A143491 → D

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 5 + 2\,x\)

\(P_{1}(x) = 26 + 18\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	5	2								7
1	26	18	3							47
2	154	142	42	4						342
3	1044	1160	465	80	5					2754
4	8028	10208	4995	1180	135	6				24552
5	69264	97720	55272	16100	2555	210	7			241128
6	663696	1018008	644028	218596	43120	4956	308	8		2592720
7	6999840	11507472	7967292	3044664	704445	101304	8862	432	9	30334320

All 15 rows

Technical details

Node id: n-f6e156a1eb27b6e0 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-6443481be5042dac

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:          5,          2;
  1:         26,         18,          3;
  2:        154,        142,         42,         4;
  3:       1044,       1160,        465,        80,         5;
  4:       8028,      10208,       4995,      1180,       135,        6;
  5:      69264,      97720,      55272,     16100,      2555,      210,       7;
  6:     663696,    1018008,     644028,    218596,     43120,     4956,     308,      8;
  7:    6999840,   11507472,    7967292,   3044664,    704445,   101304,    8862,    432,     9;
  8:   80627040,  140581872,  104901420,  44114360,  11554725,  1959678,  215670,  14880,   585,  10;
  9: 1007441280, 1848236544, 1469689848, 669240880, 193799650, 37381806, 4874331, 425040, 23760, 770, 11;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2+4*n+4+(2*n+4)*x+x^2)*P(n-2,x) + (2*n+5+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 5 + 2*x, P(1,x) = 26 + 18*x + 3*x^2.

%t Mathematica

P[0, x_] := 5 + 2*x;
P[1, x_] := 26 + 18*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 4*n + 4 + (2*n + 4)*x + x^2)*P[n - 2, x] + (2*n + 5 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 5 + 2*x; v[2] = 26 + 18*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 4*n + 4 + (2*n + 4)*x + x^2)*v[i - 2] + (2*n + 5 + 2*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 - 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 Unsigned 2-Stirling numbers of the first kind.?
  - 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.