A286724 → θ+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	2	2							4
3	8	16	3						27
4	48	144	54	4					250
5	384	1536	864	128	5				2917
6	3840	19200	14400	3200	250	6			40896
7	46080	276480	259200	76800	9000	432	7		667999
8	645120	4515840	5080320	1881600	294000	21168	686	8	12438742

All 15 rows

Technical details

Node id: n-c047d0120ec62185 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-baf7e4e24f192751

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:         2,          2;
   3:         8,         16,          3;
   4:        48,        144,         54,          4;
   5:       384,       1536,        864,        128,         5;
   6:      3840,      19200,      14400,       3200,       250,        6;
   7:     46080,     276480,     259200,      76800,      9000,      432,       7;
   8:    645120,    4515840,    5080320,    1881600,    294000,    21168,     686,     8;
   9:  10321920,   82575360,  108380160,   48168960,   9408000,   903168,   43904,  1024,    9;
  10: 185794560, 1672151040, 2508226560, 1300561920, 304819200, 36578304, 2370816, 82944, 1458, 10;

%F Formula

The row polynomials satisfy -(2-(n-1)*x)*P(n,x) = -(4*n-4-(2*n^2-4*n-2)*x-n*x^2)*P(n-1,x) - (4*x-2*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[(-(4*n - 4 - (2*n^2 - 4*n - 2)*x - n*x^2)*P[n - 1, x] - (4*x - 2*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] = (-(4*n - 4 - (2*n^2 - 4*n - 2)*x - n*x^2)*v[i - 1] - (4*x - 2*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. A generalization of unsigned Lah numbers, called L[2,1].?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Related ?

Same fingerprint ?

A079621 → (θ+1)²

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.