A344565 → 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

\(\left(-\frac{1}{2}\,n + \frac{1}{2}\,n^{2}\right)\,P_{n}(x) = \left(-4 + 3\,n + n^{2}\right)\,P_{n-1}(x) + \left(-6 + 6\,x - \frac{7}{2}\,n + \frac{7}{2}\,n x - \frac{1}{2}\,n^{2} + \frac{1}{2}\,n^{2} x\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 3\)

\(P_{1}(x) = 15\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	3				3
1	15				15
2	45	45			90
3	105	315			420
4	210	1260	210		1680
5	378	3780	1890		6048
6	630	9450	9450	630	20160
7	990	20790	34650	6930	63360

Show all 15 stored rows

Computed in your browser ?

Technical details

Node id: n-6a9acb45f04bfcf2 · revision 1 · JSON
Definition source: derived:veronese_section_2_0
Fingerprint: f6-4e15cddc85973b47

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:    3;
  1:   15;
  2:   45,    45;
  3:  105,   315;
  4:  210,  1260,    210;
  5:  378,  3780,   1890;
  6:  630,  9450,   9450,    630;
  7:  990, 20790,  34650,   6930;
  8: 1485, 41580, 103950,  41580,  1485;
  9: 2145, 77220, 270270, 180180, 19305;

%F Formula

The row polynomials satisfy ((1/2)*n^2-(1/2)*n)*P(n,x) = (n^2+3*n-4)*P(n-1,x) - ((1/2)*n^2+(7/2)*n+6-((1/2)*n^2+(7/2)*n+6)*x)*P(n-2,x) for n >= 2, with P(0,x) = 3, P(1,x) = 15.

%t Mathematica

P[0, x_] := 3;
P[1, x_] := 15;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 3*n - 4)*P[n - 1, x] - ((1/2)*n^2 + (7/2)*n + 6 - ((1/2)*n^2 + (7/2)*n + 6)*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] = 3; v[2] = 15; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 3*n - 4)*v[i - 1] - ((1/2)*n^2 + (7/2)*n + 6 - ((1/2)*n^2 + (7/2)*n + 6)*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-0442706d8c37e606
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-0442706d8c37e606
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-0442706d8c37e606
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-0442706d8c37e606

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-0442706d8c37e606

Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ?
- premise (sequence): Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
  - theorem_check liu_wang e-e2cde2e22920ce64 root: theorem check
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?
  - preserver_image reversal, shift 0 e-4dae17a76888d380
    
    Affine, reversal and Möbius substitutions (elementary) polytool ?
    
    premise (sequence): Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    
    ↵ theorem_check liu_wang e-e2cde2e22920ce64 (expanded above)

Related sequences ?

Same fingerprint (10)

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