Real-rooted atlas

A085881 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
21296108216
31201440324019206720
41680268809072010752042000268800
530240604800272160048384003780000108864013063680
6665280159667208981280021288960024948000014370048032598720745113600
7172972804843238403269185920968647680015135120000130767436805932967040110702592048709140480

All 15 rows

Technical details
Node id
n-df6b590f20115496 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d42e6339eab21b9c

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,            8;
  2:          12,           96,           108;
  3:         120,         1440,          3240,           1920;
  4:        1680,        26880,         90720,         107520,          42000;
  5:       30240,       604800,       2721600,        4838400,        3780000,        1088640;
  6:      665280,     15966720,      89812800,      212889600,      249480000,      143700480,       32598720;
  7:    17297280,    484323840,    3269185920,     9686476800,    15135120000,    13076743680,     5932967040,     1107025920;
  8:   518918400,  16605388800,  130767436800,   464950886400,   908107200000,  1046139494400,   711956044800,   265686220800,    42032390400;
  9: 17643225600, 635156121600, 5716405094400, 23712495206400, 55576160640000, 80029671321600, 72619516569600, 40649991782400, 12861911462400, 1764322560000;
%F Formula
The row polynomials satisfy (1+(3*n-1)*x+n^2*x^2)*P(n,x) = (4*n-2+(12*n^2+6*n-6)*x+(4*n^3+18*n^2+2*n-6)*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[((4*n - 2 + (12*n^2 + 6*n - 6)*x + (4*n^3 + 18*n^2 + 2*n - 6)*x^2 + (4*n^3 + 6*n^2 - 2)*x^3)*P[n - 1, x])/(1 + (3*n - 1)*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] = ((4*n - 2 + (12*n^2 + 6*n - 6)*x + (4*n^3 + 18*n^2 + 2*n - 6)*x^2 + (4*n^3 + 6*n^2 - 2)*x^3)*v[i - 1])/(1 + (3*n - 1)*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.