theorem_check / three_term_recurrence

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

62 sequences

• basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1).
  1 · 0, 2 · -2, 2, 3 · 0, -8, 8, 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 1]
• A165253: Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
  1 · 1 · 1, 1 · 1, 3, 1 … row sums 1, 1, 2, 5, 13, 34, 89, 233, …
  A165253 (−4, 0) nonnegative
• A267482: Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=n.
  1 · 1, 1 · -1, 1, 1 · -1, -2, 1, 1 … row sums 1, 2, 1, -1, -2, -1, 1, 2, …
  A267482 (−2, 2)
• A053123: Triangle of coefficients of shifted Chebyshev's S(n,x-2) = U(n,x/2-1) polynomials (exponents of x in decreasing order).
  1 · 1, -2 · 1, -4, 3 · 1, -6, 10, -4 … row sums 1, -1, 0, 1, -1, 0, 1, -1, …
  A053123 (−∞, ∞)
• A114656: Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <=…
  1 · 2, 1 · 4, 6, 1 · 8, 24, 12, 1 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A114656 (−∞, 0) nonnegative
• A085478: Triangle read by rows: T(n, k) = binomial(n + k, 2*k).
  1 · 1, 1 · 1, 3, 1 · 1, 6, 5, 1 … row sums 1, 2, 5, 13, 34, 89, 233, 610, …
  A085478 (−4, 0) nonnegative
• Triangle read by rows: coefficients of a Bessel polynomial recursion: P(x, n) = 2*(n-1)*P(x, n - 1)/x - n*P(x, n - 2) with substitution x -> 1/y.
  -2, 0, 2 · 0, -11, 0, 8 · 8, 0, -74, 0, 48 … row sums 0, -3, -18, -129, -1182, -13281, …
  (−2, 2)
• A091044: One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
  1 · 2, 2 · 3, 10, 3 · 4, 28, 28, 4 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A091044 (−∞, 0) nonnegative
• narayana
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … row sums 1, 2, 5, 14, 42, 132, 429, 1430, …
  (−∞, 0) nonnegative
• A207823: Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).
  1 · 4, 1 · 15, 8, 1 · 56, 46, 12, 1 … row sums 1, 5, 24, 115, 551, 2640, 12649, 60605, …
  A207823 (−6, −2) nonnegative
• A111125: Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
  1 · 3, 1 · 5, 5, 1 · 7, 14, 7, 1 … row sums 1, 4, 11, 29, 76, 199, 521, 1364, …
  A111125 (−4, 0) nonnegative
• A094816: Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.
  1 · 1, 1 · 1, 3, 1 · 1, 8, 6, 1 … row sums 1, 2, 5, 16, 65, 326, 1957, 13700, …
  A094816 (−∞, 0) nonnegative
• A207607: Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.
  1 · 1, 2 · 1, 5, 2 · 1, 9, 9, 2 … row sums 1, 3, 8, 21, 55, 144, 377, 987, …
  A207607 (−4, 0) nonnegative
• A053124: Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in increasing order).
  1 · -2, 4 · 3, -16, 16 · -4, 40, -96, 64 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  A053124 (0, 1)
• basis-map coefficients of Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
  1 · 0, 2 · -3, 3, 3 · 0, -12, 12, 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 1]
• A063007: T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
  1 · 1, 2 · 1, 6, 6 · 1, 12, 30, 20 … row sums 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  A063007 (−1, 0) nonnegative
• A179900: Triangle T(n,k) read by rows: coefficient of [x^k] of the polynomial p_n(x)=(5-x)*p_{n-1}(x)-p_{n-2}(x), p_0=1, p_1=5-x.
  1 · 5, -1 · 24, -10, 1 · 115, -73, 15, -1 … row sums 1, 4, 15, 56, 209, 780, 2911, 10864, …
  A179900 (3, 7)
• A137338: Triangle read by rows: T(n,k), 0 <= k <= n, gives the coefficients of the Charlier polynomials (with parameter a=1), ordered by rising powers.
  1 · -1, 1 · 0, -3, 1 · 3, 6, -6, 1 … row sums 1, 0, -2, 4, -4, -4, 44, -236, …
  A137338 (−∞, ∞)
• A124038: Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.
  1 · -2, 1 · -1, -2, 1 · 2, -2, -2, 1 … row sums 1, -1, -2, -1, 1, 2, 1, -1, …
  A124038 (−2, 3)
• Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.
  -2, 0, 1 · 0, -8, 0, 1 · 24, 0, -20, 0, 1 … row sums -1, -7, 5, 145, -5, -6095, -5815, …
  (−∞, ∞)
• A126216: Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A126216 (−∞, −1] nonnegative
• 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
• A033282: Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
  1 · 1, 2 · 1, 5, 5 · 1, 9, 21, 14 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A033282 (−1, 0) nonnegative
• A054142: Triangular array binomial(2*n-k, k), k=0..n, n >= 0.
  1 · 1, 1 · 1, 3, 1 · 1, 5, 6, 1 … row sums 1, 2, 5, 13, 34, 89, 233, 610, …
  A054142 (−∞, 0) nonnegative
• A079638: Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.
  1 · 3, 1 · 14, 9, 1 · 90, 83, 18, 1 … row sums 1, 4, 24, 192, 1920, 23040, 322560, …
  A079638 (−∞, 0) nonnegative
• A104684: Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.
  1 · 2, 1 · 6, 6, 1 · 20, 30, 12, 1 … row sums 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  A104684 (−∞, 0) nonnegative
• A172431: Even row Pascal-square read by antidiagonals.
  1 · 1, 2 · 1, 4, 3 · 1, 6, 10, 4 … row sums 1, 3, 8, 21, 55, 144, 377, 987, …
  A172431 (−∞, 0) nonnegative
• A114687: Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 …
  1 · 1, 2 · 1, 6, 4 · 1, 12, 24, 8 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A114687 (−∞, 0) nonnegative
• Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).
  1, 6, 4 · 1, 12, 20, 8 · 1, 20, 60, 56, 16 … row sums 11, 41, 153, 571, 2131, 7953, 29681, …
  (−2, 0) nonnegative
• A082985: Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
  1 · 1, 3 · 1, 5, 5 · 1, 7, 14, 7 … row sums 1, 4, 11, 29, 76, 199, 521, 1364, …
  A082985 (−∞, 0) nonnegative
• 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)
• A085841: Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ).
  1 · 3, 4 · 5, 40, 16 · 7, 140, 336, 64 … row sums 1, 7, 61, 547, 4921, 44287, 398581, …
  A085841 (−∞, 0) nonnegative
• A122076: Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.
  2 · 3, 2 · 7, 8, 2 · 18, 30, 15, 2 … row sums 2, 5, 17, 65, 257, 1025, 4097, 16385, …
  A122076 (−∞, ∞) nonnegative
• 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
• 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)
• A066325: Hermite he
  1 · 0, 1 · -1, 0, 1 · 0, -3, 0, 1 … row sums 1, 1, 0, -2, -2, 6, 16, -20, …
  A066325 (−∞, ∞)
• A008459: Square the entries of Pascal's triangle.
  1 · 1, 1 · 1, 4, 1 · 1, 9, 9, 1 … row sums 1, 2, 6, 20, 70, 252, 924, 3432, …
  A008459 (−∞, 0) nonnegative
• A367270: Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).
  1 · 1 · 1, 2 · 1, 6, 3 … row sums 1, 1, 3, 10, 35, 126, 462, 1716, …
  A367270 (−∞, 0) nonnegative
• Delannoy paths counted by number of weak peaks.
  1, 8, 4 · 1, 18, 36, 8 · 1, 32, 144, 128, 16 … row sums 13, 63, 321, 1683, 8989, 48639, 265729, …
  (−∞, 0) nonnegative
• shifted_legendre
  -1, 2 · 1, -6, 6 · -1, 12, -30, 20 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  (0, 1)
• A091042: Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.
  1 · 1, 3 · 1, 10, 5 · 1, 21, 35, 7 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A091042 (−∞, 0) nonnegative
• A145597: Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.
  1 · 3, 3 · 6, 15, 6 · 10, 45, 45, 10 … row sums 1, 6, 27, 110, 429, 1638, 6188, 23256, …
  A145597 (−∞, 0) nonnegative
• A108299: Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].
  1 · 1, -1 · 1, -1, -1 · 1, -1, -2, 1 … row sums 1, 0, -1, -1, 0, 1, 1, 0, …
  A108299 (−∞, ∞)
• A207824: Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).
  1 · 5, 1 · 24, 10, 1 · 115, 73, 15, 1 … row sums 1, 6, 35, 204, 1189, 6930, 40391, …
  A207824 (−7, −3) nonnegative
• A001263: Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … row sums 1, 2, 5, 14, 42, 132, 429, 1430, …
  A001263 (−∞, 0) nonnegative
• A103327: Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).
  1 · 3, 1 · 5, 10, 1 · 7, 35, 21, 1 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A103327 (−∞, 0) nonnegative
• A232535: Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(n,k) = (binomial(2*n,2*k) + binomial(2*n+1,2*k))/2.
  1 · 1, 2 · 1, 8, 3 · 1, 18, 25, 4 … row sums 1, 3, 12, 48, 192, 768, 3072, 12288, …
  A232535 (−∞, 0) nonnegative
• A108838: Triangle of Dyck paths counted by number of long interior inclines.
  2 · 3, 2 · 4, 8, 2 · 5, 20, 15, 2 … row sums 2, 5, 14, 42, 132, 429, 1430, 4862, …
  A108838 (−∞, 0) nonnegative
• A046716: Coefficients of a special case of Poisson-Charlier polynomials.
  1 · 1, 1 · 1, 3, 1 · 1, 6, 8, 1 … row sums 1, 2, 5, 16, 65, 326, 1957, 13700, …
  A046716 (−∞, 0) nonnegative