Real-rooted atlas

type_b_eulerian → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(3 - 10\,x + 15\,x^{2} - 6\,n + 24\,n x - 46\,n x^{2} - 8\,n^{2} x + 36\,n^{2} x^{2} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x - 8\,x^{2} + 6\,x^{3} - 4\,n x + 20\,n x^{2} - 16\,n x^{3} - 8\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-10 + 18\,x + 8\,n - 24\,n x + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-6\,x + 6\,x^{2} + 4\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2628
32346372
4764602284768
52373364504694859600
672221086706444217236106138240
7217912131477916910388923032851307472257920

All 15 rows

Technical details
Node id
n-3e0d99f9f1c80693 · 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 -1)
  • 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:     0;
  1:     1;
  2:     6,       2;
  3:    23,      46,        3;
  4:    76,     460,      228,         4;
  5:   237,    3364,     5046,       948,         5;
  6:   722,   21086,    70644,     42172,      3610,         6;
  7:  2179,  121314,   779169,   1038892,    303285,     13074,        7;
  8:  6552,  663224,  7455864,  18700056,  12426440,   1989672,    45864,      8;
  9: 19673, 3512680, 65123916, 277653176, 347066470, 130247832, 12294380, 157384, 9;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = -(6*n-3+(8*n^2-24*n+10)*x+(8*n^3-36*n^2+46*n-15)*x^2)*P(n-2,x) - ((4*n-2)*x+(8*n^2-20*n+8)*x^2-(8*n^2-16*n+6)*x^3)*d/dx P(n-2,x) + (8*n-10+(8*n^2-24*n+18)*x)*P(n-1,x) + ((4*n-6)*x-(4*n-6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 3 + (8*n^2 - 24*n + 10)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*P[n - 2, x] - ((4*n - 2)*x + (8*n^2 - 20*n + 8)*x^2 - (8*n^2 - 16*n + 6)*x^3)*D[P[n - 2, x], x] + (8*n - 10 + (8*n^2 - 24*n + 18)*x)*P[n - 1, x] + ((4*n - 6)*x - (4*n - 6)*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(6*n - 3 + (8*n^2 - 24*n + 10)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*v[i - 2] - ((4*n - 2)*x + (8*n^2 - 20*n + 8)*x^2 - (8*n^2 - 16*n + 6)*x^3)*deriv(v[i - 2], x) + (8*n - 10 + (8*n^2 - 24*n + 18)*x)*v[i - 1] + ((4*n - 6)*x - (4*n - 6)*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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. [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.