A225467 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-21 - 24\,x - 16\,x^{2}\right)\,P_{n-2}(x) + \left(-12\,x - 16\,x^{2}\right)\,P_{n-2}'(x) + \left(10 + 8\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 4\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	4							4
3	40	32						72
4	316	672	192					1180
5	2320	9920	6912	1024				20176
6	16564	127680	163200	56320	5120			368884
7	116920	1536992	3214080	1940480	399360	24576		7232408
8	821356	17842272	57433152	54010880	18887680	2580480	114688	151690508

All 15 rows

Technical details

Node id: n-f1b46de67f8e4966 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-668e26fcca6da53a

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:        4;
   3:       40,         32;
   4:      316,        672,         192;
   5:     2320,       9920,        6912,        1024;
   6:    16564,     127680,      163200,       56320,        5120;
   7:   116920,    1536992,     3214080,     1940480,      399360,      24576;
   8:   821356,   17842272,    57433152,    54010880,    18887680,    2580480,     114688;
   9:  5758240,  202835840,   968550912,  1332516864,   704471040,  160333824,   15597568,   524288;
  10: 40333924, 2277260160, 15745278720, 30483425280, 22865418240, 7710474240, 1231749120, 89653248, 2359296;

%F Formula

The row polynomials satisfy P(n,x) = -(21+24*x+16*x^2)*P(n-2,x) - (12*x+16*x^2)*d/dx P(n-2,x) + (10+8*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[-(21 + 24*x + 16*x^2)*P[n - 2, x] - (12*x + 16*x^2)*D[P[n - 2, x], x] + (10 + 8*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], 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] = 4; for(n = 3, N, my(i = n); v[i] = -(21 + 24*x + 16*x^2)*v[i - 2] - (12*x + 16*x^2)*deriv(v[i - 2], x) + (10 + 8*x)*v[i - 1] + 4*x*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, 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 read by rows, T(n, k) = 4^k*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n …?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
[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.