Root regions

Nodes grouped by the interval that contains every root of every row (proved, for all n). A node can have several proved regions; the sharpest is the smallest interval.

Nodes with all roots in (-inf, 0) (258)

• 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
• A063007: T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.
  1 · 1, 2 · 1, 6, 6 · 1, 12, 30, 20 … row sums 1, 3, 13, 63, 321, 1683, 8989, 48639, …
  A063007 (−1, 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
• 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
• A126216: Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).
  1 · 2, 1 · 5, 5, 1 · 14, 21, 9, 1 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A126216 (−∞, −1] nonnegative
• 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
• 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
• 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