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	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	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
8	6131066257801	90352555378120	449385078064860	1177336344871680	1897680292720800	2013538668486912	1442447291752320	694109974978560	215767738932480	39255988889600	3181801204736							8029186801537869
9	116490258898219	1888368407402708	10435720146172860	30757912009772640	56659311596949600	70138263618960768	60294296795246976	36267246192629760	15031819145629440	4102250838963200	664996451789824	48594782035968						286405270244451963
10	2213314919066161	39140726989801584	237934419332741208	779200437580906880	1614790380513063600	2284503443589007872	2291183278219385088	1653786426383917056	856813691300878080	311771063761203200	75809595504039936	11079610304200704	737020860878848					10158963409259090217
11	42052983462257059	805646630540082604	5342709234107916216	19246250808248399936	44317024887414078800	70534043820810618048	80846038531455445248	68080874552804585472	42326596350263377152	19251863187254297600	6241656696499288064	1368331872568786944	182044152637075456	11112006825558016				358596245714891766615
12	799006685782884121	16484769517204767128	118430054689392142788	465409337726734034816	1178832862005214496080	2084672850703958266752	2688130781170893554496	2587073233006574247936	1876479104861676387072	1024199121561928632320	415070170317202656256	121325426034432442368	24211872300731035648	2955793815598432256	166680102383370240			12604241064499707350277
13	15181127029874798299	335582808028811330820	2596351198959750822660	11053471771009933326880	30542487788316921034800	59413176245062810602432	85124141403744962559040	92164483925859207582720	76399506412225395759360	48649458274191610035200	23658999708080551406592	8644436604953311518720	2300127868569448386560	421200618722776596480	47503829179260518400	2489089528924995584		441368598673463551274547
14	288441413567621167681	6801144909383909637952	56377911748840303577760	258481186029770748567040	773743023970695332881600	1641964143499917674830848	2587773898673846861794816	3113111457051244345016320	2903181243664565038855680	2112776473622035635814400	1198722651876081271267328	525581745581161340338176	174809718011278077378560	42681662697241361776640	7220582035247598796800	756683216793198657536	37025206742759309312	15404308993208413079668449

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.