A178820 → Veronese 2 (j=1)

nonnegative coefficients

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

Recurrence

\(\left(-\frac{1}{2}\,n + \frac{1}{2}\,n^{2}\right)\,P_{n}(x) = \left(-3 + 2\,n + n^{2}\right)\,P_{n-1}(x) + \left(-3 + 3\,x - \frac{5}{2}\,n + \frac{5}{2}\,n x - \frac{1}{2}\,n^{2} + \frac{1}{2}\,n^{2} x\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	4							4
2	20							20
3	60	20						80
4	140	140						280
5	280	560	56					896
6	504	1680	504					2688
7	840	4200	2520	120				7680
8	1320	9240	9240	1320				21120
9	1980	18480	27720	7920	220			56320
10	2860	34320	72072	34320	2860			146432
11	4004	60060	168168	120120	20020	364		372736
12	5460	100100	360360	360360	100100	5460		931840
13	7280	160160	720720	960960	400400	43680	560	2293760
14	9520	247520	1361360	2333760	1361360	247520	9520	5570560

Computed in your browser ?

Technical details

Node id: n-f6e8ebcef0e02821 · revision 1 · JSON
Definition source: derived:veronese_section_2_1
Fingerprint: f6-937db0732045a112

Observed on the stored rows ?

Degrees: -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6 (half 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 ?

A082138: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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:    4;
  2:   20;
  3:   60,    20;
  4:  140,   140;
  5:  280,   560,    56;
  6:  504,  1680,   504;
  7:  840,  4200,  2520,  120;
  8: 1320,  9240,  9240, 1320;
  9: 1980, 18480, 27720, 7920, 220;

%F Formula

The row polynomials satisfy ((1/2)*n^2-(1/2)*n)*P(n,x) = (n^2+2*n-3)*P(n-1,x) - ((1/2)*n^2+(5/2)*n+3-((1/2)*n^2+(5/2)*n+3)*x)*P(n-2,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 2*n - 3)*P[n - 1, x] - ((1/2)*n^2 + (5/2)*n + 3 - ((1/2)*n^2 + (5/2)*n + 3)*x)*P[n - 2, x])/((1/2)*n^2 - (1/2)*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 2*n - 3)*v[i - 1] - ((1/2)*n^2 + (5/2)*n + 3 - ((1/2)*n^2 + (5/2)*n + 3)*x)*v[i - 2])/((1/2)*n^2 - (1/2)*n)); 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-4dba393820f57c76
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-4dba393820f57c76
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-4dba393820f57c76
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-4dba393820f57c76

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=1, shift 0 e-4dba393820f57c76

Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
- premise (sequence): Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-387009a21707feb6 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - preserver_image reversal, shift 0 e-17b3b03bdb024482
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-387009a21707feb6 (expanded above)

Related sequences ?

Same fingerprint (11)

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

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