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	x⁴	x⁵	x⁶	x⁷	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
8	1485	41580	103950	41580	1485				190080
9	2145	77220	270270	180180	19305				549120
10	3003	135135	630630	630630	135135	3003			1537536
11	4095	225225	1351350	1891890	675675	45045			4193280
12	5460	360360	2702700	5045040	2702700	360360	5460		11182080
13	7140	556920	5105100	12252240	9189180	2042040	92820		29245440
14	9180	835380	9189180	27567540	27567540	9189180	835380	9180	75202560

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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