A133087 → x ↦ x/(x+1)

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
  0: 1;
  1: 2,  3;
  2: 1,  4,   4;
  3: 2, 11,  20,   12;
  4: 1,  8,  24,   32,   16;
  5: 2, 19,  72,  136,  128,   48;
  6: 1, 12,  60,  160,  240,  192,   64;
  7: 2, 27, 156,  500,  960, 1104,  704,  192;
  8: 1, 16, 112,  448, 1120, 1792, 1792, 1024,  256;
  9: 2, 35, 272, 1232, 3584, 6944, 8960, 7424, 3584, 768;

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(6*n-8+(28*n-35)*x+(48*n-56)*x^2+(36*n-39)*x^3+(10*n-10)*x^4)*P(n-1,x) + (3+20*x+52*x^2+66*x^3+41*x^4+10*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 8 + (28*n - 35)*x + (48*n - 56)*x^2 + (36*n - 39)*x^3 + (10*n - 10)*x^4)*P[n - 1, x] + (3 + 20*x + 52*x^2 + 66*x^3 + 41*x^4 + 10*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(6*n - 8 + (28*n - 35)*x + (48*n - 56)*x^2 + (36*n - 39)*x^3 + (10*n - 10)*x^4)*v[i - 1] + (3 + 20*x + 52*x^2 + 66*x^3 + 41*x^4 + 10*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-c6d5cef6173ca46f
• Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
  All coefficients of each polynomial have the same sign (after x -> -x when alternating).
  derived from the cited steps by a stated theorem: e-c6d5cef6173ca46f
• Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
  Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
  derived from the cited steps by a stated theorem: e-c6d5cef6173ca46f

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• preserver_image mobius, shift 0 e-c6d5cef6173ca46f

  Affine, reversal and Möbius substitutions (elementary) polytool ?

  • premise (sequence): A133080 * A007318.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    • basis_map A007318 e-3537980ff71b1a89

      Geometric basis map (elementary) polytool ?

      • premise (sequence): Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • parent_provers periodic_product e-4d02ac33f5b2fbc9 root: theorem check

          Parent-row provers of polytool polytool ?

    • parent_provers periodic_product e-aa3a6b002de14e0c root: theorem check

      Parent-row provers of polytool polytool ?

    • preserver_image affine, shift 0 e-28a1cf3333f63fa4

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • ↵ parent_provers periodic_product e-4d02ac33f5b2fbc9 (expanded above)

Related sequences ?

No candidate equalities or related sequences.