A094262 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2 - 8\,x - 12\,x^{2} - 8\,x^{3} - 2\,x^{4}\right)\,P_{n-2}(x) + \left(-x - 4\,x^{2} - 6\,x^{3} - 4\,x^{4} - x^{5}\right)\,P_{n-2}'(x) + \left(3 + 8\,x + 5\,x^{2}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} + x^{3}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2	2													4
2	6	24	30	12											72
3	14	122	372	524	350	90									1472
4	30	480	2670	7320	11130	9600	4410	840							36480
5	62	1682	15180	67960	176080	282372	285572	177800	62370	9450					1078528
6	126	5544	76230	511812	2007180	5021856	8375724	9457560	7161462	3490200	990990	124740			37223424
7	254	17642	355572	3405444	18948930	67894974	165889752	285144792	349280514	303933350	183883700	73638180	17567550	1891890	1471852544

All 15 rows

Technical details

Node id: n-3a888a6c3242a89e · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-8c48c943a9ebc567

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:    0;
  1:    2,      2;
  2:    6,     24,      30,        12;
  3:   14,    122,     372,       524,        350,         90;
  4:   30,    480,    2670,      7320,      11130,       9600,        4410,          840;
  5:   62,   1682,   15180,     67960,     176080,     282372,      285572,       177800,        62370,         9450;
  6:  126,   5544,   76230,    511812,    2007180,    5021856,     8375724,      9457560,      7161462,      3490200,       990990,       124740;
  7:  254,  17642,  355572,   3405444,   18948930,   67894974,   165889752,    285144792,    349280514,    303933350,    183883700,     73638180,     17567550,      1891890;
  8:  510,  54960, 1582590,  20925840,  158632110,  773670240,  2595240270,   6233402160,  10973281770,  14312934960,  13830775530,   9783626160,   4927892970,   1674953280,    344594250,    32432400;
  9: 1022, 168962, 6828060, 121839920, 1223916260, 7856937864, 34731045804, 110777363760, 262653111360, 471418764860, 646460220692, 678232731792, 540708182300, 322104222440, 138955716900, 41026986000, 7420262850, 620269650;

%F Formula

The row polynomials satisfy P(n,x) = -(2+8*x+12*x^2+8*x^3+2*x^4)*P(n-2,x) - (x+4*x^2+6*x^3+4*x^4+x^5)*d/dx P(n-2,x) + (3+8*x+5*x^2)*P(n-1,x) + (x+2*x^2+x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2 + 2*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(2 + 8*x + 12*x^2 + 8*x^3 + 2*x^4)*P[n - 2, x] - (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*D[P[n - 2, x], x] + (3 + 8*x + 5*x^2)*P[n - 1, x] + (x + 2*x^2 + x^3)*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] = 0; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 8*x + 12*x^2 + 8*x^3 + 2*x^4)*v[i - 2] - (x + 4*x^2 + 6*x^3 + 4*x^4 + x^5)*deriv(v[i - 2], x) + (3 + 8*x + 5*x^2)*v[i - 1] + (x + 2*x^2 + x^3)*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 (−∞, 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 Triangle read by rows: T(n,k) is the number of rooted trees with k nodes which are disjoint sets of labels with union {…?
  - 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.