theorem_check / liu_wang

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

183 sequences

• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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 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