Real-rooted atlas

A317496: Triangle T(n,k) = T(n-1,k) + 3*T(n-3,k-1) for k = 0..floor(n/3) with T(0,0) = 1, T(n,k) = 0 for n or k < 0, read by rows.

A317496 nonnegative coefficients

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

Recurrence from OEIS A317496
\(P_{n}(x) = \left(1\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-3}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4Pn(1)
011
111
211
3134
4167
51910
6112922
71152743
81185473
91219027139
10124135108268
11127189270487
1213025254081904
131333249454051708
14136405151212153169

Generated b-files: A084386 (row sums), A317496 ?

Computed in your browser ?

Technical details
Node id
n-a4ffad71e3e1c62c · revision 1 · JSON
Definition source
oeis_catalog:A317496
Fingerprint
f6-40b4a235121ab3e8

Observed on the stored rows ?

  • Degrees: 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4 (irregular)
  • 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): no

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;
  2: 1;
  3: 1,  3;
  4: 1,  6;
  5: 1,  9;
  6: 1, 12,  9;
  7: 1, 15, 27;
  8: 1, 18, 54;
  9: 1, 21, 90, 27;
%F Formula
The row polynomials satisfy P(n,x) = P(n-1,x) + 3*x*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + 3*x*P[n - 3, 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; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = v[i - 1] + 3*x*v[i - 3]); 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.

A317496: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1;
  3: 1,  3;
  4: 1,  6;
  5: 1,  9;
  6: 1, 12,  9;
  7: 1, 15, 27;
  8: 1, 18, 54;
  9: 1, 21, 90, 27;
%F Formula
The row polynomials satisfy P(n,x) = P(n-1,x) + 3*x*P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + 3*x*P[n - 3, 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; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = v[i - 1] + 3*x*v[i - 3]); 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.

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-196b2df06b390016
  • 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-196b2df06b390016
  • 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-196b2df06b390016
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-196b2df06b390016

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 6 proof steps

No candidate equalities or related sequences.