Real-rooted atlas

basis-map coefficients of A026729 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 12\,x + 9\,x^{2} + 9\,n x - 18\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1 - 9\,x^{2} + 9\,n x + 27\,n x^{2} + 9\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
211213
312481106
4136243432712
5148486172820254288
6160810432010125874824064
717212158640303755248835721128512
818417011512070875183708250047139968661504
9196226824192141750489888100018811197445314413309568
1011082916362882551501102248300056450388484782969196830016187392
11112036455184042525022044967501410167961602391484519683000714492977725696
12113244557128066825040415761650310246189440876877651082565007859421925509168367525888
1311445346950401002375692841633006204110854656263063295433026000471565314306110016898135291715470336
14115663181235521447875112586766129723624018508868396456714073345002043449694198971510411675758773124873087918845952
1511687371157248202702517513496107270163480370176159591732339405366007152073929928533715281730311394374822312107616802536205232128
Technical details
Node id
n-3ab1e10847171253 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-10c18c8945c6e24a

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,  12;
   3: 1,  24,   81;
   4: 1,  36,  243,   432;
   5: 1,  48,  486,  1728,   2025;
   6: 1,  60,  810,  4320,  10125,    8748;
   7: 1,  72, 1215,  8640,  30375,   52488,   35721;
   8: 1,  84, 1701, 15120,  70875,  183708,  250047,  139968;
   9: 1,  96, 2268, 24192, 141750,  489888, 1000188, 1119744,  531441;
  10: 1, 108, 2916, 36288, 255150, 1102248, 3000564, 5038848, 4782969, 1968300;
%F Formula
The row polynomials satisfy (1+(9*n-12)*x+(9*n^2-18*n+9)*x^2)*P(n,x) = (1+9*n*x+(9*n^2+27*n-9)*x^2+27*n^2*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 9*n*x + (9*n^2 + 27*n - 9)*x^2 + 27*n^2*x^3)*P[n - 1, x])/(1 + (9*n - 12)*x + (9*n^2 - 18*n + 9)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 + 9*n*x + (9*n^2 + 27*n - 9)*x^2 + 27*n^2*x^3)*v[i - 1])/(1 + (9*n - 12)*x + (9*n^2 - 18*n + 9)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.