 1 · 1, 2 · 1, 4, 2 … row sums 1, 1, 3, 7, 17, 41, 99, 239, …
  A266213 (−∞, 0) nonnegative
• A143495: Triangle read by rows: 3-Stirling numbers of the second kind.
  1 · 3, 1 · 9, 7, 1 · 27, 37, 12, 1 … row sums 1, 4, 17, 77, 372, 1915, 10481, 60814, …
  A143495 (−∞, 0) nonnegative
• A053117: Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).
  1 · 0, 2 · -1, 0, 4 · 0, -4, 0, 8 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  A053117 (−1, 1)
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
  144, 120, 25 · 1728, 2160, 900, 125 … row sums 289, 4913, 83521, 1419857, 24137569, …
  (−∞, 0) nonnegative rational roots
• A147716: Triangle of coefficients in expansion of (14 + x)^n.
  1 · 14, 1 · 196, 28, 1 · 2744, 588, 42, 1 … row sums 1, 15, 225, 3375, 50625, 759375, …
  A147716 (−∞, 0) nonnegative rational roots
• A038271: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.
  1 · 7, 5 · 49, 70, 25 · 343, 735, 525, 125 … row sums 1, 12, 144, 1728, 20736, 248832, …
  A038271 (−∞, 0) nonnegative rational roots
• A075497: Stirling2 triangle with scaled diagonals (powers of 2).
  1 · 2, 1 · 4, 6, 1 · 8, 28, 12, 1 … row sums 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  A075497 (−∞, 0) nonnegative
• 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
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
  144, 96, 16 · 1728, 1728, 576, 64 … row sums 256, 4096, 65536, 1048576, 16777216, …
  (−∞, 0) nonnegative rational roots
• Stirling2 triangle with scaled diagonals (powers of 6).
  36, 18, 1 · 216, 252, 36, 1 · 1296, 3240, 900, 60, 1 … row sums 55, 505, 5497, 69823, 1007407, …
  (−∞, 0) nonnegative
• A130123: Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).
  1 · 0, 2 · 0, 0, 4 · 0, 0, 0, 8 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A130123 [−1, 0] nonnegative rational roots
• 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
• A122542: Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
  1 · 0, 1 · 0, 2, 1 · 0, 2, 4, 1 … row sums 1, 1, 3, 7, 17, 41, 99, 239, …
  A122542 (−∞, 0] nonnegative
• Stirling2 triangle with scaled diagonals (powers of 4).
  16, 12, 1 · 64, 112, 24, 1 · 256, 960, 400, 40, 1 … row sums 29, 201, 1657, 15821, 170389, 2032785, …
  (−∞, 0) nonnegative
• Triangle of coefficients in expansion of (1+11x)^n.
  1, 22, 121 · 1, 33, 363, 1331 … row sums 144, 1728, 20736, 248832, 2985984, …
  (−∞, 0) nonnegative rational roots
• A274496: Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).
  1 · 2 · 2, 2 · 4, 4 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A274496 (−∞, 0] nonnegative rational roots