Real-rooted atlas

A094638 → θ+1 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-4\,n x^{2} + 4\,n^{2} x + 8\,n^{2} x^{2} - n^{3} - 4\,n^{3} x - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 4\,n x + 2\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
213362473
370300420192982
4446262057005440192016126
5327624640735001086407952023040312616
6272522511049580201936480218624013063683225606988024
7253296277188812937344333748805137832047179776239178245160960176974288
8260222433052464182961408576370592112990248014110817281095949568483803136928972805008620880
92928816042420470427220306081015477971224267363120385167108484059221196827383279616107259863041857945600156673800640
10358457760583777814442668477808184290059712520810487120100610053420813452375847681229038063616734109640704258792652800408748032005368118539840
1147405779208582547916870466841244834630893989121131770248752025822603270464419660980618884857157318041639228007762944210504141619206753279836160980995276800199948997906560
1267375532160134327896934412250406611872675695513841922510228143051206616436741566081268592166377536178250056961433618213965767895041319735591116800643523571744768189566083399680255058771968008044717537198720
1310242087206402230943324313622389357237062413706544819235845710329820257920170949232445122563783014844536928062650318061247488776376937362723847109442160510976046753767910367232209065937479925765696526963179520714164561510400347706769792076800
14165836268864003919971870028804295399781771072289185105973570561336278541711942404482909306283499521125886386001272896215031952117052928031380504249733816323481811240316231680289101470409288908817417139987547422727196145814701342721824846660816076802142493684531200016067861735698662400
Technical details
Node id
n-33ade304648e36a9 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-8e9f134e0951a759

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:       13,        36,         24;
  3:       70,       300,        420,         192;
  4:      446,      2620,       5700,        5440,        1920;
  5:     3276,     24640,      73500,      108640,       79520,       23040;
  6:    27252,    251104,     958020,     1936480,     2186240,     1306368,      322560;
  7:   253296,   2771888,   12937344,    33374880,    51378320,    47179776,    23917824,     5160960;
  8:  2602224,  33052464,  182961408,   576370592,  1129902480,  1411081728,  1095949568,   483803136,    92897280;
  9: 29288160, 424204704, 2722030608, 10154779712, 24267363120, 38516710848, 40592211968, 27383279616, 10725986304, 1857945600;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(4*n^3-4*n^2)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (2*n^2-1+(4*n^2-4*n)*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[Together[(-(n^3 + (4*n^3 - 4*n^2)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (4*n^2 - 4*n)*x)*P[n - 1, x])/(n - 1)]];
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] = (-(n^3 + (4*n^3 - 4*n^2)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (4*n^2 - 4*n)*x)*v[i - 1])/(n - 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.