preserver_image / reversal, shift 0

Image under a real-rootedness preserver. All operations

164 sequences

• 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
• A007318 → ⊙ C(n+1,k+1) (shift +2) → reverse
  8 · 18, 12 · 32, 64, 16 · 50, 200, 150, 20 … row sums 8, 30, 112, 420, 1584, 6006, 22880, …
  (−∞, 0] nonnegative
• A026374: Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0, T(n,k) = T(n-1,k-1) + T(n-1,k) for odd n and 1< = k <= n-1, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for even n and 1 …
  1 · 1, 1 · 1, 3, 1 · 1, 4, 4, 1 … row sums 1, 2, 5, 10, 25, 50, 125, 250, …
  A026374 (−∞, 0] nonnegative
• A127773: Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
  A010054 → reverse
  1 · 0, 3 · 0, 0, 6 · 0, 0, 0, 10 … row sums 1, 3, 6, 10, 15, 21, 28, 36, …
  A127773 [−1, 0] nonnegative rational roots
• 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
• A010054 → reverse → reverse
  1 · 3 · 6 · 10 … row sums 1, 3, 6, 10, 15, 21, 28, 36, …
  (−∞, 0] nonnegative rational roots
• A152842 → reverse
  1 · 3, 1 · 3, 4, 1 · 9, 15, 7, 1 … row sums 1, 4, 8, 32, 64, 256, 512, 2048, …
  (−∞, 0] nonnegative rational roots
• A232535 → reverse
  1 · 2, 1 · 3, 8, 1 · 4, 25, 18, 1 … row sums 1, 3, 12, 48, 192, 768, 3072, 12288, …
  (−∞, 0] nonnegative
• A207606 → reverse
  2, 7, 4 · 2, 11, 16, 5 · 2, 15, 36, 30, 6 … row sums 13, 34, 89, 233, 610, 1597, 4181, …
  (−∞, 0] nonnegative
• A169803: Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).
  1 · 1, 1 · 1, 2 · 1, 3, 1 … row sums 1, 2, 3, 5, 8, 13, 21, 34, …
  A169803 (−∞, 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
• A130123 → reverse
  1 · 2 · 4 · 8 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  (−∞, 0] nonnegative rational roots
• A062145 → reverse
  1 · 4, 1 · 10, 10, 1 · 20, 45, 18, 1 … row sums 1, 5, 21, 84, 330, 1287, 5005, 19448, …
  (−∞, 0] nonnegative
• A023531 → 2D+1 → reverse
  1 · 1, 2 · 1, 4 · 1, 6 … row sums 1, 3, 5, 7, 9, 11, 13, 15, …
  (−∞, 0] nonnegative rational roots
• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
  A010054 → D∘x
  1 · 3 · 5 · 7 … row sums 1, 3, 5, 7, 9, 11, 13, 15, …
  A198954 (−∞, −1] [−1, 0] nonnegative rational roots
• A035607 → reverse
  1 · 2, 1 · 2, 4, 1 · 2, 8, 6, 1 … row sums 1, 3, 7, 17, 41, 99, 239, 577, …
  (−∞, 0] nonnegative
• A196347 → γ
  1 · 1 · 2 · 6 … row sums 1, 1, 2, 6, 24, 120, 720, 5040, …
  (−∞, 0] nonnegative rational roots
• 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
• A102547 → reverse
  0, 1 · 0, 0, 1 · 0, 0, 0, 1 · 0, 0, 0, 1, 1 … row sums 1, 1, 1, 2, 3, 4, 6, 9, …
  (−∞, 0] nonnegative
• basis-map coefficients of A030528, shifted → reverse
  1 · -1 · 1 · -1 … row sums 1, -1, 1, -1, 1, -1, 1, -1, …
  (−∞, 0] rational roots
• A078812 → reverse
  1 · 1, 2 · 1, 4, 3 · 1, 6, 10, 4 … row sums 1, 3, 8, 21, 55, 144, 377, 987, …
  (−∞, 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
• 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
• A106800 → reverse (shift +1) → reverse
  1 · 1 · 1, 1 · 1, 3, 1 … row sums 1, 1, 2, 5, 15, 52, 203, 877, …
  (−∞, 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
• A008277 → θ+1 → reverse
  1 · 2, 1 · 3, 6, 1 · 4, 18, 14, 1 … row sums 1, 3, 10, 37, 151, 674, 3263, 17007, …
  (−∞, 0] nonnegative
• basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0
  1 · -1, -1 · 1, 2, 1 · -1, -3, -3, -1 … row sums 1, -2, 4, -8, 16, -32, 64, -128, …
  (−∞, −1] 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
• basis-map coefficients of A021012 → reverse
  1 · -1 · 2 · -6 … row sums 1, -1, 2, -6, 24, -120, 720, -5040, …
  (−∞, 0] rational roots
• A111125 → reverse
  1 · 1, 3 · 1, 5, 5 · 1, 7, 14, 7 … row sums 1, 4, 11, 29, 76, 199, 521, 1364, …
  (−∞, 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
• 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
• A173018 → reverse (shift +1) → reverse
  1 · 1 · 1, 1 · 1, 4, 1 … row sums 1, 1, 2, 6, 24, 120, 720, 5040, …
  (−∞, 0] nonnegative
• A075497 → reverse
  1 · 1, 2 · 1, 6, 4 · 1, 12, 28, 8 … row sums 1, 3, 11, 49, 257, 1539, 10299, 75905, …
  (−∞, 0] 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
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.
  A007318 → θ+1
  1 · 1, 2 · 1, 4, 3 · 1, 6, 9, 4 … row sums 1, 3, 8, 20, 48, 112, 256, 576, …
  A103406 (−∞, 0] nonnegative rational roots
• A060920 → reverse
  1 · 1, 2 · 1, 5, 5 · 1, 9, 20, 13 … row sums 1, 3, 11, 43, 171, 683, 2731, 10923, …
  (−∞, ∞) nonnegative
• 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
• A062190 → reverse
  1 · 6, 1 · 21, 14, 1 · 56, 84, 24, 1 … row sums 1, 7, 36, 165, 715, 3003, 12376, 50388, …
  (−∞, 0] nonnegative
• A185411: A triangular decomposition of the double factorial numbers A001147.
  A185410 → reverse
  1 · 0, 1 · 0, 2, 1 · 0, 4, 10, 1 … row sums 1, 1, 3, 15, 105, 945, 10395, 135135, …
  A185411 (−∞, 0] nonnegative
• A113214 → reverse
  1 · 1, 2 · 1, 3 · 1, 4, 2 … row sums 1, 3, 4, 7, 11, 18, 29, 47, …
  (−∞, 0] nonnegative
• A023531 → (1/2)D²+2D+1 → reverse
  1 · 1, 2 · 1, 4, 1 · 1, 6, 3 … row sums 1, 3, 6, 10, 15, 21, 28, 36, …
  (−∞, 0] nonnegative
• A207823 → reverse
  1 · 1, 4 · 1, 8, 15 · 1, 12, 46, 56 … row sums 1, 5, 24, 115, 551, 2640, 12649, 60605, …
  (−1/2, 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
• A116414 → reverse
  1 · 1, 4 · 1, 8, 13 · 1, 12, 42, 40 … row sums 1, 5, 22, 95, 409, 1760, 7573, 32585, …
  (−∞, 0] nonnegative
• A249252 → reverse
  1 · 4, 1 · 8, 8, 1 · 16, 32, 12, 1 … row sums 1, 5, 17, 61, 217, 773, 2753, 9805, …
  (−∞, 0] nonnegative
• A133087 → reverse
  1 · 1, 2 · 1, 2, 1 · 1, 4, 5, 2 … row sums 1, 3, 4, 12, 16, 48, 64, 192, …
  (−∞, 0] nonnegative rational roots
• 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