A271703 → D

nonnegative coefficients

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

Recurrence

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
1								0
2	1							1
3	2	2						4
4	6	12	3					21
5	24	72	36	4				136
6	120	480	360	80	5			1045
7	720	3600	3600	1200	150	6		9276
8	5040	30240	37800	16800	3150	252	7	93289

All 15 rows

Technical details

Node id: n-afa0b50cf12c7183 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-f3c222c4860e5244

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:      2,       2;
   4:      6,      12,       3;
   5:     24,      72,      36,       4;
   6:    120,     480,     360,      80,       5;
   7:    720,    3600,    3600,    1200,     150,      6;
   8:   5040,   30240,   37800,   16800,    3150,    252,     7;
   9:  40320,  282240,  423360,  235200,   58800,   7056,   392,   8;
  10: 362880, 2903040, 5080320, 3386880, 1058400, 169344, 14112, 576, 9;

%F Formula

The row polynomials satisfy (n-2)*P(n,x) = (n^2-3*n+2+(n-1)*x)*P(n-1,x) + (n-1)*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[Together[((n^2 - 3*n + 2 + (n - 1)*x)*P[n - 1, x] + (n - 1)*x*D[P[n - 1, x], x])/(n - 2)]];
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] = ((n^2 - 3*n + 2 + (n - 1)*x)*v[i - 1] + (n - 1)*x*deriv(v[i - 1], x))/(n - 2)); 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

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: the unsigned Lah numbers T(n, k) = binomial(n-1, k-1)*n!/k! if n > 0 and k > 0, T(n, 0) = 0^n an…?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.