theorem_check

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

304 sequences

• 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
• Triangle of coefficients in expansion of (1+11x)^n.
  1, 22, 121 · 1, 33, 363, 1331 … row sums 144, 1728, 20736, 248832, 2985984, …
  (−∞, 0) nonnegative rational roots
• A271697: Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j-1,-n-1)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.
  1 · 0, 1 · 0, 1, 1 · 0, 1, 7, 1 … row sums 1, 0, 1, 2, 9, 44, 265, 1854, …
  A271697 (−∞, 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
• Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.
  9, 24, 9 · 27, 141, 141, 27 · 81, 726, 1410, 726, 81 … row sums 42, 336, 3024, 30240, 332640, 3991680, …
  (−∞, 0) nonnegative
• A038277: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*11^j.
  1 · 7, 11 · 49, 154, 121 · 343, 1617, 2541, 1331 … row sums 1, 18, 324, 5832, 104976, 1889568, …
  A038277 (−∞, 0) nonnegative rational roots
• shifted_legendre
  -1, 2 · 1, -6, 6 · -1, 12, -30, 20 … row sums 1, 1, 1, 1, 1, 1, 1, 1, …
  (0, 1)
• A113953: A Jacobsthal triangle.
  1 · 0, 1 · 0, 2, 1 · 0, 0, 4, 1 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  A113953 (−8, 0] nonnegative
• 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
• A193685: 5-Stirling numbers of the second kind.
  1 · 5, 1 · 25, 11, 1 · 125, 91, 18, 1 … row sums 1, 6, 37, 235, 1540, 10427, 73013, …
  A193685 (−∞, 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
• Triangle of coefficients in expansion of (1+5x)^n.
  1, 10, 25 · 1, 15, 75, 125 · 1, 20, 150, 500, 625 … row sums 36, 216, 1296, 7776, 46656, 279936, …
  (−∞, 0) nonnegative rational roots
• A225466: Triangle read by rows, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  1 · 2, 3 · 4, 21, 9 · 8, 117, 135, 27 … row sums 1, 5, 34, 287, 2839, 31898, 399079, …
  A225466 (−∞, 0) nonnegative
• A257616: 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) = 6*x + 2.
  1 · 2, 2 · 4, 32, 4 · 8, 312, 312, 8 … row sums 1, 4, 40, 640, 14080, 394240, 13404160, …
  A257616 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.
  64, 96, 36 · 512, 1152, 864, 216 … row sums 196, 2744, 38416, 537824, 7529536, …
  (−∞, 0) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.
  144, 288, 144 · 1728, 5184, 5184, 1728 … row sums 576, 13824, 331776, 7962624, 191102976, …
  (−∞, 0) nonnegative rational roots
• 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 (−∞, ∞)
• 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
• A113214: Riordan array (1+2x,x(1+x)).
  1 · 2, 1 · 0, 3, 1 · 0, 2, 4, 1 … row sums 1, 3, 4, 7, 11, 18, 29, 47, …
  A113214 (−4, 0] nonnegative
• 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
• A257613: 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 + 4.
  1 · 4, 4 · 16, 48, 16 · 64, 416, 416, 64 … row sums 1, 8, 80, 960, 13440, 215040, 3870720, …
  A257613 (−∞, 0) nonnegative
• A013622: Triangle of coefficients in expansion of (3+5x)^n.
  1 · 3, 5 · 9, 30, 25 · 27, 135, 225, 125 … row sums 1, 8, 64, 512, 4096, 32768, 262144, …
  A013622 (−∞, 0) [−1, 0] nonnegative rational roots
• A257606: 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) = x + 4.
  1 · 4, 4 · 16, 40, 16 · 64, 296, 296, 64 … row sums 1, 8, 72, 720, 7920, 95040, 1235520, …
  A257606 (−∞, 0) nonnegative
• A063967: Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.
  1 · 1, 1 · 2, 3, 1 · 3, 7, 5, 1 … row sums 1, 2, 6, 16, 44, 120, 328, 896, …
  A063967 (−5, −1] nonnegative
• A257610: 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) = 3*x + 2.
  1 · 2, 2 · 4, 20, 4 · 8, 132, 132, 8 … row sums 1, 4, 28, 280, 3640, 58240, 1106560, …
  A257610 (−∞, 0) nonnegative
• 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
• 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
• 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
• 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
• A035342: The convolution matrix of the double factorial of odd numbers (A001147).
  1 · 3, 1 · 15, 9, 1 · 105, 87, 18, 1 … row sums 1, 4, 25, 211, 2236, 28471, 422899, …
  A035342 (−∞, 0) nonnegative
• A111577: Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.
  1 · 1, 1 · 1, 5, 1 · 1, 21, 12, 1 … row sums 1, 2, 7, 35, 214, 1523, 12349, 112052, …
  A111577 (−∞, 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
• 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, …
  (−∞, ∞)
• 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
• A142459: Triangle read by rows: T(n,k) = (4n-4k+1) * T(n-1,k-1) + (4k-3) * T(n-1,k).
  1 · 1, 1 · 1, 10, 1 · 1, 59, 59, 1 … row sums 1, 2, 12, 120, 1680, 30240, 665280, …
  A142459 (−∞, 0) nonnegative
• A290595: Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A286718 (|S1hat[3,1]| generalized Stir…
  1 · 1, 2 · 4, 19, 4 · 28, 222, 147, 8 … row sums 1, 3, 27, 405, 8505, 229635, 7577955, …
  A290595 (−∞, 0) nonnegative
• 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
• A144696: Triangle of 2-Eulerian numbers.
  1 · 1, 2 · 1, 7, 4 · 1, 18, 33, 8 … row sums 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  A144696 (−∞, 0) nonnegative
• A008517: Second-order Eulerian triangle T(n,k), 1 <= k <= n.
  1 · 1, 2 · 1, 8, 6 · 1, 22, 58, 24 … row sums 1, 3, 15, 105, 945, 10395, 135135, …
  A008517 (−∞, 0) nonnegative
• A089231: Triangular array A066667 or A008297 unsigned and transposed.
  1 · 1, 2 · 1, 6, 6 · 1, 12, 36, 24 … row sums 1, 3, 13, 73, 501, 4051, 37633, 394353, …
  A089231 (−∞, 0) nonnegative
• A039755: Triangle of B-analogs of Stirling numbers of the second kind.
  1 · 1, 1 · 1, 4, 1 · 1, 13, 9, 1 … row sums 1, 2, 6, 24, 116, 648, 4088, 28640, …
  A039755 (−∞, 0) nonnegative
• A121574: Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
  1 · 2, 1 · 4, 5, 1 · 8, 16, 8, 1 … row sums 1, 3, 10, 33, 109, 360, 1189, 3927, …
  A121574 (−∞, 0) nonnegative
• A227450: Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.
  2 · 2, 4 · 2, 12, 8 · 2, 28, 48, 16 … row sums 2, 6, 22, 94, 454, 2430, 14214, 89918, …
  A227450 (−∞, 0) nonnegative
• A116412: Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
  1 · 3, 1 · 6, 6, 1 · 12, 21, 9, 1 … row sums 1, 4, 13, 43, 142, 469, 1549, 5116, …
  A116412 (−∞, 0) nonnegative
• type_b_eulerian
  1 · 1, 1 · 1, 6, 1 · 1, 23, 23, 1 … row sums 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  (−∞, 0) nonnegative
• A081578: Pascal-(1,3,1) array.
  1 · 1, 1 · 1, 5, 1 · 1, 9, 9, 1 … row sums 1, 2, 7, 20, 61, 182, 547, 1640, …
  A081578 (−∞, 0) nonnegative
• 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