A188440 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-4 + 2\,x + 4\,x^{2} + 3\,n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(-4 - 14\,x - 16\,x^{2} - 8\,x^{3} - 5\,n + 6\,n x + 12\,n x^{2} + 8\,n x^{3} + 7\,n^{2} + 12\,n^{2} x + 8\,n^{2} x^{2} + 2\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-48 - 140\,x - 156\,x^{2} - 80\,x^{3} - 16\,x^{4} - 36\,n - 78\,n x - 60\,n x^{2} - 16\,n x^{3} - 6\,n^{2} - 10\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	1				1
2	5	2			7
3	5	2			7
4	21	20	4		45
5	21	20	4		45
6	81	126	60	8	275
7	81	126	60	8	275

All 15 rows

Technical details

Node id: n-4f3118c2c0a5ce89 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-376115f4132a4e54

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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:   1;
  1:   1;
  2:   5,   2;
  3:   5,   2;
  4:  21,  20,   4;
  5:  21,  20,   4;
  6:  81, 126,  60,   8;
  7:  81, 126,  60,   8;
  8: 297, 648, 504, 160, 16;
  9: 297, 648, 504, 160, 16;

%F Formula

The row polynomials satisfy (n^2+3*n-4+(4*n+2)*x+4*x^2)*P(n,x) = (2*n^3+7*n^2-5*n-4+(2*n^3+12*n^2+6*n-14)*x+(8*n^2+12*n-16)*x^2+(8*n-8)*x^3)*P(n-1,x) - (6*n^2+36*n+48+(10*n^2+78*n+140)*x+(4*n^2+60*n+156)*x^2+(16*n+80)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 7*n^2 - 5*n - 4 + (2*n^3 + 12*n^2 + 6*n - 14)*x + (8*n^2 + 12*n - 16)*x^2 + (8*n - 8)*x^3)*P[n - 1, x] - (6*n^2 + 36*n + 48 + (10*n^2 + 78*n + 140)*x + (4*n^2 + 60*n + 156)*x^2 + (16*n + 80)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(n^2 + 3*n - 4 + (4*n + 2)*x + 4*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^3 + 7*n^2 - 5*n - 4 + (2*n^3 + 12*n^2 + 6*n - 14)*x + (8*n^2 + 12*n - 16)*x^2 + (8*n - 8)*x^3)*v[i - 1] - (6*n^2 + 36*n + 48 + (10*n^2 + 78*n + 140)*x + (4*n^2 + 60*n + 156)*x^2 + (16*n + 80)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(n^2 + 3*n - 4 + (4*n + 2)*x + 4*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 shift_x_plus_one of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.?
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.?
    
    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
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A274498 → x ↦ x+1 → D+1

References

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