A081580 → Veronese 2

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(2\right)\,P_{n-1}(x) + \left(-1 + x\right)\,P_{n-2}(x) + \left(10\,x\right)\,P_{n-3}(x) + \left(25\,x\right)\,P_{n-4}(x)\)

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

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

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

\(P_{3}(x) = 1 + 13\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	1	1							2
3	1	13							14
4	1	61	1						63
5	1	145	25						171
6	1	265	265	1					532
7	1	421	1585	37					2044
8	1	613	6145	613	1				7373
9	1	841	17401	6049	49				24341
10	1	1105	40105	40105	1105	1			82422
11	1	1405	80305	192637	15265	61			289674
12	1	1741	145345	712909	145345	1741	1		1007083
13	1	2113	243865	2149561	1020337	30961	73		3446911
14	1	2521	385801	5537233	5537233	385801	2521	1	11851112

Computed in your browser ?

Technical details

Node id: n-0ad24679c56fe57b · revision 1 · JSON
Definition source: derived:veronese_section_2_0
Fingerprint: f6-6aa73ad6b67815e1

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
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,   1;
  3: 1,  13;
  4: 1,  61,     1;
  5: 1, 145,    25;
  6: 1, 265,   265,    1;
  7: 1, 421,  1585,   37;
  8: 1, 613,  6145,  613,  1;
  9: 1, 841, 17401, 6049, 49;

%F Formula

The row polynomials satisfy P(n,x) = 2*P(n-1,x) - (1-x)*P(n-2,x) + 10*x*P(n-3,x) + 25*x*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1 + x, P(3,x) = 1 + 13*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[3, x_] := 1 + 13*x;
P[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] - (1 - x)*P[n - 2, x] + 10*x*P[n - 3, x] + 25*x*P[n - 4, 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 + x; v[4] = 1 + 13*x; for(n = 4, N, my(i = n + 1); v[i] = 2*v[i - 1] - (1 - x)*v[i - 2] + 10*x*v[i - 3] + 25*x*v[i - 4]); 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-7595c55a8f8b4fd6
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-7595c55a8f8b4fd6
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-7595c55a8f8b4fd6
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-7595c55a8f8b4fd6

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

veronese_section r=2, j=0, shift 0 e-7595c55a8f8b4fd6

Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
- premise (sequence): Pascal-(1,5,1) array.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang_derivative_lag e-307703197d4a988b root: theorem check
    
    Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ?
  - preserver_image reversal, shift 0 e-bf7624d7e60440c3
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Pascal-(1,5,1) array.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang_derivative_lag e-307703197d4a988b (expanded above)

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