Real-rooted atlas

A113413 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-16 - 24\,x + 4\,x^{2} + 7\,x^{3} + x^{4} + 12\,n x + 4\,n x^{2} + 6\,n x^{3} + n x^{4} + 8\,n^{2} + 4\,n^{2} x + 6\,n^{2} x^{2} + n^{2} x^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(-80 - 130\,x + 2\,x^{2} + 71\,x^{3} + 23\,x^{4} + 2\,x^{5} + 24\,n + 90\,n x + 62\,n x^{2} + 2\,n x^{3} - n x^{4} + 32\,n^{2} + 14\,n^{2} x - 14\,n^{2} x^{2} - 3\,n^{2} x^{3} - 8\,n^{3} - 6\,n^{3} x\right)\,P_{n-1}(x) + \left(-16 - 8\,x + 86\,x^{2} + 145\,x^{3} + 82\,x^{4} + 16\,x^{5} + x^{6} + 16\,n + 80\,n x + 122\,n x^{2} + 72\,n x^{3} + 15\,n x^{4} + n x^{5} + 16\,n^{2} + 32\,n^{2} x + 18\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1415
2138122
3404112194
411717284161390
53326414501422011586
6921221220499282152416350
72512723383705007166030328125114

All 15 rows

Technical details
Node id
n-206523d466339faf · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-554a0c45440599f8

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:     4,     1;
  2:    13,     8,      1;
  3:    40,    41,     12,      1;
  4:   117,   172,     84,     16,     1;
  5:   332,   641,    450,    142,    20,     1;
  6:   921,  2212,   2049,    928,   215,    24,    1;
  7:  2512,  7233,   8370,   5007,  1660,   303,   28,   1;
  8:  6761, 22728,  31648,  23696, 10370,  2700,  406,  32,  1;
  9: 18004, 69265, 112956, 101980, 55888, 19182, 4102, 524, 36, 1;
%F Formula
The row polynomials satisfy (8*n^2-16+(2*n^3+4*n^2+12*n-24)*x+(6*n^2+4*n+4)*x^2+(n^2+6*n+7)*x^3+(n+1)*x^4)*P(n,x) = -(8*n^3-32*n^2-24*n+80+(6*n^3-14*n^2-90*n+130)*x+(14*n^2-62*n-2)*x^2+(3*n^2-2*n-71)*x^3+(n-23)*x^4-2*x^5)*P(n-1,x) + (16*n^2+16*n-16+(32*n^2+80*n-8)*x+(18*n^2+122*n+86)*x^2+(2*n^2+72*n+145)*x^3+(15*n+82)*x^4+(n+16)*x^5+x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 32*n^2 - 24*n + 80 + (6*n^3 - 14*n^2 - 90*n + 130)*x + (14*n^2 - 62*n - 2)*x^2 + (3*n^2 - 2*n - 71)*x^3 + (n - 23)*x^4 - 2*x^5)*P[n - 1, x] + (16*n^2 + 16*n - 16 + (32*n^2 + 80*n - 8)*x + (18*n^2 + 122*n + 86)*x^2 + (2*n^2 + 72*n + 145)*x^3 + (15*n + 82)*x^4 + (n + 16)*x^5 + x^6)*D[P[n - 1, x], x])/(8*n^2 - 16 + (2*n^3 + 4*n^2 + 12*n - 24)*x + (6*n^2 + 4*n + 4)*x^2 + (n^2 + 6*n + 7)*x^3 + (n + 1)*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; for(n = 1, N, my(i = n + 1); v[i] = (-(8*n^3 - 32*n^2 - 24*n + 80 + (6*n^3 - 14*n^2 - 90*n + 130)*x + (14*n^2 - 62*n - 2)*x^2 + (3*n^2 - 2*n - 71)*x^3 + (n - 23)*x^4 - 2*x^5)*v[i - 1] + (16*n^2 + 16*n - 16 + (32*n^2 + 80*n - 8)*x + (18*n^2 + 122*n + 86)*x^2 + (2*n^2 + 72*n + 145)*x^3 + (15*n + 82)*x^4 + (n + 16)*x^5 + x^6)*deriv(v[i - 1], x))/(8*n^2 - 16 + (2*n^3 + 4*n^2 + 12*n - 24)*x + (6*n^2 + 4*n + 4)*x^2 + (n^2 + 6*n + 7)*x^3 + (n + 1)*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.