Real-rooted atlas

A085881 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 2\,x + 6\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(-8 - 12\,x - 6\,x^{2} - x^{3} + 16\,n + 12\,n x + 2\,n x^{2} + 24\,n^{2} x + 18\,n^{2} x^{2} + 3\,n^{2} x^{3} + 4\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
212482787
31207208102401890
41680134402268013440262553865
530240302400680400604800236250340201888110
66652807983360224532002661120015592500449064050935578305535
71729728024216192081729648012108096009459450004086482409270261086486403743509770

All 15 rows

Technical details
Node id
n-5f5f9887c1a80b31 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e1337d43fda52a72

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:          12,           48,            27;
  3:         120,          720,           810,           240;
  4:        1680,        13440,         22680,         13440,          2625;
  5:       30240,       302400,        680400,        604800,        236250,         34020;
  6:      665280,      7983360,      22453200,      26611200,      15592500,       4490640,        509355;
  7:    17297280,    242161920,     817296480,    1210809600,     945945000,     408648240,      92702610,      8648640;
  8:   518918400,   8302694400,   32691859200,   58118860800,   56756700000,   32691859200,   11124313200,   2075673600,   164189025;
  9: 17643225600, 317578060800, 1429101273600, 2964061900800, 3473510040000, 2500927228800, 1134679946400, 317578060800, 50241841650, 3445942500;
%F Formula
The row polynomials satisfy (4+(6*n-2)*x+n^2*x^2)*P(n,x) = (16*n-8+(24*n^2+12*n-12)*x+(4*n^3+18*n^2+2*n-6)*x^2+(2*n^3+3*n^2-1)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((16*n - 8 + (24*n^2 + 12*n - 12)*x + (4*n^3 + 18*n^2 + 2*n - 6)*x^2 + (2*n^3 + 3*n^2 - 1)*x^3)*P[n - 1, x])/(4 + (6*n - 2)*x + n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n - 8 + (24*n^2 + 12*n - 12)*x + (4*n^3 + 18*n^2 + 2*n - 6)*x^2 + (2*n^3 + 3*n^2 - 1)*x^3)*v[i - 1])/(4 + (6*n - 2)*x + n^2*x^2)); 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

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. [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.