A161198 → θ+1

nonnegative coefficients

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

Recurrence

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

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

\(P_{2}(x) = 1 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	1	4							5
3	3	16	12						31
4	15	92	108	32					247
5	105	704	1032	512	80				2433
6	945	6756	11400	7360	2000	192			28653
7	10395	78096	145668	111360	40400	6912	448		393279
8	135135	1056828	2127972	1836128	803600	186816	21952	1024	6169455

All 15 rows

Technical details

Node id: n-2b8d6f32cc61eaeb · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-3f072313d028fa7f

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,         4;
   3:        3,        16,        12;
   4:       15,        92,       108,        32;
   5:      105,       704,      1032,       512,        80;
   6:      945,      6756,     11400,      7360,      2000,       192;
   7:    10395,     78096,    145668,    111360,     40400,      6912,      448;
   8:   135135,   1056828,   2127972,   1836128,    803600,    186816,    21952,    1024;
   9:  2027025,  16392960,  35090064,  33216512,  16644320,   4730880,   765184,   65536,   2304;
  10: 34459425, 286788420, 645709968, 658254208, 365994720, 120371328, 24046848, 2863104, 186624, 5120;

%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) = 1, P(2,x) = 1 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
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] = 1; v[2] = 1 + 4*x; 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 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 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.