Real-rooted atlas

A038321 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(9 + 28\,x + 14\,n x\right)\,P_{n}(x) = \left(162 + 1008\,x + 1176\,x^{2} + 252\,n x + 392\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 324 + 2016\,x + 2352\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0324201623524692
158325443212700887808275080
2104976130636845722886322176307328015379088
3188956829393280137168640284497920276595200103262208832806816
4340122246348948483703553280102419251201493614080011152318464337323212844076076864
561222003213332791808933295426563226206412806273179136007025960632324250272481281079434280962292779848832
61101996057627427457433622399090237449291474468864225834448896003372461103513630601961865216155438536458243400217985024117670767448320
71983592903685554060130304518378945495042508698106593287317036144230401365846746923008165250594072166412590521453117445508353135738881057845595340805974188445116928

All 15 rows

Technical details
Node id
n-34511d415dd7604c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c7671229903995ff

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,             2016,              2352;
  1:           5832,            54432,            127008,              87808;
  2:         104976,          1306368,           4572288,            6322176,            3073280;
  3:        1889568,         29393280,         137168640,          284497920,          276595200,           103262208;
  4:       34012224,        634894848,        3703553280,        10241925120,        14936140800,         11152318464,          3373232128;
  5:      612220032,      13332791808,       93329542656,       322620641280,       627317913600,        702596063232,        425027248128,        107943428096;
  6:    11019960576,     274274574336,     2239909023744,      9291474468864,     22583444889600,      33724611035136,      30601961865216,      15543853645824,       3400217985024;
  7:   198359290368,    5554060130304,    51837894549504,    250869810659328,    731703614423040,    1365846746923008,    1652505940721664,    1259052145311744,     550835313573888,     105784559534080;
  8:  3570467226624,  111081202606080,  1166352627363840,   6450937988382720,  21951108432691200,   49170482889228288,   74362767332474880,   75543128718704640,   49575178221649920,   19041220716134400,   3258164433649664;
  9: 64268410079232, 2199407811600384, 25659757802004480, 159660715212472320, 620902781381836800, 1622625935344533504, 2944765586366005248, 3739384871575879680, 3271961762628894720, 1885080850897305600, 645116557862633472, 99522113609662464;
%F Formula
The row polynomials satisfy (9+(14*n+28)*x)*P(n,x) = (162+(252*n+1008)*x+(392*n+1176)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 2016*x + 2352*x^2.
%t Mathematica
P[0, x_] := 324 + 2016*x + 2352*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((162 + (252*n + 1008)*x + (392*n + 1176)*x^2)*P[n - 1, x])/(9 + (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] = 324 + 2016*x + 2352*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((162 + (252*n + 1008)*x + (392*n + 1176)*x^2)*v[i - 1])/(9 + (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

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.