Real-rooted atlas

A060187 → D

nonnegative coefficients

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

Recurrence

\(\left(-5 + 2\,n\right)\,P_{n}(x) = \left(9 - 42\,x + 105\,x^{2} - 6\,n + 40\,n x - 142\,n x^{2} - 8\,n^{2} x + 60\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(6\,x - 36\,x^{2} + 30\,x^{3} - 4\,n x + 36\,n x^{2} - 32\,n x^{3} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-18 + 50\,x + 8\,n - 40\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-10\,x + 10\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
211
3628
42346372
5764602284768
62373364504694859600
772221086706444217236106138240
8217912131477916910388923032851307472257920

All 15 rows

Technical details
Node id
n-f11ce5a8eedcbc04 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a7d11e539b3a8882

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     0;
   2:     1;
   3:     6,       2;
   4:    23,      46,        3;
   5:    76,     460,      228,         4;
   6:   237,    3364,     5046,       948,         5;
   7:   722,   21086,    70644,     42172,      3610,         6;
   8:  2179,  121314,   779169,   1038892,    303285,     13074,        7;
   9:  6552,  663224,  7455864,  18700056,  12426440,   1989672,    45864,      8;
  10: 19673, 3512680, 65123916, 277653176, 347066470, 130247832, 12294380, 157384, 9;
%F Formula
The row polynomials satisfy (2*n-5)*P(n,x) = -(6*n-9+(8*n^2-40*n+42)*x+(8*n^3-60*n^2+142*n-105)*x^2)*P(n-2,x) - ((4*n-6)*x+(8*n^2-36*n+36)*x^2-(8*n^2-32*n+30)*x^3)*d/dx P(n-2,x) + (8*n-18+(8*n^2-40*n+50)*x)*P(n-1,x) + ((4*n-10)*x-(4*n-10)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 9 + (8*n^2 - 40*n + 42)*x + (8*n^3 - 60*n^2 + 142*n - 105)*x^2)*P[n - 2, x] - ((4*n - 6)*x + (8*n^2 - 36*n + 36)*x^2 - (8*n^2 - 32*n + 30)*x^3)*D[P[n - 2, x], x] + (8*n - 18 + (8*n^2 - 40*n + 50)*x)*P[n - 1, x] + ((4*n - 10)*x - (4*n - 10)*x^2)*D[P[n - 1, x], x])/(2*n - 5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = (-(6*n - 9 + (8*n^2 - 40*n + 42)*x + (8*n^3 - 60*n^2 + 142*n - 105)*x^2)*v[i - 2] - ((4*n - 6)*x + (8*n^2 - 36*n + 36)*x^2 - (8*n^2 - 32*n + 30)*x^3)*deriv(v[i - 2], x) + (8*n - 18 + (8*n^2 - 40*n + 50)*x)*v[i - 1] + ((4*n - 10)*x - (4*n - 10)*x^2)*deriv(v[i - 1], x))/(2*n - 5)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.