A132393 → Veronese 2 (j=1) → basis A007318

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	1							1
3	3	1						4
4	12	6						18
5	60	37	1					98
6	360	255	15					630
7	2520	1977	178	1				4676
8	20160	17136	2044	28				39368
9	181440	164664	24093	550	1			370748
10	1814400	1739880	297945	9630	45			3861900
11	19958400	20066040	3898179	163063	1325	1		44087008
12	239500800	250953120	54105084	2768898	33000	66		547360968
13	3113510400	3383955360	796701492	48015273	763063	2723	1	7342948312
14	43589145600	48952088640	12431463408	858534495	17126473	91637	91	105848450344

Technical details

Node id: n-9a30a5b089ff9460 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-4f917058aac63e8c

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

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      0;
  1:      1;
  2:      1;
  3:      3,      1;
  4:     12,      6;
  5:     60,     37,     1;
  6:    360,    255,    15;
  7:   2520,   1977,   178,   1;
  8:  20160,  17136,  2044,  28;
  9: 181440, 164664, 24093, 550, 1;

%F Formula

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

%t Mathematica

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

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, -1]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary), Affine, reversal and Möbius substitutions (elementary)
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Geometric basis map (elementary)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from veronese_section_2_1 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for…?
  - Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
    
    from Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check
  - Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
    
    from Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)
Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from veronese_section_2_1 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for…?
  - ↑ Veronese sections of Pólya frequency sequences (above)
  - ↑ Veronese sections of Pólya frequency sequences (above)

Related ?

Same fingerprint ?

A130534 → Veronese 2 → x ↦ x+1

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
[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.