A225466 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-10 - 12\,x - 9\,x^{2}\right)\,P_{n-2}(x) + \left(-6\,x - 9\,x^{2}\right)\,P_{n-2}'(x) + \left(7 + 6\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
1															0
2	3														3
3	21	18													39
4	117	270	81												468
5	609	2862	2106	324											5901
6	3093	26550	36045	12960	1215										79863
7	15561	230958	515970	325620	69255	4374									1161738
8	77997	1941030	6714981	6622560	2398410	336798	15309								18107085
9	390369	15996222	82599426	119575764	65607570	15370236	1530900	52488							301122975
10	1952613	130311990	980576685	2004458400	1563737805	543591972	89006526	6613488	177147						5320426626
11	9764601	1054211598	11372747490	31984724340	34100261685	16501295538	3949722000	477115920	27457785	590490					99477891447
12	48826077	8492280390	129844174581	493277130720	699647164920	452786852826	148598140383	25946917920	2406541995	110421630	1948617				1961160400059
13	244136529	68231199582	1466501182146	7425256528404	13743791043840	11574466850232	5002511213700	1184099204376	157360565835	11555298810	432592974	6377292			40634456193720
14	1220694933	547314415830	16438553401725	109819596126240	261489159726795	280966984762464	155568391890912	47938046332176	8559791223984	894304781370	53273240163	1658095920	20726199		882278315418711
15	6103499241	4385839496238	183287002290210	1603228559374260	4857139200828915	6560700670265742	4561457460159600	1779766548262560	410024851866630	57150390417120	4815769184226	237356430948	6238585899	66961566	20022206057623155

Technical details

Node id: n-a600ff91ea431e5e · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-ef7369d516d6f260

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:       3;
   3:      21,        18;
   4:     117,       270,        81;
   5:     609,      2862,      2106,        324;
   6:    3093,     26550,     36045,      12960,       1215;
   7:   15561,    230958,    515970,     325620,      69255,      4374;
   8:   77997,   1941030,   6714981,    6622560,    2398410,    336798,    15309;
   9:  390369,  15996222,  82599426,  119575764,   65607570,  15370236,  1530900,   52488;
  10: 1952613, 130311990, 980576685, 2004458400, 1563737805, 543591972, 89006526, 6613488, 177147;

%F Formula

The row polynomials satisfy P(n,x) = -(10+12*x+9*x^2)*P(n-2,x) - (6*x+9*x^2)*d/dx P(n-2,x) + (7+6*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[-(10 + 12*x + 9*x^2)*P[n - 2, x] - (6*x + 9*x^2)*D[P[n - 2, x], x] + (7 + 6*x)*P[n - 1, x] + 3*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] = 3; for(n = 3, N, my(i = n); v[i] = -(10 + 12*x + 9*x^2)*v[i - 2] - (6*x + 9*x^2)*deriv(v[i - 2], x) + (7 + 6*x)*v[i - 1] + 3*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, 3^k*S_3(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >=…?
  - 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.