theorem_check

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

304 sequences

• A257607: 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 + 5.
  1 · 5, 5 · 25, 60, 25 · 125, 535, 535, 125 … row sums 1, 10, 110, 1320, 17160, 240240, …
  A257607 (−∞, 0) nonnegative
• A154227: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.
  1 · 1, 1 · 1, 8, 1 · 1, 19, 19, 1 … row sums 1, 2, 10, 40, 230, 1300, 9040, 64880, …
  A154227 (−∞, 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
• A225476: Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  1 · 1, 1 · 1, 4, 2 · 1, 13, 18, 6 … row sums 1, 2, 7, 38, 277, 2522, 27547, 351038, …
  A225476 (−2, 0) nonnegative
• A027465: Cube of lower triangular normalized binomial matrix.
  1 · 3, 1 · 9, 6, 1 · 27, 27, 9, 1 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A027465 (−∞, −1] nonnegative rational roots
• A038221: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*3^j.
  1 · 3, 3 · 9, 18, 9 · 27, 81, 81, 27 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …
  A038221 (−∞, 0) nonnegative rational roots
• 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
• A167884: Triangle read by rows: T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 8.
  1 · 1, 1 · 1, 18, 1 · 1, 179, 179, 1 … row sums 1, 2, 20, 360, 9360, 318240, 13366080, …
  A167884 (−∞, 0) nonnegative
• 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
• A144436: Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 1, and j = 4, read by rows.
  1 · 1, 1 · 1, 8, 1 · 1, 23, 23, 1 … row sums 1, 2, 10, 48, 280, 1872, 14224, 121280, …
  A144436 (−∞, 0) nonnegative
• A375853: Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.
  2 · 8, 8 · 20, 56, 20 · 40, 216, 216, 40 … row sums 2, 16, 96, 512, 2560, 12288, 57344, …
  A375853 (−∞, 0) nonnegative
• A257621: 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) = 4*n + 3.
  1 · 3, 3 · 9, 42, 9 · 27, 393, 393, 27 … row sums 1, 6, 60, 840, 15120, 332640, 8648640, …
  A257621 (−∞, 0) nonnegative
• A290315: 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 A154537 (S2[2,1] generalized Stirling2…
  1 · 1, 2 · 1, 16, 12 · 1, 66, 284, 120 … row sums 1, 3, 29, 471, 10729, 314427, 11265701, …
  A290315 (−∞, 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
• A062264: Coefficient triangle of certain polynomials N(4; m,x).
  1 · 1, 5 · 1, 12, 15 · 1, 21, 63, 35 … row sums 1, 6, 28, 120, 495, 2002, 8008, 31824, …
  A062264 (−∞, 0) nonnegative
• A178822: Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
  1 · 6, 6 · 21, 42, 21 · 56, 168, 168, 56 … row sums 1, 12, 84, 448, 2016, 8064, 29568, …
  A178822 (−∞, 0) nonnegative rational roots
• 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
• A132812: Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).
  1 · 2, 2 · 3, 9, 3 · 4, 24, 24, 4 … row sums 1, 4, 15, 56, 210, 792, 3003, 11440, …
  A132812 (−∞, 0) nonnegative
• 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
• 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
• A112007: Coefficient triangle for polynomials used for o.g.f.s for unsigned Stirling1 diagonals.
  1 · 2, 1 · 6, 8, 1 · 24, 58, 22, 1 … row sums 1, 3, 15, 105, 945, 10395, 135135, …
  A112007 (−∞, 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
• Triangle 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 + 2.
  4, 12, 4 · 8, 52, 52, 8 · 16, 196, 416, 196, 16 … row sums 20, 120, 840, 6720, 60480, 604800, …
  (−∞, 0) nonnegative
• A038326: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*12^j.
  1 · 11, 12 · 121, 264, 144 · 1331, 4356, 4752, 1728 … row sums 1, 23, 529, 12167, 279841, 6436343, …
  A038326 (−∞, 0) nonnegative rational roots
• A144443: Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 20, 1 · 1, 159, 159, 1 … row sums 1, 2, 22, 320, 6532, 171752, 5535256, …
  A144443 (−∞, 0) nonnegative
• A266213: Square array A(n,r), the number of neighbors at a sharp Manhattan distance r in a finite n-hypercube lattice, read by upwards antidiagonals; A(n,r) = Sum_{k=0..min(n,r)} binomial(r-1,k-1)*bi…
  1 · 1 · 1, 2 · 1, 4, 2 … row sums 1, 1, 3, 7, 17, 41, 99, 239, …
  A266213 (−∞, 0) nonnegative
• A062145: Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.
  1 · 1, 4 · 1, 10, 10 · 1, 18, 45, 20 … row sums 1, 5, 21, 84, 330, 1287, 5005, 19448, …
  A062145 (−∞, 0) nonnegative
• A143495: Triangle read by rows: 3-Stirling numbers of the second kind.
  1 · 3, 1 · 9, 7, 1 · 27, 37, 12, 1 … row sums 1, 4, 17, 77, 372, 1915, 10481, 60814, …
  A143495 (−∞, 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)
• A154228: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)*(2*n+3)/6)*T(n-2, k-1), read by rows.
  1 · 1, 1 · 1, 16, 1 · 1, 47, 47, 1 … row sums 1, 2, 18, 96, 1182, 11100, 187680, …
  A154228 (−∞, 0) nonnegative
• A112486: Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.
  1 · 1, 1 · 2, 5, 3 · 6, 26, 35, 15 … row sums 1, 2, 10, 82, 938, 13778, 247210, …
  A112486 [−1, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.
  144, 120, 25 · 1728, 2160, 900, 125 … row sums 289, 4913, 83521, 1419857, 24137569, …
  (−∞, 0) nonnegative rational roots
• A147716: Triangle of coefficients in expansion of (14 + x)^n.
  1 · 14, 1 · 196, 28, 1 · 2744, 588, 42, 1 … row sums 1, 15, 225, 3375, 50625, 759375, …
  A147716 (−∞, 0) nonnegative rational roots
• A038271: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j)*5^j.
  1 · 7, 5 · 49, 70, 25 · 343, 735, 525, 125 … row sums 1, 12, 144, 1728, 20736, 248832, …
  A038271 (−∞, 0) nonnegative rational roots
• A281000: Triangle read by rows: T(n,k) = binomial(2*n+1, 2*k+1)*binomial(2*n-2*k, n-k)/(n+1-k) for 0 <= k <= n.
  1 · 3, 1 · 10, 10, 1 · 35, 70, 21, 1 … row sums 1, 4, 21, 127, 835, 5798, 41835, …
  A281000 (−∞, 0) nonnegative
• A075497: Stirling2 triangle with scaled diagonals (powers of 2).
  1 · 2, 1 · 4, 6, 1 · 8, 28, 12, 1 … row sums 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  A075497 (−∞, 0) nonnegative
• A178820: Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
  1 · 4, 4 · 10, 20, 10 · 20, 60, 60, 20 … row sums 1, 8, 40, 160, 560, 1792, 5376, 15360, …
  A178820 (−∞, 0) nonnegative rational roots
• A038303: Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*1^j.
  1 · 10, 1 · 100, 20, 1 · 1000, 300, 30, 1 … row sums 1, 11, 121, 1331, 14641, 161051, …
  A038303 (−∞, 0) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.
  144, 96, 16 · 1728, 1728, 576, 64 … row sums 256, 4096, 65536, 1048576, 16777216, …
  (−∞, 0) nonnegative rational roots
• A085880: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
  1 · 1, 1 · 2, 4, 2 · 5, 15, 15, 5 … row sums 1, 2, 8, 40, 224, 1344, 8448, 54912, …
  A085880 (−∞, 0) nonnegative rational roots
• A144442: Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 17, 1 · 1, 118, 118, 1 … row sums 1, 2, 19, 238, 4141, 92292, 2512589, …
  A144442 (−∞, 0) nonnegative
• Stirling2 triangle with scaled diagonals (powers of 6).
  36, 18, 1 · 216, 252, 36, 1 · 1296, 3240, 900, 60, 1 … row sums 55, 505, 5497, 69823, 1007407, …
  (−∞, 0) nonnegative
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
  1 · 2, 1 · 4, 4, 1 · 8, 12, 6, 1 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A038207 (−∞, −1] nonnegative rational roots
• A066325: Hermite he
  1 · 0, 1 · -1, 0, 1 · 0, -3, 0, 1 … row sums 1, 1, 0, -2, -2, 6, 16, -20, …
  A066325 (−∞, ∞)
• A122542: Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
  1 · 0, 1 · 0, 2, 1 · 0, 2, 4, 1 … row sums 1, 1, 3, 7, 17, 41, 99, 239, …
  A122542 (−∞, 0] nonnegative
• 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
• Stirling2 triangle with scaled diagonals (powers of 4).
  16, 12, 1 · 64, 112, 24, 1 · 256, 960, 400, 40, 1 … row sums 29, 201, 1657, 15821, 170389, 2032785, …
  (−∞, 0) nonnegative