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)

• 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
• A165253: Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.
  1 · 1 · 1, 1 · 1, 3, 1 … row sums 1, 1, 2, 5, 13, 34, 89, 233, …
  A165253 (−4, 0) nonnegative
• 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
• A100862: Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n…
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … row sums 1, 2, 5, 14, 43, 142, 499, 1850, …
  A100862 (−∞, 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
• A154986: Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.
  1 · 1, 1 · 1, 4, 1 · 1, 11, 11, 1 … row sums 1, 2, 6, 24, 120, 720, 5040, 40320, …
  A154986 (−∞, 0) nonnegative
• Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.
  1, 7, 1 · 1, 21, 21, 1 · 1, 51, 161, 51, 1 … row sums 9, 44, 265, 1854, 14833, 133496, …
  (−∞, 0) nonnegative
• 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
• A114656: 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 peaks (1 <= k <=…
  1 · 2, 1 · 4, 6, 1 · 8, 24, 12, 1 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  A114656 (−∞, 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
• A085478: Triangle read by rows: T(n, k) = binomial(n + k, 2*k).
  1 · 1, 1 · 1, 3, 1 · 1, 6, 5, 1 … row sums 1, 2, 5, 13, 34, 89, 233, 610, …
  A085478 (−4, 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
• A144439: 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 = 2, and j = 2, read by rows.
  1 · 1, 1 · 1, 8, 1 · 1, 31, 31, 1 … row sums 1, 2, 10, 64, 532, 5448, 66440, 941056, …
  A144439 (−∞, 0) nonnegative
• A102662: Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).
  1 · 1, 3 · 1, 5, 3 · 1, 7, 11, 3 … row sums 1, 4, 9, 22, 53, 128, 309, 746, …
  A102662 (−∞, 0) nonnegative
• 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
• A059473: Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...
  1 · 2, 2 · 4, 10, 4 · 8, 32, 32, 8 … row sums 1, 4, 18, 80, 356, 1584, 7048, 31360, …
  A059473 (−∞, 0) nonnegative
• A143683: Pascal-(1,8,1) array.
  1 · 1, 1 · 1, 10, 1 · 1, 19, 19, 1 … row sums 1, 2, 12, 40, 176, 672, 2752, 10880, …
  A143683 (−∞, 0) nonnegative
• A081582: Pascal-(1,7,1) array.
  1 · 1, 1 · 1, 9, 1 · 1, 17, 17, 1 … row sums 1, 2, 11, 36, 149, 550, 2143, 8136, …
  A081582 (−∞, 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
• A081581: Pascal-(1,6,1) array.
  1 · 1, 1 · 1, 8, 1 · 1, 15, 15, 1 … row sums 1, 2, 10, 32, 124, 440, 1624, 5888, …
  A081581 (−∞, 0) nonnegative
• A091044: One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.
  1 · 2, 2 · 3, 10, 3 · 4, 28, 28, 4 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A091044 (−∞, 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+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
• A144438: Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 5, 1 · 1, 14, 14, 1 … row sums 1, 2, 7, 30, 157, 972, 6961, 56660, …
  A144438 (−∞, 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
• narayana
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … row sums 1, 2, 5, 14, 42, 132, 429, 1430, …
  (−∞, 0) nonnegative
• 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
• A144445: Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
  1 · 1, 1 · 1, 23, 1 · 1, 206, 206, 1 … row sums 1, 2, 25, 414, 9697, 293808, 10938775, …
  A144445 (−∞, 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
• A207823: Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).
  1 · 4, 1 · 15, 8, 1 · 56, 46, 12, 1 … row sums 1, 5, 24, 115, 551, 2640, 12649, 60605, …
  A207823 (−6, −2) nonnegative
• 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
• A111125: Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
  1 · 3, 1 · 5, 5, 1 · 7, 14, 7, 1 … row sums 1, 4, 11, 29, 76, 199, 521, 1364, …
  A111125 (−4, 0) nonnegative
• A094816: Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.
  1 · 1, 1 · 1, 3, 1 · 1, 8, 6, 1 … row sums 1, 2, 5, 16, 65, 326, 1957, 13700, …
  A094816 (−∞, 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
• A207607: Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.
  1 · 1, 2 · 1, 5, 2 · 1, 9, 9, 2 … row sums 1, 3, 8, 21, 55, 144, 377, 987, …
  A207607 (−4, 0) nonnegative
• 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