A188440 → basis A007318 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(-3 - 2\,x + 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(-20 - 18\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1								1
2	5	2							7
3	5	2							7
4	25	20	4						49
5	25	20	4						49
6	125	150	60	8					343
7	125	150	60	8					343
8	625	1000	600	160	16				2401
9	625	1000	600	160	16				2401
10	3125	6250	5000	2000	400	32			16807
11	3125	6250	5000	2000	400	32			16807
12	15625	37500	37500	20000	6000	960	64		117649
13	15625	37500	37500	20000	6000	960	64		117649
14	78125	218750	262500	175000	70000	16800	2240	128	823543

Technical details

Node id: n-2ef7680301a255ec · revision 0 · JSON
Definition source: derived:shift_x_plus_one
Fingerprint: f6-d56a20063c562182

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:  25,   20,   4;
  5:  25,   20,   4;
  6: 125,  150,  60,   8;
  7: 125,  150,  60,   8;
  8: 625, 1000, 600, 160, 16;
  9: 625, 1000, 600, 160, 16;

%F Formula

The row polynomials satisfy P(n,x) = (4*n-3+(2*n-2)*x)*P(n-1,x) - (20+18*x+4*x^2)*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[(4*n - 3 + (2*n - 2)*x)*P[n - 1, x] - (20 + 18*x + 4*x^2)*D[P[n - 1, x], x]];
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] = (4*n - 3 + (2*n - 2)*x)*v[i - 1] - (20 + 18*x + 4*x^2)*deriv(v[i - 1], x)); 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 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 → x ↦ x+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.