A391533 → Veronese 2
nonnegative coefficients
Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | Pn(1) |
|---|---|---|---|---|---|---|
| 0 | 4 | 1 | 5 | |||
| 1 | 4 | 5 | 9 | |||
| 2 | 4 | 13 | 1 | 18 | ||
| 3 | 4 | 25 | 7 | 36 | ||
| 4 | 4 | 41 | 26 | 1 | 72 | |
| 5 | 4 | 61 | 70 | 9 | 144 | |
| 6 | 4 | 85 | 155 | 43 | 1 | 288 |
| 7 | 4 | 113 | 301 | 147 | 11 | 576 |
Generated b-files: A005010 (row sums), A116453 (row sums) ?
Computed in your browser ?
Technical details
- Node id
n-cbf81d8f5ae9d341· revision 1 · JSON- Definition source
derived:veronese_section_2_0- Fingerprint
f6-31ecaf0e48ae96a1
Observed on the stored rows ?
- Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
- 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 ?
- A005010: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 0 ?
- A116453: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS · generated b-file 200 terms from n = 1 ?
- A288135: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
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 ExampleTriangle begins: 0: 4, 1; 1: 4, 5; 2: 4, 13, 1; 3: 4, 25, 7; 4: 4, 41, 26, 1; 5: 4, 61, 70, 9; 6: 4, 85, 155, 43, 1; 7: 4, 113, 301, 147, 11; 8: 4, 145, 532, 406, 64, 1; 9: 4, 181, 876, 966, 264, 13;
%F FormulaThe row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) for n >= 2, with P(0,x) = 4 + x, P(1,x) = 4 + 5*x.
%t MathematicaP[0, x_] := 4 + x;
P[1, x_] := 4 + 5*x;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 4 + x; v[2] = 4 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - x)*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.
Proved properties
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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-9f9d452726caee29 -
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-9f9d452726caee29 -
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-9f9d452726caee29
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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veronese_section r=2, j=0, shift 0
e-9f9d452726caee29Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
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premise (sequence):
The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
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theorem_check liu_wang
e-3cff49c1888b5e80root: theorem checkLiu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
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parent_provers periodic_product
e-5e5f630c1e2baa8aroot: theorem checkParent-row provers of polytool polytool ?
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premise (sequence):
The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.
Related sequences ?
No candidate equalities or related sequences.
References
- [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
- [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
- [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
- [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003