A133080 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2\right)\,P_{n}(x) = \left(4 - 3\,x^{2} + 4\,n x + 3\,n x^{2}\right)\,P_{n-1}(x) + \left(-8\,x - 10\,x^{2} - 3\,x^{3}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	2														4
2	4	4	1													9
3	8	16	10	2												36
4	16	32	24	8	1											81
5	32	96	112	64	18	2										324
6	64	192	240	160	60	12	1									729
7	128	512	864	800	440	144	26	2								2916
8	256	1024	1792	1792	1120	448	112	16	1							6561
9	512	2560	5632	7168	5824	3136	1120	256	34	2						26244
10	1024	5120	11520	15360	13440	8064	3360	960	180	20	1					59049
11	2048	12288	33280	53760	57600	43008	22848	8640	2280	400	42	2				236196
12	4096	24576	67584	112640	126720	101376	59136	25344	7920	1760	264	24	1			531441
13	8192	57344	184320	360448	478720	456192	321024	168960	66528	19360	4048	576	50	2		2125764
14	16384	114688	372736	745472	1025024	1025024	768768	439296	192192	64064	16016	2912	364	28	1	4782969

Technical details

Node id: n-a35a43cb83023315 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-7e6e1abd41f83340

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:   2,    2;
  2:   4,    4,    1;
  3:   8,   16,   10,    2;
  4:  16,   32,   24,    8,    1;
  5:  32,   96,  112,   64,   18,    2;
  6:  64,  192,  240,  160,   60,   12,    1;
  7: 128,  512,  864,  800,  440,  144,   26,   2;
  8: 256, 1024, 1792, 1792, 1120,  448,  112,  16,  1;
  9: 512, 2560, 5632, 7168, 5824, 3136, 1120, 256, 34, 2;

%F Formula

The row polynomials satisfy 2*P(n,x) = (4+4*n*x+(3*n-3)*x^2)*P(n-1,x) - (8*x+10*x^2+3*x^3)*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[((4 + 4*n*x + (3*n - 3)*x^2)*P[n - 1, x] - (8*x + 10*x^2 + 3*x^3)*D[P[n - 1, x], 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] = ((4 + 4*n*x + (3*n - 3)*x^2)*v[i - 1] - (8*x + 10*x^2 + 3*x^3)*deriv(v[i - 1], 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 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 basis_map_A013609 of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.?
  - Geometric basis map (elementary) 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

Related ?

Same fingerprint ?

A133080 → basis A013609

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.