Real-rooted atlas

A088996 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-4\,n - 8\,n x + 9\,n x^{2} + 4\,n^{2} + 12\,n^{2} x - 18\,n^{2} x^{2} - n^{3} - 4\,n^{3} x + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(8\,n x^{2} + 4\,n x^{3} - 12\,n^{2} x^{2} - 8\,n^{2} x^{3} + 4\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(1 - 4\,n - 4\,n x + 2\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(4\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1044
2041822
3086396167
40242617366001621
5096131452648150432019144
60480786640336901009201635280266078
70288054756339904981775146030410883883225604250567

All 15 rows

Technical details
Node id
n-f42ce8ef0e96f088 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-540b808a017f945e

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: 0,      4;
  2: 0,      4,      18;
  3: 0,      8,      63,       96;
  4: 0,     24,     261,      736,       600;
  5: 0,     96,    1314,     5264,      8150,      4320;
  6: 0,    480,    7866,    40336,     90100,     92016,     35280;
  7: 0,   2880,   54756,   339904,    981775,   1460304,   1088388,    322560;
  8: 0,  20160,  435132,  3158080,  11121225,  21532176,  23519804,  13630464,   3265920;
  9: 0, 161280, 3889296, 32226752, 133380300, 316388484, 451927588, 385521408, 181386864, 36288000;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3-4*n^2+4*n+(4*n^3-12*n^2+8*n)*x-(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + ((4*n^3-12*n^2+8*n)*x^2+(4*n^3-8*n^2+4*n)*x^3)*d/dx P(n-2,x) + (2*n^2-4*n+1+(4*n^2-4*n)*x)*P(n-1,x) - (4*n^2-4*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 + 4*n + (4*n^3 - 12*n^2 + 8*n)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + ((4*n^3 - 12*n^2 + 8*n)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - 4*n + 1 + (4*n^2 - 4*n)*x)*P[n - 1, x] - (4*n^2 - 4*n)*x^2*D[P[n - 1, x], x])/(n - 1)]];
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*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 4*n^2 + 4*n + (4*n^3 - 12*n^2 + 8*n)*x - (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + ((4*n^3 - 12*n^2 + 8*n)*x^2 + (4*n^3 - 8*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - 4*n + 1 + (4*n^2 - 4*n)*x)*v[i - 1] - (4*n^2 - 4*n)*x^2*deriv(v[i - 1], x))/(n - 1)); 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

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.