Real-rooted atlas

A111006 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 3\,x - 6\,x^{2} - 6\,x^{3} + 5\,x^{4} - 3\,n - 9\,n x - 2\,n x^{2} + 10\,n x^{3} + n^{2} + 4\,n^{2} x + 5\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-2\,x^{2} - 5\,x^{3} + x^{4} + 9\,x^{5} + 5\,x^{6} - n x - 2\,n x^{2} + 7\,n x^{3} + 18\,n x^{4} + 10\,n x^{5} + n^{2} x + 5\,n^{2} x^{2} + 9\,n^{2} x^{3} + 5\,n^{2} x^{4}\right)\,P_{n-2}(x) + \left(-x - 4\,x^{2} - 7\,x^{3} - x^{4} + 5\,x^{5} - 2\,n x - 5\,n x^{2} + 3\,n x^{3} + 10\,n x^{4} + n^{2} x + 4\,n^{2} x^{2} + 5\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1112
21528
30411318
4021625548
500943508110
6003371099813260
70001611425018521586

All 15 rows

Technical details
Node id
n-26ecae263b1ee6f4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1149272fd8ce7a1a

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: 1, 1;
  2: 1, 5,  2;
  3: 0, 4, 11,  3;
  4: 0, 2, 16, 25,   5;
  5: 0, 0,  9, 43,  50,   8;
  6: 0, 0,  3, 37, 109,  98,  13;
  7: 0, 0,  0, 16, 114, 250, 185,   21;
  8: 0, 0,  0,  4,  71, 320, 548,  343,  34;
  9: 0, 0,  0,  0,  25, 245, 817, 1151, 625, 55;
%F Formula
The row polynomials satisfy (n^2-3*n+2+(4*n^2-9*n+3)*x+(5*n^2-2*n-6)*x^2+(10*n-6)*x^3+5*x^4)*P(n,x) = ((n^2-n)*x+(5*n^2-2*n-2)*x^2+(9*n^2+7*n-5)*x^3+(5*n^2+18*n+1)*x^4+(10*n+9)*x^5+5*x^6)*P(n-2,x) + ((n^2-2*n-1)*x+(4*n^2-5*n-4)*x^2+(5*n^2+3*n-7)*x^3+(10*n-1)*x^4+5*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(((n^2 - n)*x + (5*n^2 - 2*n - 2)*x^2 + (9*n^2 + 7*n - 5)*x^3 + (5*n^2 + 18*n + 1)*x^4 + (10*n + 9)*x^5 + 5*x^6)*P[n - 2, x] + ((n^2 - 2*n - 1)*x + (4*n^2 - 5*n - 4)*x^2 + (5*n^2 + 3*n - 7)*x^3 + (10*n - 1)*x^4 + 5*x^5)*P[n - 1, x])/(n^2 - 3*n + 2 + (4*n^2 - 9*n + 3)*x + (5*n^2 - 2*n - 6)*x^2 + (10*n - 6)*x^3 + 5*x^4)]];
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 + x; for(n = 2, N, my(i = n + 1); v[i] = (((n^2 - n)*x + (5*n^2 - 2*n - 2)*x^2 + (9*n^2 + 7*n - 5)*x^3 + (5*n^2 + 18*n + 1)*x^4 + (10*n + 9)*x^5 + 5*x^6)*v[i - 2] + ((n^2 - 2*n - 1)*x + (4*n^2 - 5*n - 4)*x^2 + (5*n^2 + 3*n - 7)*x^3 + (10*n - 1)*x^4 + 5*x^5)*v[i - 1])/(n^2 - 3*n + 2 + (4*n^2 - 9*n + 3)*x + (5*n^2 - 2*n - 6)*x^2 + (10*n - 6)*x^3 + 5*x^4)); 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.