A161198 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 20\,x - 4\,x^{2} + 20\,n - 8\,n x - 4\,n^{2}\right)\,P_{n-2}(x) + \left(-8 + 4\,x + 4\,n\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	2							2
3	8	8						16
4	46	72	24					142
5	352	688	384	64				1488
6	3378	7600	5520	1600	160			18258
7	39048	97112	83520	32320	5760	384		258144
8	528414	1418648	1377096	642880	155680	18816	896	4142430

All 15 rows

Technical details

Node id: n-b96b9868e74ecc25 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-8ede4fc391b0c029

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 -2)
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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:         0;
   2:         2;
   3:         8,         8;
   4:        46,        72,        24;
   5:       352,       688,       384,        64;
   6:      3378,      7600,      5520,      1600,       160;
   7:     39048,     97112,     83520,     32320,      5760,      384;
   8:    528414,   1418648,   1377096,    642880,    155680,    18816,     896;
   9:   8196480,  23393376,  24912384,  13315456,   3942400,   655872,   57344,   2048;
  10: 143394210, 430473312, 493690656, 292795776, 100309440, 20611584, 2505216, 165888, 4608;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^2-20*n+25+(8*n-20)*x+4*x^2)*P(n-2,x) + (4*n-8+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 20*n + 25 + (8*n - 20)*x + 4*x^2)*P[n - 2, x] + (4*n - 8 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 20*n + 25 + (8*n - 20)*x + 4*x^2)*v[i - 2] + (4*n - 8 + 4*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - 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 ?

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
[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.