A166973 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 8\,x - x^{2}\right)\,P_{n-2}(x) + \left(20\,x - 5\,x^{2}\right)\,P_{n-2}'(x) + \left(-3 + 2\,x\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	2							3
3	1	14	3						18
4	1	86	54	4					145
5	1	518	723	136	5				1383
6	1	3110	8730	3140	275	6			15262
7	1	18662	100695	61880	9700	486	7		191431
8	1	111974	1135638	1124340	281050	24276	784	8	2678071

All 15 rows

Technical details

Node id: n-c5f90b5ed3d10fcd · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-3389634cb786b734

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,       2;
   3: 1,      14,         3;
   4: 1,      86,        54,         4;
   5: 1,     518,       723,       136,         5;
   6: 1,    3110,      8730,      3140,       275,        6;
   7: 1,   18662,    100695,     61880,      9700,      486,       7;
   8: 1,  111974,   1135638,   1124340,    281050,    24276,     784,      8;
   9: 1,  671846,  12659979,  19503624,   7307475,   968436,   52626,   1184,    9;
  10: 1, 4031078, 140267538, 328937956, 177836505, 33948306, 2761248, 102768, 1701, 10;

%F Formula

The row polynomials satisfy P(n,x) = (4+8*x-x^2)*P(n-2,x) + (20*x-5*x^2)*d/dx P(n-2,x) - (3-2*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[(4 + 8*x - x^2)*P[n - 2, x] + (20*x - 5*x^2)*D[P[n - 2, x], x] - (3 - 2*x)*P[n - 1, x] + 5*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] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = (4 + 8*x - x^2)*v[i - 2] + (20*x - 5*x^2)*deriv(v[i - 2], x) - (3 - 2*x)*v[i - 1] + 5*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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m…?
  - 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.