A038333 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(19 + 28\,x + 14\,n x\right)\,P_{n}(x) = \left(361 + 1064\,x + 588\,x^{2} + 266\,n x + 196\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 361 + 1064\,x + 588\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	361	1064	588								2013
1	6859	30324	33516	10976							81675
2	130321	768208	1273608	834176	192080						3198393
3	2476099	18244940	40330920	39623360	18247600	3226944					122149863
4	47045881	415984632	1149431220	1505687680	1040113200	367871616	52706752				4578840981
5	893871739	9220992676	30574870452	50064115360	46111685200	24463462464	7009998016	843308032			169182303939
6	16983563041	200227269536	774563384784	1521949106944	1752244037600	1239482098176	532759849216	128182820864	13282101504		6179674231665
7	322687697779	4279857886332	18921476971152	43375549547904	59926746085920	52987859697024	30367311405312	10959631183872	2271239357184	206610467840	223618970300319

All 15 rows

Technical details

Node id: n-57d381e60984b331 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-7e918e3a6020597f

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:             361,             1064,               588;
  1:            6859,            30324,             33516,             10976;
  2:          130321,           768208,           1273608,            834176,            192080;
  3:         2476099,         18244940,          40330920,          39623360,          18247600,           3226944;
  4:        47045881,        415984632,        1149431220,        1505687680,        1040113200,         367871616,          52706752;
  5:       893871739,       9220992676,       30574870452,       50064115360,       46111685200,       24463462464,        7009998016,         843308032;
  6:     16983563041,     200227269536,      774563384784,     1521949106944,     1752244037600,     1239482098176,      532759849216,      128182820864,       13282101504;
  7:    322687697779,    4279857886332,    18921476971152,    43375549547904,    59926746085920,    52987859697024,    30367311405312,    10959631183872,     2271239357184,     206610467840;
  8:   6131066257801,   90352555378120,   449385078064860,  1177336344871680,  1897680292720800,  2013538668486912,  1442447291752320,   694109974978560,   215767738932480,   39255988889600,   3181801204736;
  9: 116490258898219, 1888368407402708, 10435720146172860, 30757912009772640, 56659311596949600, 70138263618960768, 60294296795246976, 36267246192629760, 15031819145629440, 4102250838963200, 664996451789824, 48594782035968;

%F Formula

The row polynomials satisfy (19+(14*n+28)*x)*P(n,x) = (361+(266*n+1064)*x+(196*n+588)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 361 + 1064*x + 588*x^2.

%t Mathematica

P[0, x_] := 361 + 1064*x + 588*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((361 + (266*n + 1064)*x + (196*n + 588)*x^2)*P[n - 1, x])/(19 + (14*n + 28)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 361 + 1064*x + 588*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((361 + (266*n + 1064)*x + (196*n + 588)*x^2)*v[i - 1])/(19 + (14*n + 28)*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by 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 basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*7^j.?
    
    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
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
    
    Parent-row provers of polytool polytool ? 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.