A133087 → x ↦ x+1

OEIS matches ?

• A133125: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

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:   3,    1;
  2:   4,    4,    1;
  3:  12,   16,    7,    1;
  4:  16,   32,   24,    8,    1;
  5:  48,  112,  104,   48,   11,    1;
  6:  64,  192,  240,  160,   60,   12,   1;
  7: 192,  640,  912,  720,  340,   96,  15,   1;
  8: 256, 1024, 1792, 1792, 1120,  448, 112,  16,  1;
  9: 768, 3328, 6400, 7168, 5152, 2464, 784, 160, 19, 1;

The row polynomials satisfy P(n,x) = -(5*n-8+(2*n-3)*x)*P(n-1,x) + (10+9*x+2*x^2)*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[-(5*n - 8 + (2*n - 3)*x)*P[n - 1, x] + (10 + 9*x + 2*x^2)*D[P[n - 1, x], x]];
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] = -(5*n - 8 + (2*n - 3)*x)*v[i - 1] + (10 + 9*x + 2*x^2)*deriv(v[i - 1], x)); 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 (-inf, -1]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-52dd73548014190c
• 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-52dd73548014190c, e-f0f6d4d5dc8cc42f
• 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-52dd73548014190c, e-f0f6d4d5dc8cc42f
• Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-f0f6d4d5dc8cc42f

Proof chains

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

• basis_map A007318 e-f0f6d4d5dc8cc42f

  Geometric basis map (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)
• definition_equality e-dcae3f88ad03ec5b root: theorem check

  Ore-algebra identity of two definitions polytool ?

• preserver_image affine, shift 0 e-52dd73548014190c

  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 (expanded above)
    • ↵ parent_provers periodic_product e-aa3a6b002de14e0c (expanded above)
    • ↵ preserver_image affine, shift 0 e-28a1cf3333f63fa4 (expanded above)

Related sequences ?

No candidate equalities or related sequences.