theorem_check

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

304 sequences

• 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
• A112493: Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.
  1 · 1, 1 · 1, 4, 3 · 1, 11, 25, 15 … row sums 1, 2, 8, 52, 472, 5504, 78416, 1320064, …
  A112493 [−1, 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
• A099097: Riordan array (1, 3+x).
  1 · 0, 3 · 0, 1, 9 · 0, 0, 6, 27 … row sums 1, 3, 10, 33, 109, 360, 1189, 3927, …
  A099097 (−4/9, 0] nonnegative
• A249248: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 3 · 1, 7, 3 · 1, 12, 22, 3 … row sums 1, 4, 11, 38, 131, 490, 1897, 7714, …
  A249248 (−∞, 0) nonnegative
• 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
• A116414: Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).
  1 · 4, 1 · 13, 8, 1 · 40, 42, 12, 1 … row sums 1, 5, 22, 95, 409, 1760, 7573, 32585, …
  A116414 (−8, 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
• A033282: Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.
  1 · 1, 2 · 1, 5, 5 · 1, 9, 21, 14 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A033282 (−1, 0) nonnegative
• A054142: Triangular array binomial(2*n-k, k), k=0..n, n >= 0.
  1 · 1, 1 · 1, 3, 1 · 1, 5, 6, 1 … row sums 1, 2, 5, 13, 34, 89, 233, 610, …
  A054142 (−∞, 0) nonnegative
• A028246: Triangular array a(n,k) = (1/k)*Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*i^n; n >= 1, 1 <= k <= n, read by rows.
  1 · 1, 1 · 1, 3, 2 · 1, 7, 12, 6 … row sums 1, 2, 6, 26, 150, 1082, 9366, 94586, …
  A028246 [−1, 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
• A079638: Matrix product of unsigned Lah-triangle |A008297(n,k)| and unsigned Stirling1-triangle |A008275(n,k)|.
  1 · 3, 1 · 14, 9, 1 · 90, 83, 18, 1 … row sums 1, 4, 24, 192, 1920, 23040, 322560, …
  A079638 (−∞, 0) nonnegative
• A081580: Pascal-(1,5,1) array.
  1 · 1, 1 · 1, 7, 1 · 1, 13, 13, 1 … row sums 1, 2, 9, 28, 101, 342, 1189, 4088, …
  A081580 (−∞, 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
• A144441: Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 14, 1 · 1, 83, 83, 1 … row sums 1, 2, 16, 168, 2416, 44160, 981184, …
  A144441 (−∞, 0) nonnegative
• 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
• A154233: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.
  1 · 1, 1 · 1, 7, 1 · 1, 19, 19, 1 … row sums 1, 2, 9, 40, 251, 1662, 13615, 118640, …
  A154233 (−∞, 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
• A104684: Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.
  1 · 2, 1 · 6, 6, 1 · 20, 30, 12, 1 … row sums 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  A104684 (−∞, 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
• A172431: Even row Pascal-square read by antidiagonals.
  1 · 1, 2 · 1, 4, 3 · 1, 6, 10, 4 … row sums 1, 3, 8, 21, 55, 144, 377, 987, …
  A172431 (−∞, 0) nonnegative
• A143685: Pascal-(1,9,1) array.
  1 · 1, 1 · 1, 11, 1 · 1, 21, 21, 1 … row sums 1, 2, 13, 44, 205, 806, 3457, 14168, …
  A143685 (−∞, 0) nonnegative
• 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
• A008288: Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
  1 · 1, 1 · 1, 3, 1 · 1, 5, 5, 1 … row sums 1, 2, 5, 12, 29, 70, 169, 408, …
  A008288 [−3−2√2, −3+2√2] nonnegative
• 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