A271703 → θ+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	0	2							2
3	0	4	3						7
4	0	12	18	4					34
5	0	48	108	48	5				209
6	0	240	720	480	100	6			1546
7	0	1440	5400	4800	1500	180	7		13327
8	0	10080	45360	50400	21000	3780	294	8	130922

All 15 rows

Technical details

Node id: n-2f500c4b2eb2b60e · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-06d4fc90c866b1f9

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: 0,      2;
   3: 0,      4,       3;
   4: 0,     12,      18,       4;
   5: 0,     48,     108,      48,       5;
   6: 0,    240,     720,     480,     100,       6;
   7: 0,   1440,    5400,    4800,    1500,     180,      7;
   8: 0,  10080,   45360,   50400,   21000,    3780,    294,     8;
   9: 0,  80640,  423360,  564480,  294000,   70560,   8232,   448,   9;
  10: 0, 725760, 4354560, 6773760, 4233600, 1270080, 197568, 16128, 648, 10;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n - 4 - (n^2 - 3*n - 2)*x - n*x^2)*P[n - 1, x] - (2*x - n*x^2)*D[P[n - 1, x], x])/(-2 + (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; for(n = 2, N, my(i = n); v[i] = (-(2*n - 4 - (n^2 - 3*n - 2)*x - n*x^2)*v[i - 1] - (2*x - n*x^2)*deriv(v[i - 1], x))/(-2 + (n - 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 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

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] 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

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.