Real-rooted atlas

A059364 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(9\,n x^{2} + 4\,n^{2} x - 18\,n^{2} x^{2} - n^{3} - 4\,n^{3} x + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4\,n x^{3} - 4\,n^{2} x^{2} - 8\,n^{2} x^{3} + 4\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-1 - 4\,n x + 2\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
26281852
32418426196565
41201304296123366007321
5720102243243640336218504320109886
6504088848365076628336531100219024352801872704
74032085190443199649569856111212257105680236905232256035700561
836288089573765421394814756819221971422519206763298694428276572163265920752501817
93628800102637440723526344234310508842723199504768172676333977277214367352323476921763628800017373878478
1039916800127416384010268132184386368465288360653645011397130671610163762858259425609472220082942164691664000439084800435999183588
11479001600170463052801547531412486644411877121667349237100269198321796029260552740932173370385536109141543746035517803760067715065440574801920011815534310229
1262270208002445940454402472041078784119391357868803413381235190063807576650568820077777898897411525277734447196463417308207864320808006037480956672104176253952080951270400343869507766305
13871782912003748121717760417629509320962242795170593287203870685976001532291040335520227714850923655224300721331927681880178006194550104848415335960041149531002439210799099250278417027430975360122049607680010696172908496310
141307674368000611038100736007445100980908804403626201310976157119204646368003750736890796041663355884684941800780903690239313287126051070292760848224166111164000239337512082775048475852862824192202976346644697629477649793536019615115520000354114526830409440
Technical details
Node id
n-8225d29aced81e4e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-34fef7dd77edd667

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:       2,         4;
  2:       6,        28,        18;
  3:      24,       184,       261,         96;
  4:     120,      1304,      2961,       2336,        600;
  5:     720,     10224,     32436,      40336,      21850,       4320;
  6:    5040,     88848,    365076,     628336,     531100,     219024,      35280;
  7:   40320,    851904,   4319964,    9569856,   11121225,    7105680,    2369052,     322560;
  8:  362880,   8957376,  54213948,  147568192,  219714225,  192067632,   98694428,   27657216,   3265920;
  9: 3628800, 102637440, 723526344, 2343105088, 4272319950, 4768172676, 3339772772, 1436735232, 347692176, 36288000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(4*n^3-4*n^2)*x-(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + ((4*n^3-4*n^2)*x^2+(4*n^3-8*n^2+4*n)*x^3)*d/dx P(n-2,x) + (2*n^2-1+(4*n^2-4*n)*x)*P(n-1,x) - (4*n^2-4*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (4*n^3 - 4*n^2)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + ((4*n^3 - 4*n^2)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 1 + (4*n^2 - 4*n)*x)*P[n - 1, x] - (4*n^2 - 4*n)*x^2*D[P[n - 1, x], 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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (4*n^3 - 4*n^2)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + ((4*n^3 - 4*n^2)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 1 + (4*n^2 - 4*n)*x)*v[i - 1] - (4*n^2 - 4*n)*x^2*deriv(v[i - 1], x))/(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.