Real-rooted atlas

A356546 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4\,n - 2\,n x + 6\,n^{2} x + n^{3} x^{2}\right)\,P_{n}(x) = \left(-16 - 24\,x - 12\,x^{2} - 2\,x^{3} + 32\,n + 24\,n x + 4\,n x^{2} + 48\,n^{2} x + 36\,n^{2} x^{2} + 6\,n^{2} x^{3} + 8\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
2249654174
316096010803202520
411208960151208960175035910
5806480640181440161280630009072503496
659136709632199584023654401386000399168452766960492
7439296615014420756736307507202402400010378368235435221964895073264

All 15 rows

Technical details
Node id
n-2edcc7c310a160d3 · 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:        4,         8;
  2:       24,        96,         54;
  3:      160,       960,       1080,        320;
  4:     1120,      8960,      15120,       8960,       1750;
  5:     8064,     80640,     181440,     161280,      63000,       9072;
  6:    59136,    709632,    1995840,    2365440,    1386000,     399168,      45276;
  7:   439296,   6150144,   20756736,   30750720,   24024000,   10378368,    2354352,    219648;
  8:  3294720,  52715520,  207567360,  369008640,  360360000,  207567360,   70630560,  13178880,  1042470;
  9: 24893440, 448081920, 2016368640, 4182097920, 4900896000, 3528645120, 1600959360, 448081920, 70887960, 4862000;
%F Formula
The row polynomials satisfy (4*n+(6*n^2-2*n)*x+n^3*x^2)*P(n,x) = (32*n-16+(48*n^2+24*n-24)*x+(8*n^3+36*n^2+4*n-12)*x^2+(4*n^3+6*n^2-2)*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[((32*n - 16 + (48*n^2 + 24*n - 24)*x + (8*n^3 + 36*n^2 + 4*n - 12)*x^2 + (4*n^3 + 6*n^2 - 2)*x^3)*P[n - 1, x])/(4*n + (6*n^2 - 2*n)*x + n^3*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] = ((32*n - 16 + (48*n^2 + 24*n - 24)*x + (8*n^3 + 36*n^2 + 4*n - 12)*x^2 + (4*n^3 + 6*n^2 - 2)*x^3)*v[i - 1])/(4*n + (6*n^2 - 2*n)*x + n^3*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.