theorem_check / liu_wang

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

183 sequences

• Stirling2 triangle with scaled diagonals (powers of 9).
  81, 27, 1 · 729, 567, 54, 1 … row sums 109, 1351, 19612, 333451, 6493069, …
  (−∞, 0) nonnegative
• A038220: Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
  1 · 3, 2 · 9, 12, 4 · 27, 54, 36, 8 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A038220 (−∞, −1] nonnegative rational roots
• A156366: Triangle of coefficients of p(n, x) where p(n, x) is defined as p(n, x) = (1-3*x)^(n+1)*PolyLog(-n, 3*x)/(3*x), read by rows.
  1, 12, 9 · 1, 33, 99, 27 · 1, 78, 594, 702, 81 … row sums 22, 160, 1456, 15904, 202672, 2951680, …
  A156366 (−∞, 0) nonnegative
• A142460: Triangle read by rows: T(n,k) (1<=k<=n) 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 = 5.
  1 · 1, 1 · 1, 12, 1 · 1, 83, 83, 1 … row sums 1, 2, 14, 168, 2856, 62832, 1696464, …
  A142460 (−∞, 0) nonnegative
• A028326: Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).
  2 · 2, 2 · 2, 4, 2 · 2, 6, 6, 2 … row sums 2, 4, 8, 16, 32, 64, 128, 256, …
  A028326 (−∞, 0) nonnegative rational roots
• A126351: Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.
  1 · 1, 2 · 1, 5, 4 · 1, 9, 19, 8 … row sums 1, 3, 10, 37, 151, 674, 3263, 17007, …
  A126351 (−∞, 0) nonnegative
• A142462: Triangle read by rows: T(n,k) (1<=k<=n) 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 = 7.
  1 · 1, 1 · 1, 16, 1 · 1, 143, 143, 1 … row sums 1, 2, 18, 288, 6624, 198720, 7352640, …
  A142462 (−∞, 0) nonnegative
• A257611: 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(n) = 2*n + 3.
  1 · 3, 3 · 9, 30, 9 · 27, 213, 213, 27 … row sums 1, 6, 48, 480, 5760, 80640, 1290240, …
  A257611 (−∞, 0) nonnegative
• A145596: 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). shifted by x -> x + 1
  1 · 2, 2 · 3, 8, 3 · 4, 20, 20, 4 … row sums 1, 4, 14, 48, 165, 572, 2002, 7072, …
  A145596 (−∞, 0) nonnegative
• A038281: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*3^j.
  64, 48, 9 · 512, 576, 216, 27 … row sums 121, 1331, 14641, 161051, 1771561, …
  A038281 (−∞, 0) nonnegative rational roots
• A075498: Stirling2 triangle with scaled diagonals (powers of 3).
  1 · 3, 1 · 9, 9, 1 · 27, 63, 18, 1 … row sums 1, 4, 19, 109, 742, 5815, 51193, …
  A075498 (−∞, 0) nonnegative
• A060187: Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).
  1 · 1, 1 · 1, 6, 1 · 1, 23, 23, 1 … row sums 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  A060187 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*6^j.
  121, 132, 36 · 1331, 2178, 1188, 216 … row sums 289, 4913, 83521, 1419857, 24137569, …
  (−∞, 0) nonnegative rational roots
• 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+8*x+x^2)/(1-x…
  1, 8, 1 · 1, 19, 19, 1 · 1, 42, 114, 42, 1 … row sums 10, 40, 200, 1200, 8400, 67200, 604800, …
  (−∞, 0) nonnegative
• A038255: Triangle whose (i,j)-th entry is binomial(i,j)*6^(i-j).
  1 · 6, 1 · 36, 12, 1 · 216, 108, 18, 1 … row sums 1, 7, 49, 343, 2401, 16807, 117649, …
  A038255 (−∞, 0) nonnegative rational roots
• A286718: Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.
  1 · 1, 1 · 4, 5, 1 · 28, 39, 12, 1 … row sums 1, 2, 10, 80, 880, 12320, 209440, …
  A286718 (−∞, 0) nonnegative rational roots
• Triangle read by rows: (i,j)-th entry is binomial(i,j)*3^(i-j)*8^j.
  9, 48, 64 · 27, 216, 576, 512 … row sums 121, 1331, 14641, 161051, 1771561, …
  (−∞, 0) nonnegative rational roots
• A039756: Triangle of B-analogs of Stirling numbers of 2nd kind.
  1 · 1, 1 · 1, 4, 1 · 1, 9, 13, 1 … row sums 1, 2, 6, 24, 116, 648, 4088, 28640, …
  A039756 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*12^j.
  100, 240, 144 · 1000, 3600, 4320, 1728 … row sums 484, 10648, 234256, 5153632, 113379904, …
  (−∞, 0) nonnegative rational roots
• A109692: Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).
  1 · 1, 1 · 1, 4, 3 · 1, 9, 23, 15 … row sums 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  A109692 (−∞, 0) nonnegative rational roots
• Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m = 0.
  1, 11, 1 · 1, 111, 30, 1 · 1, 1111, 681, 58, 1 … row sums 13, 143, 1852, 27563, 473725, 9290396, …
  (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*5^j.
  121, 110, 25 · 1331, 1815, 825, 125 … row sums 256, 4096, 65536, 1048576, 16777216, …
  (−∞, 0) nonnegative rational roots
• A257612: 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) = 4*x + 2.
  1 · 2, 2 · 4, 24, 4 · 8, 184, 184, 8 … row sums 1, 4, 32, 384, 6144, 122880, 2949120, …
  A257612 (−∞, 0) nonnegative
• A013611: Triangle of coefficients in expansion of (1+4x)^n.
  1 · 1, 4 · 1, 8, 16 · 1, 12, 48, 64 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A013611 (−∞, 0) [−1, 0] nonnegative rational roots
• Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).
  4, 16, 8 · 8, 72, 144, 48 · 16, 256, 1152, 1536, 384 … row sums 28, 272, 3344, 49472, 852928, 16758016, …
  (−∞, 0) nonnegative
• A038322: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*8^j.
  1 · 11, 8 · 121, 176, 64 · 1331, 2904, 2112, 512 … row sums 1, 19, 361, 6859, 130321, 2476099, …
  A038322 (−∞, 0) nonnegative rational roots
• A145324: Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....
  1 · 1, 2 · 1, 5, 6 · 1, 9, 26, 24 … row sums 1, 3, 12, 60, 360, 2520, 20160, 181440, …
  A145324 (−∞, 0) [−1, 0] nonnegative rational roots
• A282629: Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].
  1 · 1, 3 · 1, 15, 9 · 1, 63, 108, 27 … row sums 1, 4, 25, 199, 1876, 20257, 245017, …
  A282629 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*11^j.
  144, 264, 121 · 1728, 4752, 4356, 1331 … row sums 529, 12167, 279841, 6436343, 148035889, …
  (−∞, 0) nonnegative rational roots
• A144699: Triangle of 5-Eulerian numbers.
  1 · 1, 5 · 1, 16, 25 · 1, 39, 171, 125 … row sums 1, 6, 42, 336, 3024, 30240, 332640, …
  A144699 (−∞, 0) nonnegative
• A280580: Triangle read by rows: T(n,k) = binomial(2*n,2*k)*binomial(2*n-2*k,n-k)/(n+1-k) for 0<=k<=n.
  1 · 1, 1 · 2, 6, 1 · 5, 30, 15, 1 … row sums 1, 2, 9, 51, 323, 2188, 15511, 113634, …
  A280580 (−∞, 0) nonnegative
• A038279: Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*1^j.
  1 · 8, 1 · 64, 16, 1 · 512, 192, 24, 1 … row sums 1, 9, 81, 729, 6561, 59049, 531441, …
  A038279 (−∞, 0) nonnegative rational roots
• A142461: Triangle read by rows: T(n,k) (1 <= k <= n) 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 = 6.
  1 · 1, 1 · 1, 14, 1 · 1, 111, 111, 1 … row sums 1, 2, 16, 224, 4480, 116480, 3727360, …
  A142461 (−∞, 0) nonnegative
• A008277: Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
  1 · 1, 1 · 1, 3, 1 · 1, 7, 6, 1 … row sums 1, 2, 5, 15, 52, 203, 877, 4140, …
  A008277 (−∞, 0) nonnegative
• A038231: Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
  1 · 4, 1 · 16, 8, 1 · 64, 48, 12, 1 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A038231 (−∞, −1] nonnegative rational roots
• A038291: Triangle whose (i,j)-th entry is binomial(i,j)*9^(i-j)*1^j.
  1 · 9, 1 · 81, 18, 1 · 729, 243, 27, 1 … row sums 1, 10, 100, 1000, 10000, 100000, …
  A038291 (−∞, 0) nonnegative rational roots
• A257615: 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 + 5.
  1 · 5, 5 · 25, 70, 25 · 125, 715, 715, 125 … row sums 1, 10, 120, 1680, 26880, 483840, …
  A257615 (−∞, 0) nonnegative
• A257623: 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(n) = 5*n + 3.
  1 · 3, 3 · 9, 48, 9 · 27, 501, 501, 27 … row sums 1, 6, 66, 1056, 22176, 576576, …
  A257623 (−∞, 0) nonnegative
• A290319: Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.
  1 · 1, 1 · 5, 6, 1 · 45, 59, 15, 1 … row sums 1, 2, 12, 120, 1680, 30240, 665280, …
  A290319 (−∞, 0) nonnegative rational roots
• A257609: 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 + 2.
  1 · 2, 2 · 4, 16, 4 · 8, 88, 88, 8 … row sums 1, 4, 24, 192, 1920, 23040, 322560, …
  A257609 (−∞, 0) nonnegative
• A142075: Triangle T(n, k) = 2^(k-1) * E(n, k-1) where E(n,k) are the Eulerian numbers A173018, read by rows.
  1 · 1, 2 · 1, 8, 4 · 1, 22, 44, 8 … row sums 1, 3, 13, 75, 541, 4683, 47293, 545835, …
  A142075 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*10^(i-j)*3^j.
  100, 60, 9 · 1000, 900, 270, 27 … row sums 169, 2197, 28561, 371293, 4826809, …
  (−∞, 0) nonnegative rational roots
• A257625: 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) = 6*n + 3.
  1 · 3, 3 · 9, 54, 9 · 27, 621, 621, 27 … row sums 1, 6, 72, 1296, 31104, 933120, …
  A257625 (−∞, 0) nonnegative
• A144084: T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.
  1 · 1, 1 · 1, 4, 2 · 1, 9, 18, 6 … row sums 1, 2, 7, 34, 209, 1546, 13327, 130922, …
  A144084 (−∞, 0) nonnegative
• A156365: T(n, k) = E(n, k)*2^k where E(n,k) are the Eulerian numbers A173018, for n > 0 and 0 <= k <= n-1, additionally T(0,0) = 1.
  1, 8, 4 · 1, 22, 44, 8 · 1, 52, 264, 208, 16 … row sums 13, 75, 541, 4683, 47293, 545835, …
  A156365 (−∞, 0) nonnegative
• A285061: Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).
  1 · 1, 4 · 1, 24, 16 · 1, 124, 240, 64 … row sums 1, 5, 41, 429, 5329, 75989, 1215481, …
  A285061 (−∞, 0) nonnegative
• A144697: Triangle of 3-Eulerian numbers.
  1 · 1, 3 · 1, 10, 9 · 1, 25, 67, 27 … row sums 1, 4, 20, 120, 840, 6720, 60480, …
  A144697 (−∞, 0) nonnegative
• A062196: Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k).
  1 · 1, 3 · 1, 8, 6 · 1, 15, 30, 10 … row sums 1, 4, 15, 56, 210, 792, 3003, 11440, …
  A062196 (−∞, 0) nonnegative
• A094638: Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first …
  1 · 1, 1 · 1, 3, 2 · 1, 6, 11, 6 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A094638 (−∞, 0) nonnegative rational roots
• A038323: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*9^j.
  1 · 11, 9 · 121, 198, 81 · 1331, 3267, 2673, 729 … row sums 1, 20, 400, 8000, 160000, 3200000, …
  A038323 (−∞, 0) nonnegative rational roots