Real-rooted atlas

A145324 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
21101829
31187896193
41282136166001458
514046523205220432012366
61548856660255204816835280116568
7170153316100921202931604848483225601210392
8188247834496273245128805635630285309568326592013736880
91108379867536704445456699618590348460298886299856036288000169249680
1011305580123240163306513865670772001302797371206326184248062704004390848002250538560
111154792021252034816653738180627131951013384817604409069544924118272011081854080574801920032131010880
121180109233498006935115916957808382769015369514480239843652847292774280014323076697616290885888080951270400490309787520
131208147035536961305661520820571223359746411878194489610855421834444601786944012663180060962350433299968255173563008012204960768007964894851200
14123819383847756234384154432247825979574601588272244564254995384642255254841840862561285185623072000672448407462767545604244084087040019615115520000137245875379200
Technical details
Node id
n-b009dd14facb07f0 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4f1e76425f53aff9

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: 1,   4;
  2: 1,  10,   18;
  3: 1,  18,   78,    96;
  4: 1,  28,  213,   616,    600;
  5: 1,  40,  465,  2320,   5220,    4320;
  6: 1,  54,  885,  6660,  25520,   48168,    35280;
  7: 1,  70, 1533, 16100,  92120,  293160,   484848,   322560;
  8: 1,  88, 2478, 34496, 273245, 1288056,  3563028,  5309568,  3265920;
  9: 1, 108, 3798, 67536, 704445, 4566996, 18590348, 46029888, 62998560, 36288000;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n+1+(2*n^2+2*n)*x+(n^3+n^2)*x^2)*P(n-2,x) + (2*n+1+(2*n^2+2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (2*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*P[n - 2, x] + (2*n + 1 + (2*n^2 + 2*n)*x)*P[n - 1, x])/(n)]];
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] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 1 + (2*n^2 + 2*n)*x + (n^3 + n^2)*x^2)*v[i - 2] + (2*n + 1 + (2*n^2 + 2*n)*x)*v[i - 1])/(n)); 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.