Real-rooted atlas

A156992 → basis A008277 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-2\,n - 8\,n x + 9\,n x^{2} - 2\,n^{2} - 8\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-4\,n x + 8\,n x^{2} + 4\,n x^{3} - 4\,n^{2} x + 8\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-n x^{2} + 4\,n x^{3} - n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-2}''(x) + \left(3 + 4\,x + 3\,n + 4\,n x\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2} + 2\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
22412054198
3192182419443844344
419203120059400268803000122400
52304060768018468001440000360000259204303440
6322560134064006128136073382400308700004898880246960184408560
751609603320755202202318720380685312023990400006299596806914880025804809447137280

All 15 rows

Technical details
Node id
n-2bc5bb45e09155dc · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-fefd2b9ccc27c8c7

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,          120,            54;
  3:        192,         1824,          1944,            384;
  4:       1920,        31200,         59400,          26880,           3000;
  5:      23040,       607680,       1846800,        1440000,         360000,         25920;
  6:     322560,     13406400,      61281360,       73382400,       30870000,       4898880,        246960;
  7:    5160960,    332075520,    2202318720,     3806853120,     2399040000,     629959680,      69148800,      2580480;
  8:   92897280,   9151833600,   86008003200,   206545489920,   183082032000,   70779018240,   12695719680,   1021870080,    29393280;
  9: 1857945600, 278270899200, 3646236384000, 11856305664000, 14212195200000, 7590912537600, 1979038656000, 257789952000, 15872371200, 362880000;
%F Formula
The row polynomials satisfy P(n,x) = -(2*n^2+2*n+(8*n^2+8*n)*x-(9*n^2+9*n)*x^2)*P(n-2,x) - ((4*n^2+4*n)*x-(8*n^2+8*n)*x^2-(4*n^2+4*n)*x^3)*d/dx P(n-2,x) - ((n^2+n)*x^2-(4*n^2+4*n)*x^3)*d^2/dx^2 P(n-2,x) + (3*n+3+(4*n+4)*x)*P(n-1,x) + ((2*n+2)*x-(4*n+4)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(2*n^2 + 2*n + (8*n^2 + 8*n)*x - (9*n^2 + 9*n)*x^2)*P[n - 2, x] - ((4*n^2 + 4*n)*x - (8*n^2 + 8*n)*x^2 - (4*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] - ((n^2 + n)*x^2 - (4*n^2 + 4*n)*x^3)*D[P[n - 2, x], {x, 2}] + (3*n + 3 + (4*n + 4)*x)*P[n - 1, x] + ((2*n + 2)*x - (4*n + 4)*x^2)*D[P[n - 1, x], x]];
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] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(2*n^2 + 2*n + (8*n^2 + 8*n)*x - (9*n^2 + 9*n)*x^2)*v[i - 2] - ((4*n^2 + 4*n)*x - (8*n^2 + 8*n)*x^2 - (4*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) - ((n^2 + n)*x^2 - (4*n^2 + 4*n)*x^3)*deriv(deriv(v[i - 2], x), x) + (3*n + 3 + (4*n + 4)*x)*v[i - 1] + ((2*n + 2)*x - (4*n + 4)*x^2)*deriv(v[i - 1], x)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.