preserver_image / reversal, shift 0

Image under a real-rootedness preserver. All operations

164 sequences

• A100861: Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
  1 · 1 · 1, 1 · 1, 3 … row sums 1, 1, 2, 4, 10, 26, 76, 232, …
  A100861 A144299 (−∞, 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
• A121574 → reverse
  1 · 1, 2 · 1, 5, 4 · 1, 8, 16, 8 … row sums 1, 3, 10, 33, 109, 360, 1189, 3927, …
  (−∞, 0] nonnegative
• A237765: Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.
  A007318 → ×(−(1/2)n+(1/2)n²)
  1, 2, 1 · 3, 9, 9, 3 · 6, 24, 36, 24, 6 … row sums 0, 0, 4, 24, 96, 320, 960, 2688, …
  A237765 (−∞, 0] nonnegative rational roots
• A059473 → γ
  1 · 2 · 4, 2 · 8, 8 … row sums 1, 2, 6, 16, 44, 120, 328, 896, …
  (−∞, 0] nonnegative
• A172431 → reverse
  1 · 2, 1 · 3, 4, 1 · 4, 10, 6, 1 … row sums 1, 3, 8, 21, 55, 144, 377, 987, …
  (−∞, 0] nonnegative
• A135838: Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).
  1 · 2, 2 · 2, 4, 2 · 4, 12, 12, 4 … row sums 1, 4, 8, 32, 64, 256, 512, 2048, …
  A135838 (−∞, 0] nonnegative rational roots
• A178820: Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
  1 · 4, 4 · 10, 20, 10 · 20, 60, 60, 20 … row sums 1, 8, 40, 160, 560, 1792, 5376, 15360, …
  A178820 (−∞, 0) nonnegative rational roots
• A008277 → θ+2 (shift +1) → reverse
  2 · 3, 2 · 4, 9, 2 · 5, 24, 21, 2 … row sums 2, 5, 15, 52, 203, 877, 4140, 21147, …
  (−∞, 0] nonnegative
• A085880: Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
  1 · 1, 1 · 2, 4, 2 · 5, 15, 15, 5 … row sums 1, 2, 8, 40, 224, 1344, 8448, 54912, …
  A085880 (−∞, 0) nonnegative rational roots
• A132393 → reverse
  1 · 1 · 1, 1 · 1, 3, 2 … row sums 1, 1, 2, 6, 24, 120, 720, 5040, …
  (−∞, 0] nonnegative rational roots
• A111910: Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).
  A091044 → ⊙ C(n+2,k+1)
  1 · 1, 1 · 1, 5, 1 · 1, 14, 14, 1 … row sums 1, 2, 7, 30, 146, 772, 4331, 25398, …
  A111910 (−∞, 0] nonnegative
• 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
• A008459: Square the entries of Pascal's triangle.
  1 · 1, 1 · 1, 4, 1 · 1, 9, 9, 1 … row sums 1, 2, 6, 20, 70, 252, 924, 3432, …
  A008459 (−∞, 0) nonnegative
• A272866: Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.
  1 · 1, 3, 1 · 1, 6, 11, 6, 1 · 1, 9, 30, 45, 30, 9, 1 … row sums 1, 5, 25, 125, 625, 3125, 15625, 78125, …
  A272866 (−∞, 0] nonnegative
• A113953: A Jacobsthal triangle.
  1 · 0, 1 · 0, 2, 1 · 0, 0, 4, 1 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  A113953 (−8, 0] nonnegative
• A145597: Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.
  1 · 3, 3 · 6, 15, 6 · 10, 45, 45, 10 … row sums 1, 6, 27, 110, 429, 1638, 6188, 23256, …
  A145597 (−∞, 0) nonnegative
• A010054 → reverse → reverse
  1 · 4 · 7 · 10 … row sums 1, 4, 7, 10, 13, 16, 19, 22, …
  (−∞, 0] nonnegative rational roots
• A007318 → 2θ+1 → reverse
  1 · 3, 1 · 5, 6, 1 · 7, 15, 9, 1 … row sums 1, 4, 12, 32, 80, 192, 448, 1024, …
  (−∞, 0] nonnegative rational roots
• A001263: Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.
  1 · 1, 1 · 1, 3, 1 · 1, 6, 6, 1 … row sums 1, 2, 5, 14, 42, 132, 429, 1430, …
  A001263 (−∞, 0) nonnegative
• A114656 → reverse
  1 · 1, 2 · 1, 6, 4 · 1, 12, 24, 8 … row sums 1, 3, 11, 45, 197, 903, 4279, 20793, …
  (−∞, 0] nonnegative
• A125662 → reverse
  1 · 1, 3 · 1, 6, 8 · 1, 9, 25, 21 … row sums 1, 4, 15, 56, 209, 780, 2911, 10864, …
  (−1, 0] nonnegative
• A126351 → reverse
  1 · 2, 1 · 4, 5, 1 · 8, 19, 9, 1 … row sums 1, 3, 10, 37, 151, 674, 3263, 17007, …
  (−∞, 0] nonnegative
• basis-map coefficients of A026729, shifted → reverse
  1 · -1 · 1 · -1 … row sums 1, -1, 1, -1, 1, -1, 1, -1, …
  (−∞, 0] rational roots
• A108838 → reverse
  2 · 2, 3 · 2, 8, 4 · 2, 15, 20, 5 … row sums 2, 5, 14, 42, 132, 429, 1430, 4862, …
  (−∞, 0] nonnegative
• A023531 → basis A049020 → reverse
  1 · 1, 1 · 1, 3, 2 · 1, 6, 10, 5 … row sums 1, 2, 6, 22, 94, 454, 2430, 14214, …
  (−∞, 0] nonnegative
• A052553: Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals.
  1 · 1 · 1, 1 · 1, 2 … row sums 1, 1, 2, 3, 5, 8, 13, 21, …
  A052553 (−∞, 0] nonnegative
• A007318: Binomial
  1 · 1, 1 · 1, 2, 1 · 1, 3, 3, 1 … row sums 1, 2, 4, 8, 16, 32, 64, 128, …
  A007318 (−∞, −1] nonnegative rational roots
• A143494 → reverse
  1 · 1, 2 · 1, 5, 4 · 1, 9, 19, 8 … row sums 1, 3, 10, 37, 151, 674, 3263, 17007, …
  (−∞, 0] nonnegative
• A046716: Coefficients of a special case of Poisson-Charlier polynomials.
  1 · 1, 1 · 1, 3, 1 · 1, 6, 8, 1 … row sums 1, 2, 5, 16, 65, 326, 1957, 13700, …
  A046716 (−∞, 0) nonnegative
• A131431: 3n + 1 preceded by n zeros.
  A010054 → reverse
  1 · 0, 4 · 0, 0, 7 · 0, 0, 0, 10 … row sums 1, 4, 7, 10, 13, 16, 19, 22, …
  A131431 [−1, 0] nonnegative rational roots
• A123519 → reverse
  4, 6, 1 · 8, 20, 12, 1 · 16, 56, 60, 20, 1 … row sums 11, 41, 153, 571, 2131, 7953, 29681, …
  (−∞, 0] nonnegative
• A038208: Triangle whose (i,j)-th entry is binomial(i,j)*2^i.
  1 · 2, 2 · 4, 8, 4 · 8, 24, 24, 8 … row sums 1, 4, 16, 64, 256, 1024, 4096, 16384, …
  A038208 (−∞, −1] nonnegative rational roots
• A127733: Triangle read by rows: n-th row is n^2 preceded by (n-1) zeros.
  A023531 → (θ+1)²
  1 · 0, 4 · 0, 0, 9 · 0, 0, 0, 16 … row sums 1, 4, 9, 16, 25, 36, 49, 64, …
  A127733 [−1, 0] nonnegative rational roots
• A116412: Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
  1 · 3, 1 · 6, 6, 1 · 12, 21, 9, 1 … row sums 1, 4, 13, 43, 142, 469, 1549, 5116, …
  A116412 (−∞, 0) nonnegative
• A131689 → reverse
  1 · 1 · 2, 1 · 6, 6, 1 … row sums 1, 1, 3, 13, 75, 541, 4683, 47293, …
  (−∞, −1] nonnegative
• A127648: Triangle read by rows: row n consists of n zeros followed by n+1.
  A010054 → reverse
  1 · 0, 2 · 0, 0, 3 · 0, 0, 0, 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  A127648 [−1, 0] nonnegative rational roots
• A085478 → reverse
  1 · 1, 1 · 1, 3, 1 · 1, 5, 6, 1 … row sums 1, 2, 5, 13, 34, 89, 233, 610, …
  (−∞, 0] nonnegative
• A081578: Pascal-(1,3,1) array.
  1 · 1, 1 · 1, 5, 1 · 1, 9, 9, 1 … row sums 1, 2, 7, 20, 61, 182, 547, 1640, …
  A081578 (−∞, 0) nonnegative
• A039756 → reverse
  1 · 1, 1 · 1, 4, 1 · 1, 13, 9, 1 … row sums 1, 2, 6, 24, 116, 648, 4088, 28640, …
  (−∞, 0] nonnegative
• A007318 → D → γ
  1 · 2 · 3 · 4 … row sums 1, 2, 3, 4, 5, 6, 7, 8, …
  (−∞, 0] nonnegative rational roots
• A344565: Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
  3 · 15, 15 · 45, 90, 45 · 105, 315, 315, 105 … row sums 3, 30, 180, 840, 3360, 12096, 40320, …
  A344565 (−∞, 0) nonnegative rational roots
• delannoy
  1 · 1, 1 · 1, 3, 1 · 1, 5, 5, 1 … row sums 1, 2, 5, 12, 29, 70, 169, 408, …
  [−3−2√2, −3+2√2] nonnegative
• A128099: Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).
  1 · 1 · 1, 2 · 1, 4 … row sums 1, 1, 3, 5, 11, 21, 43, 85, …
  A128099 (−∞, 0] nonnegative
• A249251: Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
  1 · 1, 3 · 1, 6, 6 · 1, 9, 21, 12 … row sums 1, 4, 13, 43, 142, 469, 1549, 5116, …
  A249251 (−∞, 0) nonnegative
• A207607 → reverse
  1 · 2, 1 · 2, 5, 1 · 2, 9, 9, 1 … row sums 1, 3, 8, 21, 55, 144, 377, 987, …
  (−∞, 0] nonnegative
• A133080 → reverse
  1 · 1, 1 · 1 · 1, 1 … row sums 1, 2, 1, 2, 1, 2, 1, 2, …
  (−∞, 0] nonnegative rational roots
• A023531 → (θ+1)² → reverse
  1 · 4 · 9 · 16 … row sums 1, 4, 9, 16, 25, 36, 49, 64, …
  (−∞, 0] nonnegative rational roots
• A122542 → reverse
  1 · 1 · 1, 2 · 1, 4, 2 … row sums 1, 1, 3, 7, 17, 41, 99, 239, …
  (−∞, 0] nonnegative
• A081579: Pascal-(1,4,1) array.
  1 · 1, 1 · 1, 6, 1 · 1, 11, 11, 1 … row sums 1, 2, 8, 24, 80, 256, 832, 2688, …
  A081579 (−∞, 0) nonnegative