Real-rooted atlas

A152842: Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

A152842 nonnegative coefficients only rational roots

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

Recurrence from OEIS A152842
\(P_{n}(x) = \left(1 + 4\,x + 3\,x^{2}\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1134
21438
31715932
4182224964
511146908127256
61125713617110827512
711593307579621351812048

Show all 15 stored rows

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

Computed in your browser ?

Technical details
Node id
n-7196be931c2ed996 · revision 1 · JSON
Definition source
oeis_catalog:A152842
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: 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
The row polynomials satisfy P(n,x) = (1+4*x+3*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x + 3*x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x + 3*x^2)*v[i - 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)

Generated with polytool c48629374597.

A152842: 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
The row polynomials satisfy P(n,x) = (1+4*x+3*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[(1 + 4*x + 3*x^2)*P[n - 2, x]];
Table[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; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 4*x + 3*x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #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-596b986e3a00db44
  • 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-596b986e3a00db44
  • 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-596b986e3a00db44

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

Used in 13 proof steps

Same fingerprint (1)

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