A075498 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-18 - 6\,x - x^{2}\right)\,P_{n-2}(x) + \left(-9\,x - 3\,x^{2}\right)\,P_{n-2}'(x) + \left(9 + 2\,x\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	1														1
2	9	2													11
3	63	36	3												102
4	405	450	90	4											949
5	2511	4860	1755	180	5										9311
6	15309	48762	28350	5040	315	6									97782
7	92583	469476	413343	113400	11970	504	7								1101283
8	557685	4410450	5664330	2252124	357210	24948	756	8							13267511
9	3352671	40809420	74587635	41334300	9244935	952560	47250	1080	9						170329860
10	20135709	373990122	956265750	719464680	218076705	31097682	2245320	83160	1485	10					2321360623
11	120873303	3406182516	12036174183	12066991200	4824711540	914743368	90168309	4811400	138105	1980	11				33464795915
12	725416965	30897389250	149543363970	197053100244	101928546720	24999264576	3231240012	232957296	9555975	218790	2574	12			508621056384
13	4353033231	279527337180	1840866451515	3155187655620	2081030216265	647298812160	106715568960	9982692720	548756208	17837820	333333	3276	13		8125528698301
14	26119793709	2524452101082	22509688423950	49782303436320	41402528462295	16090511314878	3316355602560	391493353680	27693215550	1198377180	31621590	491400	4095	14	136072376198303

Technical details

Node id: n-6cd380772964c61d · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-f58e35c5faa04e2f

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 -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:       1;
  2:       9,        2;
  3:      63,       36,        3;
  4:     405,      450,       90,        4;
  5:    2511,     4860,     1755,      180,       5;
  6:   15309,    48762,    28350,     5040,     315,      6;
  7:   92583,   469476,   413343,   113400,   11970,    504,     7;
  8:  557685,  4410450,  5664330,  2252124,  357210,  24948,   756,    8;
  9: 3352671, 40809420, 74587635, 41334300, 9244935, 952560, 47250, 1080, 9;

%F Formula

The row polynomials satisfy P(n,x) = -(18+6*x+x^2)*P(n-2,x) - (9*x+3*x^2)*d/dx P(n-2,x) + (9+2*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(18 + 6*x + x^2)*P[n - 2, x] - (9*x + 3*x^2)*D[P[n - 2, x], x] + (9 + 2*x)*P[n - 1, x] + 3*x*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(18 + 6*x + x^2)*v[i - 2] - (9*x + 3*x^2)*deriv(v[i - 2], x) + (9 + 2*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 - 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Stirling2 triangle with scaled diagonals (powers of 3).?
  - 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.