Real-rooted atlas

A102365 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,x + n\right)\,P_{n}(x) = \left(4 - 11\,x^{2} - 12\,x^{3} + 4\,x^{4} - 8\,n - 9\,n x + 9\,n x^{2} + 11\,n x^{3} - 2\,n x^{4} + 5\,n^{2} + 9\,n^{2} x + n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2 + 2\,x + 7\,x^{2} - 4\,x^{3} - 5\,x^{4} + 2\,x^{5} + 3\,n + n x - 6\,n x^{2} - n x^{3} + 3\,n x^{4} - n^{2} - n^{2} x + n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2 + 4\,x - 11\,x^{2} + 2\,x^{3} - 5\,n - 4\,n x + 5\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2 + 2\,x + 2\,x^{2} - 2\,x^{3} + n - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2314
3149124
48687211195
56579233694511995
65995108546287134093124570
7635411413541089453490045001891353430

All 15 rows

Technical details
Node id
n-3567764ab15a8d76 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-f61857259f979bad

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:        1;
  2:        3,        1;
  3:       14,        9,        1;
  4:       86,       87,       21,        1;
  5:      657,      923,      369,       45,       1;
  6:     5995,    10854,     6287,     1340,      93,      1;
  7:    63541,   141354,   108945,    34900,    4500,    189,     1;
  8:   766701,  2027629,  1966179,   862485,  172960,  14469,   381,   1;
  9: 10372418, 31831473, 37359445, 21081569, 5941410, 799897, 45347, 765, 1;
%F Formula
The row polynomials satisfy (n-2+2*x)*P(n,x) = -(n^3-5*n^2+8*n-4+(2*n^3-9*n^2+9*n)*x+(n^3-n^2-9*n+11)*x^2+(3*n^2-11*n+12)*x^3+(2*n-4)*x^4)*P(n-2,x) - (n^2-3*n+2+(n^2-n-2)*x-(n^2-6*n+7)*x^2-(n^2-n-4)*x^3-(3*n-5)*x^4-2*x^5)*d/dx P(n-2,x) + (2*n^2-5*n+2+(2*n^2-4*n+4)*x+(5*n-11)*x^2+2*x^3)*P(n-1,x) + (n-2+2*x-(n-2)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 5*n^2 + 8*n - 4 + (2*n^3 - 9*n^2 + 9*n)*x + (n^3 - n^2 - 9*n + 11)*x^2 + (3*n^2 - 11*n + 12)*x^3 + (2*n - 4)*x^4)*P[n - 2, x] - (n^2 - 3*n + 2 + (n^2 - n - 2)*x - (n^2 - 6*n + 7)*x^2 - (n^2 - n - 4)*x^3 - (3*n - 5)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (2*n^2 - 5*n + 2 + (2*n^2 - 4*n + 4)*x + (5*n - 11)*x^2 + 2*x^3)*P[n - 1, x] + (n - 2 + 2*x - (n - 2)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 2 + 2*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 5*n^2 + 8*n - 4 + (2*n^3 - 9*n^2 + 9*n)*x + (n^3 - n^2 - 9*n + 11)*x^2 + (3*n^2 - 11*n + 12)*x^3 + (2*n - 4)*x^4)*v[i - 2] - (n^2 - 3*n + 2 + (n^2 - n - 2)*x - (n^2 - 6*n + 7)*x^2 - (n^2 - n - 4)*x^3 - (3*n - 5)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (2*n^2 - 5*n + 2 + (2*n^2 - 4*n + 4)*x + (5*n - 11)*x^2 + 2*x^3)*v[i - 1] + (n - 2 + 2*x - (n - 2)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 2 + 2*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. [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.