Real-rooted atlas

A152842 → reverse

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(7 + 3\,x - 4\,n - 2\,n x\right)\,P_{n-1}(x) + \left(6 + 8\,x + 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
23418
39157132
4924228164
527819046111256
62710817113657121512
781351621579307931512048

Show all 15 stored rows

Generated b-files: A094015 (row sums), A152842 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-5426b03c17935143 · revision 1 · JSON
Definition source
derived:reversal
Fingerprint
f6-7ccdd1ee4a60f495

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

OEIS matches ?

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.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   3,    1;
  2:   3,    4,    1;
  3:   9,   15,    7,    1;
  4:   9,   24,   22,    8,    1;
  5:  27,   81,   90,   46,   11,    1;
  6:  27,  108,  171,  136,   57,   12,   1;
  7:  81,  351,  621,  579,  307,   93,  15,   1;
  8:  81,  432,  972, 1200,  886,  400, 108,  16,  1;
  9: 243, 1377, 3348, 4572, 3858, 2086, 724, 156, 19, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(4*n-7+(2*n-3)*x)*P(n-1,x) + (6+8*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4*n - 7 + (2*n - 3)*x)*P[n - 1, x] + (6 + 8*x + 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(4*n - 7 + (2*n - 3)*x)*v[i - 1] + (6 + 8*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)

Generated with polytool c48629374597.

A152842: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1,  3;
  2: 1,  4,   3;
  3: 1,  7,  15,   9;
  4: 1,  8,  22,  24,    9;
  5: 1, 11,  46,  90,   81,   27;
  6: 1, 12,  57, 136,  171,  108,   27;
  7: 1, 15,  93, 307,  579,  621,  351,   81;
  8: 1, 16, 108, 400,  886, 1200,  972,  432,   81;
  9: 1, 19, 156, 724, 2086, 3858, 4572, 3348, 1377, 243;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = -(4*n-7+(2*n-3)*x)*P(n-1,x) + (6+8*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4*n - 7 + (2*n - 3)*x)*P[n - 1, x] + (6 + 8*x + 2*x^2)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = -(4*n - 7 + (2*n - 3)*x)*v[i - 1] + (6 + 8*x + 2*x^2)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • 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-67f427bf8180eb6f
  • 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-67f427bf8180eb6f
  • 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-67f427bf8180eb6f

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

Same fingerprint (1)

Equal up to content, sign, reversal and a small index shift.