A128813 → D+1

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	5	10	3						18
3	11	64	81	18					174
4	21	274	991	1008	180				2474
5	36	889	7006	20193	18000	2700			48824
6	57	2380	34642	213372	536553	437400	56700		1281104
7	85	5544	133818	1495828	7926453	18182484	13891500	1587600	43223312

All 15 rows

Technical details

Node id: n-7a8f239d989ba059 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-cc8071de9b089857

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 0)
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:   1;
  1:   2,     1;
  2:   5,    10,       3;
  3:  11,    64,      81,       18;
  4:  21,   274,     991,     1008,       180;
  5:  36,   889,    7006,    20193,     18000,       2700;
  6:  57,  2380,   34642,   213372,    536553,     437400,       56700;
  7:  85,  5544,  133818,  1495828,   7926453,   18182484,    13891500,     1587600;
  8: 121, 11628,  431682,  7886332,  74845449,  357991416,   768479724,   558835200,    57153600;
  9: 166, 22473, 1213872, 33700942, 516854394, 4382220861, 19453665024, 39712710780, 27776649600, 2571912000;

%F Formula

The row polynomials satisfy (4*n-4)*P(n,x) = -(4*n+4+(4*n^3-4*n)*x+(n^5-n^4-n^3+n^2)*x^2)*P(n-2,x) + (8*n+(4*n^3-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n + 4 + (4*n^3 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*P[n - 2, x] + (8*n + (4*n^3 - 4*n)*x)*P[n - 1, x])/(4*n - 4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n + 4 + (4*n^3 - 4*n)*x + (n^5 - n^4 - n^3 + n^2)*x^2)*v[i - 2] + (8*n + (4*n^3 - 4*n)*x)*v[i - 1])/(4*n - 4)); 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 Triangle of coefficients of (x+1)*(x+3)*(x+6)*...*(x+n(n+1)/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

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.