A118933 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6 - 24\,x + 11\,n + 44\,n x - 6\,n^{2} - 24\,n^{2} x + n^{3} + 4\,n^{3} x\right)\,P_{n-4}(x) + \left(1\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 1\)

\(P_{2}(x) = 1\)

\(P_{3}(x) = 1\)

Rows ?

n	x⁰	x¹	P_n(1)
0	1		1
1	1		1
2	1		1
3	1		1
4	7	24	31
5	31	120	151
6	91	360	451
7	211	840	1051

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-08ddd13f3e918f7a · revision 1 · JSON
Definition source: derived:basis_map_A013611
Fingerprint: f6-fc52cbab43e69f98

Observed on the stored rows ?

Degrees: 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3 (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 ?

No OEIS matches.

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;
  4:     7,    24;
  5:    31,   120;
  6:    91,   360;
  7:   211,   840;
  8:  1681, 11760,  20160;
  9: 12097, 93744, 181440;

%F Formula

The row polynomials satisfy P(n,x) = (n^3-6*n^2+11*n-6+(4*n^3-24*n^2+44*n-24)*x)*P(n-4,x) + P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1, P(3,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1;
P[3, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(n^3 - 6*n^2 + 11*n - 6 + (4*n^3 - 24*n^2 + 44*n - 24)*x)*P[n - 4, x] + P[n - 1, 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; v[4] = 1; for(n = 4, N, my(i = n + 1); v[i] = (n^3 - 6*n^2 + 11*n - 6 + (4*n^3 - 24*n^2 + 44*n - 24)*x)*v[i - 4] + v[i - 1]); 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-986b4ce84a8ba732
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-986b4ce84a8ba732
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-986b4ce84a8ba732
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-986b4ce84a8ba732

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

basis_map A013611 e-986b4ce84a8ba732

Geometric basis map (elementary) polytool ?
- premise (sequence): Triangle, read by rows, where T(n,k) = n!/(k!*(n-4*k)!*4^k) for n>=4*k>=0.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - parent_provers paper_theorem e-281e0bfa773c0053 root: theorem check
    
    Parent-row provers of polytool polytool ?

Related sequences ?

No candidate equalities or related sequences.