Real-rooted atlas

A102547 → reverse

A277627 (tail) nonnegative coefficients

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

Recurrence from OEIS A277627
\(P_{n}(x) = \left(x\right)\,P_{n-1}(x) + \left(x^{2}\right)\,P_{n-3}(x)\)
\(P_{0}(x) = x\)
\(P_{1}(x) = x^{2}\)
\(P_{2}(x) = x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0011
10011
200011
3000112
40000213
500000314
6000001416
70000003519
8000000066113
900000001107119
10000000004158128
1100000000010219141
120000000001202810160
1300000000005353611188
1400000000000155645121129

Generated b-files: A000930 (row sums), A068921 (row sums), A102547 (reversed triangle) ?

Computed in your browser ?

Technical details
Node id
n-67a75673619cc276 · revision 1 · JSON
Definition source
oeis_catalog:A277627
Fingerprint
f6-00788a960ffaf339

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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: 0, 1;
  1: 0, 0, 1;
  2: 0, 0, 0, 1;
  3: 0, 0, 0, 1, 1;
  4: 0, 0, 0, 0, 2, 1;
  5: 0, 0, 0, 0, 0, 3, 1;
  6: 0, 0, 0, 0, 0, 1, 4, 1;
  7: 0, 0, 0, 0, 0, 0, 3, 5,  1;
  8: 0, 0, 0, 0, 0, 0, 0, 6,  6, 1;
  9: 0, 0, 0, 0, 0, 0, 0, 1, 10, 7, 1;
%F Formula
The row polynomials satisfy P(n,x) = x*P(n-1,x) + x^2*P(n-3,x) for n >= 3, with P(0,x) = x, P(1,x) = x^2, P(2,x) = x^3.
%t Mathematica
P[0, x_] := x;
P[1, x_] := x^2;
P[2, x_] := x^3;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x^2*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] = x; v[2] = x^2; v[3] = x^3; for(n = 3, N, my(i = n + 1); v[i] = x*v[i - 1] + x^2*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.

A102547: reversed triangle, offset 0, from the row P0, low zeros dropped
%e Example
Triangle begins:
  0: 1;
  1: 1;
  2: 1;
  3: 1, 1;
  4: 1, 2;
  5: 1, 3;
  6: 1, 4,  1;
  7: 1, 5,  3;
  8: 1, 6,  6;
  9: 1, 7, 10, 1;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = x*P(n-1,x) + x^2*P(n-3,x) for n >= 3, with P(0,x) = x, P(1,x) = x^2, P(2,x) = x^3.
%t Mathematica
P[0, x_] := x;
P[1, x_] := x^2;
P[2, x_] := x^3;
P[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] + x^2*P[n - 3, x]];
Table[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = x; v[2] = x^2; v[3] = x^3; for(n = 3, N, my(i = n + 1); v[i] = x*v[i - 1] + x^2*v[i - 3]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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-0e9e334768ea0a35
  • 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-0e9e334768ea0a35
  • 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-0e9e334768ea0a35

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 11 proof steps

Same fingerprint (1)

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