Real-rooted atlas

A038321 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(9 + 14\,x + 7\,n x\right)\,P_{n}(x) = \left(162 + 504\,x + 294\,x^{2} + 126\,n x + 98\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 324 + 1008\,x + 588\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
032410085881920
1583227216317521097675776
210497665318411430727902721920802883584
31889568146966403429216035562240172872003226944106954752
4340122243174474249258883201280240640933508800348509952527067523892314112
56122200326666395904233323856644032758016039207369600219561269766641050752843308032139586437120
611019960576137137287168559977255936116143430860814114653056001053894094848478155654144121436356608132821015044947802324992
71983592903682777030065152129594736373763135872633241645731475901440426827108413442582040532377698363448852482151700443648206610467840173722837188608
83570467226624555406013030402915881568409608063672485478401371944277043200153657759028838411619182395699205901806931148801936530399283203718988421120031818012047366051711999279104
96426841007923210997039058001926414939450501120199575894015590403880642383636480050707060479516672460119622869688322921394430918656012781100635269120368179853690880062999663853772848594782035968209417382672728064
10115683138142617621594185786621952138562692130824192478982145637416960104777344358184960015646750090822287361656430642330877952126204239415685939269017943430453248026508949465743360068039636962074624104964729197690887370208608788487205759403792793600
11208229648656711684210866228391280642947606359873896448112081822079155568642724210953312808960045766744015655190528553721100436322058244921965337211751628832300397525452120064155077354374598656005307091683041820672122808733161298329617246288144565043211112006825558016246725201985865252864
1237481336758208102481626022273430978566189973355735182540825676926512679275724868650116023482785792012814688324383453347841744221466374414483456177190752139623058636813566166960689890426887815898660479772262403343467760316347023361031593358554905968642173032306215195443228002257200406200321666801023833702408411715297611555536896
136746640616477458432157421614384474030080128560985080653791232057773084653528370380801685048302394577469440034599658475835324039168523266439912324345036805980187884712278228992052326643991232434503680351715439721589751808001805472590570827392614469632551702456152883201955729075593675898880378030472205483704320450036276435099648002489089528924995584285703172213613535428608
141214395310965942517763022494996181901377536264468312165916370534401279896029247397743820804044115925746985926656009058819673673248475709441507007346947494113705984191366012310792903327744018837591836843676421324801447057809140254978867200866626843473997148454912401083497806147440607232140812493442744664719360362909253317264356147206480522380665434931200716857784330398728192370252067427593093129671406556917033397649408
Technical details
Node id
n-33e9f5dc77541f85 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-69e7a304998295a4

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:            324,             1008,              588;
  1:           5832,            27216,            31752,             10976;
  2:         104976,           653184,          1143072,            790272,            192080;
  3:        1889568,         14696640,         34292160,          35562240,          17287200,           3226944;
  4:       34012224,        317447424,        925888320,        1280240640,         933508800,         348509952,          52706752;
  5:      612220032,       6666395904,      23332385664,       40327580160,       39207369600,       21956126976,        6641050752,         843308032;
  6:    11019960576,     137137287168,     559977255936,     1161434308608,     1411465305600,     1053894094848,      478155654144,      121436356608,       13282101504;
  7:   198359290368,    2777030065152,   12959473637376,    31358726332416,    45731475901440,    42682710841344,    25820405323776,     9836344885248,     2151700443648,     206610467840;
  8:  3570467226624,   55540601303040,  291588156840960,   806367248547840,  1371944277043200,  1536577590288384,  1161918239569920,   590180693114880,   193653039928320,   37189884211200,   3181801204736;
  9: 64268410079232, 1099703905800192, 6414939450501120, 19957589401559040, 38806423836364800, 50707060479516672, 46011962286968832, 29213944309186560, 12781100635269120, 3681798536908800, 629996638537728, 48594782035968;
%F Formula
The row polynomials satisfy (9+(7*n+14)*x)*P(n,x) = (162+(126*n+504)*x+(98*n+294)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1008*x + 588*x^2.
%t Mathematica
P[0, x_] := 324 + 1008*x + 588*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((162 + (126*n + 504)*x + (98*n + 294)*x^2)*P[n - 1, x])/(9 + (7*n + 14)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 324 + 1008*x + 588*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((162 + (126*n + 504)*x + (98*n + 294)*x^2)*v[i - 1])/(9 + (7*n + 14)*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

Proof chains ? JSON

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.