A133080 → D → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 - 12\,n + 13\,n^{2} - 6\,n^{3} + n^{4}\right)\,P_{n}(x) = \left(12 + 2\,x - 40\,n - 9\,n x + 51\,n^{2} + 28\,n^{2} x - 31\,n^{3} - 30\,n^{3} x + 9\,n^{4} + 13\,n^{4} x - n^{5} - 2\,n^{5} x\right)\,P_{n-1}(x) + \left(4\,x + x^{2} - 12\,n x - 4\,n x^{2} + 13\,n^{2} x + 10\,n^{2} x^{2} - 6\,n^{3} x - 8\,n^{3} x^{2} + n^{4} x + 2\,n^{4} x^{2}\right)\,P_{n-1}'(x)\)

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

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

\(P_{2}(x) = 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	0	4						4
3	0	4	9					13
4	0	0	0	16				16
5	0	0	0	16	25			41
6	0	0	0	0	0	36		36
7	0	0	0	0	0	36	49	85

All 15 rows

Technical details

Node id: n-270fa0d4650ef4be · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-14a58a693c7ea773

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 0;
  1: 1;
  2: 0, 4;
  3: 0, 4, 9;
  4: 0, 0, 0, 16;
  5: 0, 0, 0, 16, 25;
  6: 0, 0, 0,  0,  0, 36;
  7: 0, 0, 0,  0,  0, 36, 49;
  8: 0, 0, 0,  0,  0,  0,  0, 64;
  9: 0, 0, 0,  0,  0,  0,  0, 64, 81;

%F Formula

The row polynomials satisfy (n^4-6*n^3+13*n^2-12*n+4)*P(n,x) = -(n^5-9*n^4+31*n^3-51*n^2+40*n-12+(2*n^5-13*n^4+30*n^3-28*n^2+9*n-2)*x)*P(n-1,x) + ((n^4-6*n^3+13*n^2-12*n+4)*x+(2*n^4-8*n^3+10*n^2-4*n+1)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 4*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 9*n^4 + 31*n^3 - 51*n^2 + 40*n - 12 + (2*n^5 - 13*n^4 + 30*n^3 - 28*n^2 + 9*n - 2)*x)*P[n - 1, x] + ((n^4 - 6*n^3 + 13*n^2 - 12*n + 4)*x + (2*n^4 - 8*n^3 + 10*n^2 - 4*n + 1)*x^2)*D[P[n - 1, x], x])/(n^4 - 6*n^3 + 13*n^2 - 12*n + 4)]];
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; v[3] = 4*x; for(n = 3, N, my(i = n + 1); v[i] = (-(n^5 - 9*n^4 + 31*n^3 - 51*n^2 + 40*n - 12 + (2*n^5 - 13*n^4 + 30*n^3 - 28*n^2 + 9*n - 2)*x)*v[i - 1] + ((n^4 - 6*n^3 + 13*n^2 - 12*n + 4)*x + (2*n^4 - 8*n^3 + 10*n^2 - 4*n + 1)*x^2)*deriv(v[i - 1], x))/(n^4 - 6*n^3 + 13*n^2 - 12*n + 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from derivative of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
  - Laguerre’s theorem on differential operators [SP14] polytool ? step ›
    
    from Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
    
    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

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.