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, -1] (188)

• 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
• A038207 → D
  1 · 4, 2 · 12, 12, 3 · 32, 48, 24, 4 … row sums 0, 1, 6, 27, 108, 405, 1458, 5103, …
  (−∞, −1] nonnegative rational roots
• A128100: Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).
  A011973 → basis A007318
  1 · 1 · 2, 1 · 3, 2 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  A128100 (−∞, −1] nonnegative
• A075497 → x ↦ x+1
  1 · 3, 1 · 11, 8, 1 · 49, 55, 15, 1 … row sums 1, 4, 20, 120, 832, 6496, 56128, …
  (−∞, −1] nonnegative
• A094305: Triangle read by rows: T(n,k) = ((n+1)(n+2)/2) * binomial(n,k) (0 <= k <= n).
  A007318 → ×(1+(3/2)n+(1/2)n²)
  1 · 3, 3 · 6, 12, 6 · 10, 30, 30, 10 … row sums 1, 6, 24, 80, 240, 672, 1792, 4608, …
  A094305 (−∞, −1] nonnegative rational roots
• 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
• A154602: Exponential Riordan array [exp(sinh(x)*exp(x)), sinh(x)*exp(x)].
  basis-map coefficients of A026729 → basis A154602
  1 · 1, 1 · 3, 4, 1 · 11, 19, 9, 1 … row sums 1, 2, 8, 40, 240, 1664, 12992, 112256, …
  A154602 (−∞, −1] nonnegative
• A100861 → basis A007318
  1 · 1 · 2, 1 · 4, 3 … row sums 1, 1, 3, 7, 25, 81, 331, 1303, …
  (−∞, −1] nonnegative
• A126216 → basis A007318
  1 · 3, 1 · 11, 7, 1 · 45, 42, 12, 1 … row sums 1, 4, 19, 100, 562, 3304, 20071, …
  (−∞, −1] nonnegative
• A008288 → basis A007318
  1 · 2, 1 · 5, 5, 1 · 12, 18, 8, 1 … row sums 1, 3, 11, 39, 139, 495, 1763, 6279, …
  (−∞, −1] nonnegative
• A102365 → x ↦ x+1
  1 · 1 · 2, 1 · 7, 7, 1 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  (−∞, −1] nonnegative
• A103327 → basis A007318
  1 · 4, 1 · 16, 12, 1 · 64, 80, 24, 1 … row sums 1, 5, 29, 169, 985, 5741, 33461, …
  (−∞, −1] nonnegative
• A059344 → basis A007318
  1 · 1 · 3, 2 · 7, 6 … row sums 1, 1, 5, 13, 73, 281, 1741, 8485, …
  (−∞, −1] nonnegative
• A094816 → x ↦ x+1
  1 · 2, 1 · 5, 5, 1 · 16, 23, 9, 1 … row sums 1, 3, 11, 49, 261, 1631, 11743, 95901, …
  (−∞, −1] nonnegative
• A034870 → x ↦ x+1
  1 · 4, 4, 1 · 16, 32, 24, 8, 1 … row sums 1, 9, 81, 729, 6561, 59049, 531441, …
  (−∞, −1] nonnegative rational roots
• A152198 → x ↦ x+1
  1 · 1 · 2, 1 · 2, 1 … row sums 1, 1, 3, 3, 9, 9, 27, 27, …
  (−∞, −1] nonnegative rational roots
• A055356 → x ↦ x+1
  1 · 1 · 2, 1 · 7, 8, 2 … row sums 1, 1, 3, 17, 143, 1597, 22243, 371241, …
  (−∞, −1] nonnegative
• A085880 → x ↦ x+1
  1 · 2, 1 · 8, 8, 2 · 40, 60, 30, 5 … row sums 1, 3, 18, 135, 1134, 10206, 96228, …
  (−∞, −1] nonnegative rational roots
• 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
• A008277 → θ+1 → x ↦ x+1
  1 · 3, 2 · 10, 12, 3 · 37, 62, 30, 4 … row sums 0, 1, 5, 25, 133, 761, 4677, 30745, …
  (−∞, −1] nonnegative
• A059340: Triangle T(n,k) of numbers with e.g.f. exp((exp((1+x)*y)-1)/(1+x)), k=0..n-1.
  A008278 → x ↦ x+1 (shift +1)
  1 · 2, 1 · 5, 5, 1 · 15, 23, 10, 1 … row sums 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  A059340 (−∞, −1] nonnegative
• A318772 → x ↦ (1/243)x → x ↦ x+1
  1 · 3 · 9 · 27 … row sums 1, 3, 9, 27, 83, 255, 783, 2403, …
  (−∞, −1] nonnegative
• A118394 → basis A007318
  1 · 1 · 1 · 7, 6 … row sums 1, 1, 1, 13, 49, 121, 1681, 10501, …
  (−∞, −1] 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
• basis-map coefficients of A026729, shifted → basis A007318
  1 · -1, -1 · 1, 2, 1 · -1, -3, -3, -1 … row sums 1, -2, 4, -8, 16, -32, 64, -128, …
  (−∞, −1] rational roots
• A391533 → Pₙ + Pₙ₊₁
  8, 12, 6, 1 · 8, 20, 18, 7, 1 · 8, 28, 38, 25, 8, 1 … row sums 27, 54, 108, 216, 432, 864, 1728, 3456, …
  (−∞, −1] nonnegative rational roots
• A081582 → x ↦ x+1
  1 · 2, 1 · 11, 11, 1 · 36, 54, 20, 1 … row sums 1, 3, 23, 111, 655, 3519, 19727, …
  (−∞, −1] nonnegative
• A120910 → basis A007318
  3 · 9, 3 · 27, 18, 3 · 81, 81, 27, 3 … row sums 3, 12, 48, 192, 768, 3072, 12288, …
  (−∞, −1] nonnegative rational roots
• A152828 → basis A007318
  1 · 1 · 1 · 2, 1 … row sums 1, 1, 1, 3, 3, 3, 9, 9, …
  (−∞, −1] nonnegative rational roots
• basis-map coefficients of A026729 → basis A007318
  1 · 1, 1 · 1, 2, 1 · 1, 3, 3, 1 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, −1] nonnegative rational roots
• basis-map coefficients of A021012 → x ↦ x+1
  1 · -1, -1 · 2, 4, 2 · -6, -18, -18, -6 … row sums 1, -2, 8, -48, 384, -3840, 46080, …
  (−∞, −1] rational roots
• A062264 → x ↦ x+1
  1 · 6, 5 · 28, 42, 15 · 120, 252, 168, 35 … row sums 1, 11, 85, 575, 3649, 22363, 134245, …
  (−∞, −1] nonnegative
• A099097 → basis A007318
  1 · 3, 3 · 10, 19, 9 · 33, 93, 87, 27 … row sums 1, 6, 38, 240, 1516, 9576, 60488, …
  (−∞, −1] nonnegative
• A118933 → x ↦ x+1
  1 · 1 · 1 · 1 … row sums 1, 1, 1, 1, 13, 61, 181, 421, …
  (−∞, −1] nonnegative
• A169803 → x ↦ x+1
  1 · 2, 1 · 3, 2 · 5, 5, 1 … row sums 1, 3, 5, 11, 21, 43, 85, 171, …
  (−∞, −1] nonnegative
• A007318 → ×(1+(3/2)n+(1/2)n²) → x ↦ x+1
  1 · 6, 3 · 24, 24, 6 · 80, 120, 60, 10 … row sums 1, 9, 54, 270, 1215, 5103, 20412, …
  (−∞, −1] nonnegative rational roots
• A038207: Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).
  1 · 2, 1 · 4, 4, 1 · 8, 12, 6, 1 … row sums 1, 3, 9, 27, 81, 243, 729, 2187, …
  A038207 (−∞, −1] nonnegative rational roots
• A007318 → −D²+(−1+n)D+1+n → x ↦ x+1
  1 · 4, 2 · 14, 14, 3 · 44, 66, 30, 4 … row sums 1, 6, 31, 144, 621, 2538, 9963, 37908, …
  (−∞, −1] nonnegative
• A118931 → basis A007318
  1 · 1 · 1 · 3, 2 … row sums 1, 1, 1, 5, 17, 41, 241, 1261, …
  (−∞, −1] nonnegative
• A317496 → x ↦ x+1
  1 · 1 · 1 · 4, 3 … row sums 1, 1, 1, 7, 13, 19, 61, 139, …
  (−∞, −1] nonnegative
• A103633 → basis A007318
  1 · 1, 1 · 2, 3, 1 · 2, 5, 4, 1 … row sums 1, 2, 6, 12, 36, 72, 216, 432, …
  (−∞, −1] nonnegative rational roots
• A143683 → x ↦ x+1
  1 · 2, 1 · 12, 12, 1 · 40, 60, 22, 1 … row sums 1, 3, 25, 123, 769, 4275, 25129, …
  (−∞, −1] nonnegative
• A114687 → basis A007318
  1 · 3, 2 · 11, 14, 4 · 45, 84, 48, 8 … row sums 1, 5, 29, 185, 1257, 8925, 65445, …
  (−∞, −1] nonnegative
• A121574 → basis A007318
  1 · 3, 1 · 10, 7, 1 · 33, 35, 11, 1 … row sums 1, 4, 18, 80, 356, 1584, 7048, 31360, …
  (−∞, −1] nonnegative
• A137597: Triangle read by rows: A008277 * A007318.
  A007318 → basis A049020
  1 · 2, 1 · 5, 5, 1 · 15, 22, 9, 1 … row sums 1, 3, 11, 47, 227, 1215, 7107, 44959, …
  A137597 (−∞, −1] nonnegative
• A081578 → x ↦ x+1
  1 · 2, 1 · 7, 7, 1 · 20, 30, 12, 1 … row sums 1, 3, 15, 63, 279, 1215, 5319, 23247, …
  (−∞, −1] nonnegative
• A143495 → x ↦ x+1
  1 · 4, 1 · 17, 9, 1 · 77, 64, 15, 1 … row sums 1, 5, 27, 157, 979, 6517, 46107, …
  (−∞, −1] nonnegative
• A049020: Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.
  A023531 → basis A049020
  1 · 1, 1 · 2, 3, 1 · 5, 10, 6, 1 … row sums 1, 2, 6, 22, 94, 454, 2430, 14214, …
  A049020 (−∞, −1] nonnegative
• A086645 → x ↦ x+1
  1 · 2, 1 · 8, 8, 1 · 32, 48, 18, 1 … row sums 1, 3, 17, 99, 577, 3363, 19601, 114243, …
  (−∞, −1] nonnegative
• A367270 → x ↦ x+1
  1 · 1 · 3, 2 · 10, 12, 3 … row sums 1, 1, 5, 25, 129, 681, 3653, 19825, …
  (−∞, −1] nonnegative