A133080 → θ+1 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-n + n^{2}\right)\,P_{n}(x) = \left(-2 - 2\,x - n + n x + 6\,n^{2} + 3\,n^{2} x - 3\,n^{3} - 2\,n^{3} x\right)\,P_{n-1}(x) + \left(-1 - 2\,x - x^{2} - n - n x + 3\,n^{2} + 5\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	3	2							5
2	3	6	3						12
3	7	18	15	4					44
4	5	20	30	20	5				80
5	11	50	90	80	35	6			272
6	7	42	105	140	105	42	7		448
7	15	98	273	420	385	210	63	8	1472

All 15 rows

Technical details

Node id: n-9f2940070dd72ce9 · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-22c6fc872c4f0677

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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:  3,   2;
  2:  3,   6,   3;
  3:  7,  18,  15,    4;
  4:  5,  20,  30,   20,    5;
  5: 11,  50,  90,   80,   35,    6;
  6:  7,  42, 105,  140,  105,   42,    7;
  7: 15,  98, 273,  420,  385,  210,   63,   8;
  8:  9,  72, 252,  504,  630,  504,  252,  72,  9;
  9: 19, 162, 612, 1344, 1890, 1764, 1092, 432, 99, 10;

%F Formula

The row polynomials satisfy (n^2-n)*P(n,x) = -(3*n^3-6*n^2+n+2+(2*n^3-3*n^2-n+2)*x)*P(n-1,x) + (3*n^2-n-1+(5*n^2-n-2)*x+(2*n^2-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(3*n^3 - 6*n^2 + n + 2 + (2*n^3 - 3*n^2 - n + 2)*x)*P[n - 1, x] + (3*n^2 - n - 1 + (5*n^2 - n - 2)*x + (2*n^2 - 1)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n^3 - 6*n^2 + n + 2 + (2*n^3 - 3*n^2 - n + 2)*x)*v[i - 1] + (3*n^2 - n - 1 + (5*n^2 - n - 2)*x + (2*n^2 - 1)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); 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 Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from theta_plus_one of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
  - Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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.