A111577 → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	1							1
3	5	2						7
4	21	24	3					48
5	85	210	66	4				365
6	341	1640	975	140	5			3101
7	1365	12162	12210	3120	255	6		29118
8	5461	87864	140343	56840	7980	420	7	298915

All 15 rows

Technical details

Node id: n-875f8968dcda360b · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-814590f13101ba28

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 -2)
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:     0;
   2:     1;
   3:     5,       2;
   4:    21,      24,        3;
   5:    85,     210,       66,        4;
   6:   341,    1640,      975,      140,       5;
   7:  1365,   12162,    12210,     3120,     255,      6;
   8:  5461,   87864,   140343,    56840,    7980,    420,     7;
   9: 21845,  625970,  1535226,   926044,  198730,  17556,   644,   8;
  10: 87381, 4425480, 16291215, 14085540, 4337235, 572040, 34650, 936, 9;

%F Formula

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

%t Mathematica

P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(4 + 2*x + x^2)*P[n - 2, x] - (3*x + 3*x^2)*D[P[n - 2, x], x] + (5 + 2*x)*P[n - 1, x] + 3*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] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(4 + 2*x + x^2)*v[i - 2] - (3*x + 3*x^2)*deriv(v[i - 2], x) + (5 + 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, 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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 ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.?
  - 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.