A122431 → reverse
palindromic γ-positive nonnegative coefficients only rational roots
Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means
Rows ?
| n | x0 | x1 | x2 | x3 | Pn(1) |
|---|---|---|---|---|---|
| 0 | 1 | 3 | 3 | 1 | 8 |
| 1 | 1 | 3 | 3 | 1 | 8 |
| 2 | 1 | 3 | 3 | 1 | 8 |
| 3 | 1 | 3 | 3 | 1 | 8 |
| 4 | 1 | 3 | 3 | 1 | 8 |
| 5 | 1 | 3 | 3 | 1 | 8 |
| 6 | 1 | 3 | 3 | 1 | 8 |
| 7 | 1 | 3 | 3 | 1 | 8 |
| 8 | 1 | 3 | 3 | 1 | 8 |
| 9 | 1 | 3 | 3 | 1 | 8 |
| 10 | 1 | 3 | 3 | 1 | 8 |
| 11 | 1 | 3 | 3 | 1 | 8 |
| 12 | 1 | 3 | 3 | 1 | 8 |
| 13 | 1 | 3 | 3 | 1 | 8 |
| 14 | 1 | 3 | 3 | 1 | 8 |
Computed in your browser ?
Technical details
- Node id
n-1bc415e3a319469d· revision 1 · JSON- Definition source
derived:reversal- Fingerprint
f6-5659ab0d98136444
Observed on the stored rows ?
- Degrees: 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 (affine: slope 0, intercept 3)
- Palindromic rows: yes
- γ-positive (gamma_positive): yes
- Nonnegative coefficients (
nonn): yes - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes
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 ExampleTriangle begins: 0: 1, 3, 3, 1; 1: 1, 3, 3, 1; 2: 1, 3, 3, 1; 3: 1, 3, 3, 1; 4: 1, 3, 3, 1; 5: 1, 3, 3, 1; 6: 1, 3, 3, 1; 7: 1, 3, 3, 1; 8: 1, 3, 3, 1; 9: 1, 3, 3, 1;
%F FormulaThe row polynomials satisfy P(n,x) = P(n-1,x) for n >= 1, with P(0,x) = 1 + 3*x + 3*x^2 + x^3.
%t MathematicaP[0, x_] := 1 + 3*x + 3*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1 + 3*x + 3*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = 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-d15d6831373644cf -
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-d15d6831373644cf -
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-d15d6831373644cf
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
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preserver_image reversal, shift 0
e-d15d6831373644cfAffine, reversal and Möbius substitutions (elementary) polytool ?
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premise (sequence):
Riordan array ((1+x)^3,x).
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
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parent_provers periodic_product
e-73f8c6c9e8d957afroot: theorem checkParent-row provers of polytool polytool ?
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premise (sequence):
Riordan array ((1+x)^3,x).
Related sequences ?
Same fingerprint (1)
Equal up to content, sign, reversal and a small index shift.