Real-rooted atlas

A079510 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9\,x^{2} + 9\,n^{2} x^{2}\right)\,P_{n-4}(x) + \left(-4\,x^{3} + 4\,n^{2} x^{3}\right)\,P_{n-4}'(x) + \left(-9\,x^{2} + 9\,n x^{2} + 18\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-4\,x^{3} + 4\,n x^{3} + 8\,n^{2} x^{3}\right)\,P_{n-3}'(x) + \left(4\,x + 4\,n x + 9\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x^{2} - 4\,n x^{2} + 4\,n x^{3} + 4\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(4\,x + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,x^{2} - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 8\,x\)
\(P_{2}(x) = 12\,x + 54\,x^{2}\)
\(P_{3}(x) = 180\,x^{2} + 384\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1088
20125466
300180384564
400135208030005215
50003360231002592052380
6000168059500263088246960571228
7000063000951552314697625804806742008

All 15 rows

Technical details
Node id
n-2c096deb92ac6bc1 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d5c7a04bc225be4d

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,  8;
  2: 0, 12,  54;
  3: 0,  0, 180,  384;
  4: 0,  0, 135, 2080,  3000;
  5: 0,  0,   0, 3360, 23100,   25920;
  6: 0,  0,   0, 1680, 59500,  263088,   246960;
  7: 0,  0,   0,    0, 63000,  951552,  3146976,   2580480;
  8: 0,  0,   0,    0, 23625, 1587600, 14879340,  39896064,  29393280;
  9: 0,  0,   0,    0,     0, 1247400, 34560680, 235445760, 537593760, 362880000;
%F Formula
The row polynomials satisfy P(n,x) = (9*n^2-9)*x^2*P(n-4,x) + (4*n^2-4)*x^3*d/dx P(n-4,x) + (18*n^2+9*n-9)*x^2*P(n-3,x) + (8*n^2+4*n-4)*x^3*d/dx P(n-3,x) + ((4*n+4)*x+(9*n^2+9*n)*x^2)*P(n-2,x) - ((4*n+4)*x^2-(4*n^2+4*n)*x^3)*d/dx P(n-2,x) + (4*n+4)*x*P(n-1,x) - (4*n+4)*x^2*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 8*x, P(2,x) = 12*x + 54*x^2, P(3,x) = 180*x^2 + 384*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 8*x;
P[2, x_] := 12*x + 54*x^2;
P[3, x_] := 180*x^2 + 384*x^3;
P[n_, x_] := P[n, x] = Expand[(9*n^2 - 9)*x^2*P[n - 4, x] + (4*n^2 - 4)*x^3*D[P[n - 4, x], x] + (18*n^2 + 9*n - 9)*x^2*P[n - 3, x] + (8*n^2 + 4*n - 4)*x^3*D[P[n - 3, x], x] + ((4*n + 4)*x + (9*n^2 + 9*n)*x^2)*P[n - 2, x] - ((4*n + 4)*x^2 - (4*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] + (4*n + 4)*x*P[n - 1, 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] = 8*x; v[3] = 12*x + 54*x^2; v[4] = 180*x^2 + 384*x^3; for(n = 4, N, my(i = n + 1); v[i] = (9*n^2 - 9)*x^2*v[i - 4] + (4*n^2 - 4)*x^3*deriv(v[i - 4], x) + (18*n^2 + 9*n - 9)*x^2*v[i - 3] + (8*n^2 + 4*n - 4)*x^3*deriv(v[i - 3], x) + ((4*n + 4)*x + (9*n^2 + 9*n)*x^2)*v[i - 2] - ((4*n + 4)*x^2 - (4*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) + (4*n + 4)*x*v[i - 1] - (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

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.