A391533 → Veronese 2 (j=1) → D

nonnegative coefficients

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

Recurrence

\(\left(-24 - 6\,x + 56\,n + 10\,n x - 3\,n x^{2} - 48\,n^{2} + 4\,n^{2} x + 2\,n^{2} x^{2} + 16\,n^{3} - 8\,n^{3} x + n^{3} x^{2}\right)\,P_{n}(x) = \left(-32\,x + 8\,x^{2} + 8\,n + 66\,n x - 2\,n x^{3} - 92\,n^{2} x + 27\,n^{2} x^{2} - n^{2} x^{3} + 16\,n^{3} + 8\,n^{3} x - 7\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(16\,x - 20\,x^{2} + 4\,x^{3} - 32\,n x + 32\,n x^{2} + 2\,n x^{3} - 2\,n x^{4} + 32\,n^{2} x - 48\,n^{2} x^{2} + 18\,n^{2} x^{3} - 2\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0					0
1	1				1
2	6				6
3	19	2			21
4	44	16			60
5	85	68	3		156
6	146	208	30		384
7	231	518	159	4	912

All 15 rows

Technical details

Node id: n-992e25cda0fbb245 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-80c779f6eeda7c77

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:   6;
  3:  19,    2;
  4:  44,   16;
  5:  85,   68,    3;
  6: 146,  208,   30;
  7: 231,  518,  159,   4;
  8: 344, 1120,  600,  48;
  9: 489, 2184, 1818, 304, 5;

%F Formula

The row polynomials satisfy (16*n^3-48*n^2+56*n-24-(8*n^3-4*n^2-10*n+6)*x+(n^3+2*n^2-3*n)*x^2)*P(n,x) = (16*n^3+8*n+(8*n^3-92*n^2+66*n-32)*x-(7*n^3-27*n^2-8)*x^2+(n^3-n^2-2*n)*x^3)*P(n-1,x) + ((32*n^2-32*n+16)*x-(48*n^2-32*n+20)*x^2+(18*n^2+2*n+4)*x^3-(2*n^2+2*n)*x^4)*d/dx 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[Together[((16*n^3 + 8*n + (8*n^3 - 92*n^2 + 66*n - 32)*x - (7*n^3 - 27*n^2 - 8)*x^2 + (n^3 - n^2 - 2*n)*x^3)*P[n - 1, x] + ((32*n^2 - 32*n + 16)*x - (48*n^2 - 32*n + 20)*x^2 + (18*n^2 + 2*n + 4)*x^3 - (2*n^2 + 2*n)*x^4)*D[P[n - 1, x], x])/(16*n^3 - 48*n^2 + 56*n - 24 - (8*n^3 - 4*n^2 - 10*n + 6)*x + (n^3 + 2*n^2 - 3*n)*x^2)]];
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] = ((16*n^3 + 8*n + (8*n^3 - 92*n^2 + 66*n - 32)*x - (7*n^3 - 27*n^2 - 8)*x^2 + (n^3 - n^2 - 2*n)*x^3)*v[i - 1] + ((32*n^2 - 32*n + 16)*x - (48*n^2 - 32*n + 20)*x^2 + (18*n^2 + 2*n + 4)*x^3 - (2*n^2 + 2*n)*x^4)*deriv(v[i - 1], x))/(16*n^3 - 48*n^2 + 56*n - 24 - (8*n^3 - 4*n^2 - 10*n + 6)*x + (n^3 + 2*n^2 - 3*n)*x^2)); 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.