theorem_check / liu_wang

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

183 sequences

• A106800: Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
  1 · 1 · 1, 1 · 1, 3, 1 … row sums 1, 1, 2, 5, 15, 52, 203, 877, …
  A106800 (−∞, 0) nonnegative
• A028338: Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
  1 · 1, 1 · 3, 4, 1 · 15, 23, 9, 1 … row sums 1, 2, 8, 48, 384, 3840, 46080, 645120, …
  A028338 (−∞, 0) nonnegative rational roots
• A055356: Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.
  1 · 1 · 1, 1 · 1, 4, 2 … row sums 1, 1, 2, 7, 36, 245, 2076, 21059, …
  A055356 (−∞, 0) nonnegative
• A120910: Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.
  3 · 6, 3 · 12, 12, 3 · 24, 36, 18, 3 … row sums 3, 9, 27, 81, 243, 729, 2187, 6561, …
  A120910 (−∞, 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+12*x+x^2)/(1-…
  1, 12, 1 · 1, 27, 27, 1 · 1, 58, 162, 58, 1 … row sums 14, 56, 280, 1680, 11760, 94080, …
  (−∞, 0) nonnegative
• A049374: A triangle of numbers related to triangle A030527.
  1 · 6, 1 · 42, 18, 1 · 336, 276, 36, 1 … row sums 1, 7, 61, 649, 8245, 122215, 2069425, …
  A049374 (−∞, 0) nonnegative
• A185410: A decomposition of the double factorials A001147.
  1 · 1 · 1, 2 · 1, 10, 4 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  A185410 (−∞, 0) nonnegative
• A013620: Triangle of coefficients in expansion of (2+3x)^n.
  1 · 2, 3 · 4, 12, 9 · 8, 36, 54, 27 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A013620 (−∞, 0) [−1, 0] nonnegative rational roots
• A225468: Triangle read by rows, S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  1 · 2, 1 · 4, 7, 1 · 8, 39, 15, 1 … row sums 1, 3, 12, 63, 405, 3024, 25515, 239355, …
  A225468 (−∞, 0) nonnegative
• A062190: Coefficient triangle of certain polynomials N(5; m,x).
  1 · 1, 6 · 1, 14, 21 · 1, 24, 84, 56 … row sums 1, 7, 36, 165, 715, 3003, 12376, 50388, …
  A062190 (−∞, 0) nonnegative
• A257617: 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) = 7*x + 2.
  1 · 2, 2 · 4, 36, 4 · 8, 388, 388, 8 … row sums 1, 4, 44, 792, 19800, 633600, 24710400, …
  A257617 (−∞, 0) nonnegative
• A049029: Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.
  1 · 5, 1 · 45, 15, 1 · 585, 255, 30, 1 … row sums 1, 6, 61, 871, 15996, 358891, 9509641, …
  A049029 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.
  144, 168, 49 · 1728, 3024, 1764, 343 … row sums 361, 6859, 130321, 2476099, 47045881, …
  (−∞, 0) nonnegative rational roots
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*3^j.
  144, 72, 9 · 1728, 1296, 324, 27 … row sums 225, 3375, 50625, 759375, 11390625, …
  (−∞, 0) nonnegative rational roots
• A257626: 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 + 6.
  1 · 6, 6 · 36, 108, 36 · 216, 1404, 1404, 216 … row sums 1, 12, 180, 3240, 68040, 1632960, …
  A257626 (−∞, 0) nonnegative
• A142963: Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-1-p)*(2*p+2)!/(p+1)!.
  1 · 1, 2 · 1, 10, 4 · 1, 30, 72, 8 … row sums 1, 3, 15, 111, 1071, 12663, 177255, …
  A142963 (−∞, 0) nonnegative
• A257627: 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) = 7*x + 3.
  1 · 3, 3 · 9, 60, 9 · 27, 753, 753, 27 … row sums 1, 6, 78, 1560, 42120, 1432080, …
  A257627 (−∞, 0) nonnegative
• Stirling2 triangle with scaled diagonals (powers of 10).
  100, 30, 1 · 1000, 700, 60, 1 … row sums 131, 1761, 27601, 506651, 10674211, …
  (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*8^j.
  144, 192, 64 · 1728, 3456, 2304, 512 … row sums 400, 8000, 160000, 3200000, 64000000, …
  (−∞, 0) nonnegative rational roots
• Stirling2 triangle with scaled diagonals (powers of 8).
  64, 24, 1 · 512, 448, 48, 1 · 4096, 7680, 1600, 80, 1 … row sums 89, 1009, 13457, 210105, 3747753, …
  (−∞, 0) nonnegative
• A114655: Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisti…
  2 · 4, 2 · 8, 12, 2 · 16, 48, 24, 2 … row sums 2, 6, 22, 90, 394, 1806, 8558, 41586, …
  A114655 (−∞, 0) nonnegative
• A013610: Triangle of coefficients in expansion of (1+3*x)^n.
  1 · 1, 3 · 1, 6, 9 · 1, 9, 27, 27 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A013610 (−∞, 0) [−1, 0] nonnegative rational roots
• A257619: 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) = 9*x + 2.
  1 · 2, 2 · 4, 44, 4 · 8, 564, 564, 8 … row sums 1, 4, 52, 1144, 35464, 1418560, …
  A257619 (−∞, 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+10*x+x^2)/(1-…
  1, 10, 1 · 1, 23, 23, 1 · 1, 50, 138, 50, 1 … row sums 12, 48, 240, 1440, 10080, 80640, …
  (−∞, 0) nonnegative
• A143494: Triangle read by rows: 2-Stirling numbers of the second kind.
  1 · 2, 1 · 4, 5, 1 · 8, 19, 9, 1 … row sums 1, 3, 10, 37, 151, 674, 3263, 17007, …
  A143494 (−∞, 0) nonnegative
• A013609: Triangle of coefficients in expansion of (1+2*x)^n.
  1 · 1, 2 · 1, 4, 4 · 1, 6, 12, 8 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A013609 (−∞, 0) [−1, 0] nonnegative rational roots
• A166973: Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.
  1 · 1, 1 · 1, 7, 1 · 1, 43, 18, 1 … row sums 1, 2, 9, 63, 536, 5307, 60389, 775988, …
  A166973 (−∞, 0) nonnegative
• A156992: Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.
  1 · 2, 2 · 6, 12, 6 · 24, 72, 72, 24 … row sums 1, 4, 24, 192, 1920, 23040, 322560, …
  A156992 (−∞, 0) nonnegative rational roots
• A120909: Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs (i.e., subwords of maximal length) of identical letters (1 <= k <= n).
  3 · 3, 6 · 3, 12, 12 · 3, 18, 36, 24 … row sums 3, 9, 27, 81, 243, 729, 2187, 6561, …
  A120909 (−∞, 0) nonnegative rational roots
• Stirling2 triangle with scaled diagonals (powers of 5).
  25, 15, 1 · 125, 175, 30, 1 · 625, 1875, 625, 50, 1 … row sums 41, 331, 3176, 35451, 447981, 6282416, …
  (−∞, 0) nonnegative
• A178821: Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
  1 · 5, 5 · 15, 30, 15 · 35, 105, 105, 35 … row sums 1, 10, 60, 280, 1120, 4032, 13440, …
  A178821 (−∞, 0) nonnegative rational roots
• A102365: Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.
  1 · 1 · 1, 1 · 1, 5, 1 … row sums 1, 1, 2, 7, 35, 226, 1787, 16717, …
  A102365 (−∞, 0) nonnegative
• A038243: Triangle whose (i,j)-th entry is 5^(i-j)*binomial(i,j).
  1 · 5, 1 · 25, 10, 1 · 125, 75, 15, 1 … row sums 1, 6, 36, 216, 1296, 7776, 46656, …
  A038243 (−∞, −1] nonnegative rational roots
• A161198: Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).
  1 · 1, 2 · 3, 8, 4 · 15, 46, 36, 8 … row sums 1, 3, 15, 105, 945, 10395, 135135, …
  A161198 (−∞, 0) nonnegative rational roots
• A225469: Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
  1 · 3, 1 · 9, 10, 1 · 27, 79, 21, 1 … row sums 1, 4, 20, 128, 1008, 9280, 96704, …
  A225469 (−∞, 0) nonnegative
• Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*7^j.
  121, 154, 49 · 1331, 2541, 1617, 343 … row sums 324, 5832, 104976, 1889568, 34012224, …
  (−∞, 0) nonnegative rational roots
• A144698: Triangle of 4-Eulerian numbers.
  1 · 1, 4 · 1, 13, 16 · 1, 32, 113, 64 … row sums 1, 5, 30, 210, 1680, 15120, 151200, …
  A144698 (−∞, 0) nonnegative
• A059364: Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.
  1 · 2, 1 · 6, 7, 2 · 24, 46, 29, 6 … row sums 1, 3, 15, 105, 945, 10395, 135135, …
  A059364 (−∞, −1] nonnegative rational roots
• A290316: 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 A282629 (S2[3,1] generalized Stirling2…
  1 · 1, 6 · 1, 48, 90 · 1, 234, 2214, 2160 … row sums 1, 7, 139, 4609, 214297, 12820999, …
  A290316 (−∞, 0) nonnegative
• A038325: Triangle read by rows whose (i,j)-th entry is binomial(i,j)*11^(i-j)*11^j.
  1 · 11, 11 · 121, 242, 121 · 1331, 3993, 3993, 1331 … row sums 1, 22, 484, 10648, 234256, 5153632, …
  A038325 (−∞, 0) nonnegative rational roots
• A038327: Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.
  1 · 12, 1 · 144, 24, 1 · 1728, 432, 36, 1 … row sums 1, 13, 169, 2197, 28561, 371293, …
  A038327 (−∞, 0) nonnegative rational roots
• A225470: Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
  1 · 2, 1 · 10, 7, 1 · 80, 66, 15, 1 … row sums 1, 3, 18, 162, 1944, 29160, 524880, …
  A225470 (−∞, 0) nonnegative rational roots
• A219512: Triangle of third-order Eulerian numbers: 3-Stirling permutations enumerated by ascents.
  1 · 1, 3 · 1, 12, 15 · 1, 33, 141, 105 … row sums 1, 4, 28, 280, 3640, 58240, 1106560, …
  A219512 (−∞, 0) nonnegative
• 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
• 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
• 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
• 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