parent_provers

Shared parent provers on the node's own rule. All operations

135 sequences

• A085881: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by A001147(n).
  1 · 1, 1 · 3, 6, 3 · 15, 45, 45, 15 … row sums 1, 2, 12, 120, 1680, 30240, 665280, …
  A085881 (−∞, 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
• A059344: Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.
  1 · 1 · 1, 2 · 1, 6 … row sums 1, 1, 3, 7, 25, 81, 331, 1303, …
  A059344 (−∞, 0] 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
• basis-map coefficients of Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle. shifted by x -> -x + 0
  1 · 0, 0, 1 · 0, 0, 0, 0, 1 · 0, 0, 0, 0, 0, 0, 1 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  [−1, 0] [0, ∞) nonnegative rational roots
• basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).
  1 · 0, -1 · 0, 0, 2 · 0, 0, 0, -6 … row sums 1, -1, 2, -6, 24, -120, 720, -5040, …
  [−1, 0] rational roots
• 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
• A272867: Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
  1 · 1, 4, 1 · 1, 8, 18, 8, 1 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …
  A272867 (−∞, 0] nonnegative
• 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
• 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
• A257620: Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 3*n + 3.
  1 · 3, 3 · 9, 36, 9 · 27, 297, 297, 27 … row sums 1, 6, 54, 648, 9720, 174960, 3674160, …
  A257620 (−∞, 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
• 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
• A008955: Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.
  1 · 1, 1 · 1, 5, 4 · 1, 14, 49, 36 … row sums 1, 2, 10, 100, 1700, 44200, 1635400, …
  A008955 (−∞, 0) nonnegative rational roots
• A133080: Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
  1 · 1, 1 · 0, 0, 1 · 0, 0, 1, 1 … row sums 1, 2, 1, 2, 1, 2, 1, 2, …
  A133080 (−∞, 0] nonnegative rational roots
• 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
• 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
• A128813: Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/2).
  1 · 1, 1 · 1, 4, 3 · 1, 10, 27, 18 … row sums 1, 2, 8, 56, 616, 9856, 216832, …
  A128813 (−∞, 0) nonnegative rational roots
• A200536: Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.
  1 · 1, 3, 2 · 1, 6, 13, 12, 4 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …
  A200536 (−∞, 0] nonnegative rational roots
• 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
• A173018: Euler's triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
  1 · 1 · 1, 1 · 1, 4, 1 … row sums 1, 1, 2, 6, 24, 120, 720, 5040, …
  A173018 (−∞, 0) nonnegative
• 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
• 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
• A118933: Triangle, read by rows, where T(n,k) = n!/(k!*(n-4*k)!*4^k) for n>=4*k>=0.
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 7, 31, 91, 211, …
  A118933 (−∞, 0] nonnegative
• A136572: Triangle read by rows: row n consists of n zeros followed by n!.
  1 · 0, 1 · 0, 0, 2 · 0, 0, 0, 6 … row sums 1, 1, 2, 6, 24, 120, 720, 5040, …
  A136572 [−1, 0] nonnegative rational roots
• A269947: Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+(n-1)^3*T(n-1,k), for n>=0 and 0<=k<=n.
  1 · 0, 1 · 0, 1, 1 · 0, 8, 9, 1 … row sums 1, 1, 2, 18, 504, 32760, 4127760, …
  A269947 (−∞, 0] nonnegative rational roots
• A027467: Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).
  1 · 15, 1 · 225, 30, 1 · 3375, 675, 45, 1 … row sums 1, 16, 256, 4096, 65536, 1048576, …
  A027467 (−∞, 0) nonnegative rational roots
• A008292: Eulerian
  1 · 1, 1 · 1, 4, 1 · 1, 11, 11, 1 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A008292 (−∞, 0) nonnegative