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)

• 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
• 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
• A154227: Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.
  1 · 1, 1 · 1, 8, 1 · 1, 19, 19, 1 … row sums 1, 2, 10, 40, 230, 1300, 9040, 64880, …
  A154227 (−∞, 0) nonnegative
• A114687: Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 …
  1 · 1, 2 · 1, 6, 4 · 1, 12, 24, 8 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A114687 (−∞, 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
• Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).
  1, 6, 4 · 1, 12, 20, 8 · 1, 20, 60, 56, 16 … row sums 11, 41, 153, 571, 2131, 7953, 29681, …
  (−2, 0) nonnegative
• 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
• A144436: Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 1, and j = 4, read by rows.
  1 · 1, 1 · 1, 8, 1 · 1, 23, 23, 1 … row sums 1, 2, 10, 48, 280, 1872, 14224, 121280, …
  A144436 (−∞, 0) nonnegative
• 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
• 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
• A082985: Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
  1 · 1, 3 · 1, 5, 5 · 1, 7, 14, 7 … row sums 1, 4, 11, 29, 76, 199, 521, 1364, …
  A082985 (−∞, 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
• A078812: Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).
  1 · 2, 1 · 3, 4, 1 · 4, 10, 6, 1 … row sums 1, 3, 8, 21, 55, 144, 377, 987, …
  A078812 (−4, 0) nonnegative
• 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
• A085841: Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ).
  1 · 3, 4 · 5, 40, 16 · 7, 140, 336, 64 … row sums 1, 7, 61, 547, 4921, 44287, 398581, …
  A085841 (−∞, 0) nonnegative
• A110441: Triangular array formed by the Mersenne numbers.
  1 · 3, 1 · 7, 6, 1 · 15, 23, 9, 1 … row sums 1, 4, 14, 48, 164, 560, 1912, 6528, …
  A110441 (−6, 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