Real-rooted atlas

A122542 → D+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-x^{2} - 4\,x^{3} - 6\,x^{4} - 4\,x^{5} - x^{6}\right)\,P_{n-4}(x) + \left(-2\,x - 10\,x^{2} - 18\,x^{3} - 14\,x^{4} - 4\,x^{5}\right)\,P_{n-3}(x) + \left(-1 - 4\,x - 7\,x^{2} - 6\,x^{3} - 2\,x^{4}\right)\,P_{n-2}(x) + \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 2\,x\)
\(P_{3}(x) = 2 + 8\,x + 7\,x^{2}\)
\(P_{4}(x) = 2 + 16\,x + 33\,x^{2} + 20\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2123
328717
4216332071
52269212457301
62361904244251581235
72463301066173013804314985
82565122208510064964305116019839

All 15 rows

Technical details
Node id
n-770d3c7f19a43d37 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-608c159058702e92

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,  2;
   3: 2,  8,    7;
   4: 2, 16,   33,   20;
   5: 2, 26,   92,  124,    57;
   6: 2, 36,  190,  424,   425,   158;
   7: 2, 46,  330, 1066,  1730,  1380,   431;
   8: 2, 56,  512, 2208,  5100,  6496,  4305,  1160;
   9: 2, 66,  736, 4012, 12232, 21932, 22984, 13032,  3089;
  10: 2, 76, 1002, 6640, 25412, 59664, 87332, 77744, 38529, 8154;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(x^2+4*x^3+6*x^4+4*x^5+x^6)*P(n-4,x) - (2*x+10*x^2+18*x^3+14*x^4+4*x^5)*P(n-3,x) - (1+4*x+7*x^2+6*x^3+2*x^4)*P(n-2,x) + (2+8*x+10*x^2+4*x^3)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 2 + 8*x + 7*x^2, P(4,x) = 2 + 16*x + 33*x^2 + 20*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[3, x_] := 2 + 8*x + 7*x^2;
P[4, x_] := 2 + 16*x + 33*x^2 + 20*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(x^2 + 4*x^3 + 6*x^4 + 4*x^5 + x^6)*P[n - 4, x] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*P[n - 3, x] - (1 + 4*x + 7*x^2 + 6*x^3 + 2*x^4)*P[n - 2, x] + (2 + 8*x + 10*x^2 + 4*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 2 + 8*x + 7*x^2; v[4] = 2 + 16*x + 33*x^2 + 20*x^3; for(n = 5, N, my(i = n); v[i] = (-(x^2 + 4*x^3 + 6*x^4 + 4*x^5 + x^6)*v[i - 4] - (2*x + 10*x^2 + 18*x^3 + 14*x^4 + 4*x^5)*v[i - 3] - (1 + 4*x + 7*x^2 + 6*x^3 + 2*x^4)*v[i - 2] + (2 + 8*x + 10*x^2 + 4*x^3)*v[i - 1])/(1 + 2*x + x^2)); 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

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.