parent_provers / root_domain

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

32 sequences

• A204420: Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.
  1 · 0, 1 · 0, 6, 3 · 0, 120, 90, 15 … row sums 1, 1, 9, 225, 11025, 893025, 108056025, …
  A204420 (−∞, 0] nonnegative rational roots
• basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3. shifted by x -> x + 1
  1 · 0, 1 · 0, 3, 1 · 0, 18, 9, 1 … row sums 1, 1, 4, 28, 280, 3640, 58240, 1106560, …
  (−∞, 0] nonnegative rational roots
• Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+8*x+x^2)/(1-x…
  1, 8, 1 · 1, 19, 19, 1 · 1, 42, 114, 42, 1 … row sums 10, 40, 200, 1200, 8400, 67200, 604800, …
  (−∞, 0) nonnegative
• 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
• A090582: T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n - k + 1, j)*(n - k + 1 - j)^n. Triangle read by rows, T(n, k) for 1 <= k <= n.
  1 · 2, 1 · 6, 6, 1 · 24, 36, 14, 1 … row sums 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  A090582 (−∞, 0] nonnegative
• 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
• A257609: Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 2.
  1 · 2, 2 · 4, 16, 4 · 8, 88, 88, 8 … row sums 1, 4, 24, 192, 1920, 23040, 322560, …
  A257609 (−∞, 0) nonnegative
• 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
• Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+12*x+x^2)/(1-…
  1, 12, 1 · 1, 27, 27, 1 · 1, 58, 162, 58, 1 … row sums 14, 56, 280, 1680, 11760, 94080, …
  (−∞, 0) nonnegative
• A269944: Triangle read by rows, Stirling cycle numbers of order 2, 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)^2*T(n-1, k), for 0 <= k <= n.
  1 · 0, 1 · 0, 1, 1 · 0, 4, 5, 1 … row sums 1, 1, 2, 10, 100, 1700, 44200, 1635400, …
  A269944 (−∞, 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 T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+10*x+x^2)/(1-…
  1, 10, 1 · 1, 23, 23, 1 · 1, 50, 138, 50, 1 … row sums 12, 48, 240, 1440, 10080, 80640, …
  (−∞, 0) nonnegative
• A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
  1 · 2, 2 · 6, 12, 6 · 24, 72, 72, 24 … row sums 1, 4, 24, 192, 1920, 23040, 322560, …
  A156992 (−∞, 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
• A059364: Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.
  1 · 2, 1 · 6, 7, 2 · 24, 46, 29, 6 … row sums 1, 3, 15, 105, 945, 10395, 135135, …
  A059364 (−∞, −1] 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
• A088996: Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.
  1 · 0, 1 · 0, 1, 2 · 0, 2, 7, 6 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  A088996 [−1, 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
• Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x…
  1, 6, 1 · 1, 15, 15, 1 · 1, 34, 90, 34, 1 … row sums 8, 32, 160, 960, 6720, 53760, 483840, …
  (−∞, 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
• 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
• 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
• 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
• 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 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
• 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
• 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
• 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
• 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