A391533 → Veronese 2 (j=1) → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(-24 - 30\,x + 162\,x^{2} - 78\,x^{3} + 12\,x^{4} + 152\,n - 294\,n x + 123\,n x^{2} - 30\,n x^{3} + 4\,n x^{4} - 48\,n^{2} + 68\,n^{2} x - 30\,n^{2} x^{2} + 4\,n^{2} x^{3} + 16\,n^{3} - 8\,n^{3} x + n^{3} x^{2}\right)\,P_{n}(x) = \left(96 - 408\,x + 668\,x^{2} - 276\,x^{3} + 40\,x^{4} + 104\,n - 142\,n x - 210\,n x^{2} + 134\,n x^{3} - 34\,n x^{4} + 4\,n x^{5} - 28\,n^{2} x + 59\,n^{2} x^{2} - 29\,n^{2} x^{3} + 4\,n^{2} x^{4} + 16\,n^{3} + 8\,n^{3} x - 7\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(80\,x - 372\,x^{2} + 504\,x^{3} - 280\,x^{4} + 76\,x^{5} - 8\,x^{6} - 32\,n x + 160\,n x^{2} - 190\,n x^{3} + 70\,n x^{4} - 8\,n x^{5} + 32\,n^{2} x - 48\,n^{2} x^{2} + 18\,n^{2} x^{3} - 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	P_n(1)
0	4					4
1	9	1				10
2	18	6				24
3	35	21	1			57
4	64	60	8			132
5	109	153	37	1		300
6	174	354	134	10		672
7	263	749	418	57	1	1488

All 15 rows

Technical details

Node id: n-e060923914b88ef8 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-9ccd1ecaa9787a99

Observed on the stored rows ?

Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (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:   4;
  1:   9,    1;
  2:  18,    6;
  3:  35,   21,    1;
  4:  64,   60,    8;
  5: 109,  153,   37,   1;
  6: 174,  354,  134,  10;
  7: 263,  749,  418,  57,  1;
  8: 380, 1464, 1160, 248, 12;
  9: 529, 2673, 2910, 910, 81, 1;

%F Formula

The row polynomials satisfy (16*n^3-48*n^2+152*n-24-(8*n^3-68*n^2+294*n+30)*x+(n^3-30*n^2+123*n+162)*x^2+(4*n^2-30*n-78)*x^3+(4*n+12)*x^4)*P(n,x) = (16*n^3+104*n+96+(8*n^3-28*n^2-142*n-408)*x-(7*n^3-59*n^2+210*n-668)*x^2+(n^3-29*n^2+134*n-276)*x^3+(4*n^2-34*n+40)*x^4+4*n*x^5)*P(n-1,x) + ((32*n^2-32*n+80)*x-(48*n^2-160*n+372)*x^2+(18*n^2-190*n+504)*x^3-(2*n^2-70*n+280)*x^4-(8*n-76)*x^5-8*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4.

%t Mathematica

P[0, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((16*n^3 + 104*n + 96 + (8*n^3 - 28*n^2 - 142*n - 408)*x - (7*n^3 - 59*n^2 + 210*n - 668)*x^2 + (n^3 - 29*n^2 + 134*n - 276)*x^3 + (4*n^2 - 34*n + 40)*x^4 + 4*n*x^5)*P[n - 1, x] + ((32*n^2 - 32*n + 80)*x - (48*n^2 - 160*n + 372)*x^2 + (18*n^2 - 190*n + 504)*x^3 - (2*n^2 - 70*n + 280)*x^4 - (8*n - 76)*x^5 - 8*x^6)*D[P[n - 1, x], x])/(16*n^3 - 48*n^2 + 152*n - 24 - (8*n^3 - 68*n^2 + 294*n + 30)*x + (n^3 - 30*n^2 + 123*n + 162)*x^2 + (4*n^2 - 30*n - 78)*x^3 + (4*n + 12)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 4; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 + 104*n + 96 + (8*n^3 - 28*n^2 - 142*n - 408)*x - (7*n^3 - 59*n^2 + 210*n - 668)*x^2 + (n^3 - 29*n^2 + 134*n - 276)*x^3 + (4*n^2 - 34*n + 40)*x^4 + 4*n*x^5)*v[i - 1] + ((32*n^2 - 32*n + 80)*x - (48*n^2 - 160*n + 372)*x^2 + (18*n^2 - 190*n + 504)*x^3 - (2*n^2 - 70*n + 280)*x^4 - (8*n - 76)*x^5 - 8*x^6)*deriv(v[i - 1], x))/(16*n^3 - 48*n^2 + 152*n - 24 - (8*n^3 - 68*n^2 + 294*n + 30)*x + (n^3 - 30*n^2 + 123*n + 162)*x^2 + (4*n^2 - 30*n - 78)*x^3 + (4*n + 12)*x^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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from veronese_section_2_1 of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n…?
  - Veronese sections of Pólya frequency sequences [ASW52; Edr53; AW24] polytool ? step ›
    
    from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from reversal of The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from The truncated (4,4,1)-Pascal triangle, given by 4*C(n-2, k) + 4*C(n-2, k-1) + C(n-2, k-2) for n>=2.?
    
    ↑ Liu–Wang theorem (above)
    
    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
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

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
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.