Real-rooted atlas

A225470 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 + 12\,x - 4\,x^{2} + 18\,n - 12\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 4\,x + 6\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
218361266
316239621632806
41944540037809608012164
5291608877672900244803600192219108
652488017146081578528635040126000120964484591600
7110224803810628838292912175452484233600556416376321024109795600
8264539520958640832103334875252320038414546952023514624220147211059223042951028000
9714256704026941460544307763388001688200704052356780009840216961143072008017920311040512088084714400
10214277011200836814084480100411454563258853044800019927535760042086278080572526662450181120027371520844800112642891353030400
1170711413696002847197283264035646222259296220991435723528047412920800186710803488028713511411229646093312203233536088704000223027224576103521905491200
12254561089305600105327558745344013686799198325768906257612961283449547240796808652968026560014693449522752172356819148813986778291277096448002754385925750784532484015191638617600
13992788248291840042095992267906560565383436358307843838421781010252815679798642347840420254886814963277494711867372810080418694860893328719644166114934425602770332364882523750414536704114688167714507921497600
14416971064282572800180774320518374912025008984095086126081762906064386521600754512087503865600214138569203919360423537263433123846005083551667200618790043139840464224400640002508825166848951035904002398556160361267202457607506196028811110400
Technical details
Node id
n-6ce02af0438033a9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-09d18f264aa0068a

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:          3,           4;
  2:         18,          36,          12;
  3:        162,         396,         216,          32;
  4:       1944,        5400,        3780,         960,         80;
  5:      29160,       88776,       72900,       24480,       3600,       192;
  6:     524880,     1714608,     1578528,      635040,     126000,     12096,       448;
  7:   11022480,    38106288,    38292912,    17545248,    4233600,    556416,     37632,    1024;
  8:  264539520,   958640832,  1033348752,   523200384,  145469520,  23514624,   2201472,  110592,   2304;
  9: 7142567040, 26941460544, 30776338800, 16882007040, 5235678000, 984021696, 114307200, 8017920, 311040, 5120;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(12*n-12)*x+4*x^2)*P(n-2,x) + (6*n-3+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (12*n - 12)*x + 4*x^2)*P[n - 2, x] + (6*n - 3 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (12*n - 12)*x + 4*x^2)*v[i - 2] + (6*n - 3 + 4*x)*v[i - 1]); 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.