A176230 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + n x\right)\,P_{n}(x) = \left(3 - 6\,x + x^{2} - 6\,n - n x + n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 2\,x^{2} + 2\,n x^{2}\right)\,P_{n-1}'(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	3	12	3													18
3	15	90	45	4												154
4	105	840	630	112	5											1692
5	945	9450	9450	2520	225	6										22596
6	10395	124740	155925	55440	7425	396	7									354328
7	135135	1891890	2837835	1261260	225225	18018	637	8								6370008
8	2027025	32432400	56756700	30270240	6756750	720720	38220	960	9							129003024
9	34459425	620269650	1240539300	771891120	206756550	27567540	1949220	73440	1377	10						2903507632
10	654729075	13094581500	29462808375	20951330400	6547290750	1047566520	92587950	4651200	130815	1900	11					71855678496
11	13749310575	302484832650	756212081625	604969665300	216060594750	40331311020	4277563290	268606800	10072755	219450	2541	12				1938364260768
12	316234143225	7589619437400	20871453452850	18552403069200	7454090518875	1590205977360	196767911340	14827095360	695020095	20189400	350658	3312	13			56586317169088
13	7905853580625	205552193096250	616656579288750	602953099749000	269175490959375	64602117830250	9135653026500	803134332000	45176306175	1640388750	37987950	538200	4225	14		1776830977088064
14	213458046676875	5976825306952500	19424682247595625	20719661064102000	10174833558264375	2713288948870500	431659605502125	43369253928000	2846107289025	124013389500	3589861275	67813200	798525	5292	15	59700751811048832

Technical details

Node id: n-5a42344e07f62baa · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-6ab721de33bab167

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:        1,         2;
  2:        3,        12,          3;
  3:       15,        90,         45,         4;
  4:      105,       840,        630,       112,         5;
  5:      945,      9450,       9450,      2520,       225,        6;
  6:    10395,    124740,     155925,     55440,      7425,      396,       7;
  7:   135135,   1891890,    2837835,   1261260,    225225,    18018,     637,     8;
  8:  2027025,  32432400,   56756700,  30270240,   6756750,   720720,   38220,   960,    9;
  9: 34459425, 620269650, 1240539300, 771891120, 206756550, 27567540, 1949220, 73440, 1377, 10;

%F Formula

The row polynomials satisfy -(3-n*x)*P(n,x) = -(6*n-3-(2*n^2-n-6)*x-(n+1)*x^2)*P(n-1,x) - (6*x-(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 3 - (2*n^2 - n - 6)*x - (n + 1)*x^2)*P[n - 1, x] - (6*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(-3 + n*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(6*n - 3 - (2*n^2 - n - 6)*x - (n + 1)*x^2)*v[i - 1] - (6*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(-3 + n*x)); 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, 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 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].?
  - 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

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.