parent_provers / paper_theorem

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

18 sequences

• A102547: Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318).
  1 · 1 · 1 · 1, 1 … row sums 1, 1, 1, 2, 3, 4, 6, 9, …
  A102547 (−∞, 0] nonnegative
• A109692: Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).
  1 · 1, 1 · 1, 4, 3 · 1, 9, 23, 15 … row sums 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  A109692 (−∞, 0) nonnegative rational roots
• A145324: Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....
  1 · 1, 2 · 1, 5, 6 · 1, 9, 26, 24 … row sums 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  A145324 (−∞, 0) [−1, 0] nonnegative rational roots
• A094638: Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first …
  1 · 1, 1 · 1, 3, 2 · 1, 6, 11, 6 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A094638 (−∞, 0) nonnegative rational roots
• A169803: Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).
  1 · 1, 1 · 1, 2 · 1, 3, 1 … row sums 1, 2, 3, 5, 8, 13, 21, 34, …
  A169803 (−∞, 0] nonnegative
• A118931: Triangle, read by rows, where T(n,k) = n!/(k!*(n-3*k)!*3^k) for n>=3*k>=0.
  1 · 1 · 1 · 1, 2 … row sums 1, 1, 1, 3, 9, 21, 81, 351, …
  A118931 (−∞, 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
• A100861: Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
  1 · 1 · 1, 1 · 1, 3 … row sums 1, 1, 2, 4, 10, 26, 76, 232, …
  A100861 A144299 (−∞, 0] nonnegative
• A318772: Triangle read by rows: T(0,0) = 1; T(n,k) = T(n-1,k) + 3 * T(n-4,k-1) for k = 0..floor(n/4); T(n,k)=0 for n or k < 0.
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 4, 7, 10, 13, …
  A318772 (−∞, 0] nonnegative
• A317496: Triangle T(n,k) = T(n-1,k) + 3*T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.
  1 · 1 · 1 · 1, 3 … row sums 1, 1, 1, 4, 7, 10, 22, 43, …
  A317496 (−∞, 0] nonnegative
• A118394: Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.
  1 · 1 · 1 · 1, 6 … row sums 1, 1, 1, 7, 25, 61, 481, 2731, …
  A118394 (−∞, 0] nonnegative
• 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
• 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
• 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
• 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
• 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
• 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
• 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