theorem_check / three_term_recurrence

Theorem check of the node's own rule. All operations

62 sequences

• Triangle, T(n, k) = (1/2)*(n+2)! * [x^k]( p(x, n) ), where p(x,0) = 1, p(x,1) = -x, P(x, n) = (1/(n+1))*( (2*n-x)*P(x, n-1) - n*P(x, n-2) ), read by rows.
  -8, -16, 4 · -60, -65, 50, -5 … row sums -20, -80, -192, 784, 19072, 229536, …
  (−∞, ∞)
• A123202: Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial.
  1 · 1, -2 · 2, -8, 7 · 6, -36, 63, -34 … row sums 1, -1, 1, -1, 1, -1, 1, -1, …
  A123202 (0, 1)
• A039991: Triangle of coefficients of cos(x)^n in polynomial for cos(nx).
  1 · 1 · 2, 0, -1 · 4, 0, -3 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  A039991 (−∞, ∞)
• A207606: Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.
  4, 7, 2 · 5, 16, 11, 2 · 6, 30, 36, 15, 2 … row sums 13, 34, 89, 233, 610, 1597, 4181, …
  A207606 (−4, 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)
• 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
• A358735: Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.
  1 · 1, 1 · 1, 4, 2 · 1, 10, 16, 6 … row sums 1, 2, 7, 33, 191, 1304, 10241, 90865, …
  A358735 (−2, 0) nonnegative
• A086645: Triangle read by rows: T(n, k) = binomial(2n, 2k).
  1 · 1, 1 · 1, 6, 1 · 1, 15, 15, 1 … row sums 1, 2, 8, 32, 128, 512, 2048, 8192, …
  A086645 (−∞, 0) nonnegative
• 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)
• A060920: Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).
  1 · 2, 1 · 5, 5, 1 · 13, 20, 9, 1 … row sums 1, 3, 11, 43, 171, 683, 2731, 10923, …
  A060920 (−∞, −1] nonnegative
• A069466: Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.
  1 · 2, 2 · 6, 24, 6 · 20, 180, 180, 20 … row sums 1, 4, 36, 400, 4900, 63504, 853776, …
  A069466 (−∞, 0) nonnegative
• A327997: Triangle read by rows: coefficients of the polynomials given by KummerU(-n, 1 - n - x, 3).
  1 · 3, 1 · 9, 7, 1 · 27, 38, 12, 1 … row sums 1, 4, 17, 78, 393, 2208, 13977, 100026, …
  A327997 (−∞, 0) nonnegative