Real-rooted atlas

A060187 → γ → D+1

nonnegative coefficients

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

Recurrence

\(\left(-7 + 8\,x + 2\,n\right)\,P_{n}(x) = \left(15 - 234\,x + 1696\,x^{2} - 2656\,x^{3} + 768\,x^{4} - 6\,n + 124\,n x - 1288\,n x^{2} + 1440\,n x^{3} - 256\,n x^{4} - 16\,n^{2} x + 352\,n^{2} x^{2} - 192\,n^{2} x^{3} - 32\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(10\,x - 196\,x^{2} + 928\,x^{3} - 1344\,x^{4} + 512\,x^{5} - 4\,n x + 112\,n x^{2} - 480\,n x^{3} + 384\,n x^{4} - 16\,n^{2} x^{2} + 64\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-26 + 270\,x - 344\,x^{2} + 64\,x^{3} + 8\,n - 124\,n x + 80\,n x^{2} + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-14\,x + 72\,x^{2} - 64\,x^{3} + 4\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
111
211
3549
4212041
57323280385
623321849763393
77171604419376390440041
8217310172431569688640508233

All 15 rows

Technical details
Node id
n-5dadc9a0b740e8b3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-81cf55bca2009298

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
   3:     5,       4;
   4:    21,      20;
   5:    73,     232,       80;
   6:   233,    2184,      976;
   7:   717,   16044,    19376,     3904;
   8:  2173,  101724,   315696,    88640;
   9:  6545,  591184,  3945696,  2636032,   354560;
  10: 19665, 3256976, 41223648, 64935168, 12933376;
%F Formula
The row polynomials satisfy (2*n-7+8*x)*P(n,x) = -(6*n-15+(16*n^2-124*n+234)*x+(32*n^3-352*n^2+1288*n-1696)*x^2+(192*n^2-1440*n+2656)*x^3+(256*n-768)*x^4)*P(n-2,x) - ((4*n-10)*x+(16*n^2-112*n+196)*x^2-(64*n^2-480*n+928)*x^3-(384*n-1344)*x^4-512*x^5)*d/dx P(n-2,x) + (8*n-26+(16*n^2-124*n+270)*x+(80*n-344)*x^2+64*x^3)*P(n-1,x) + ((4*n-14)*x-(16*n-72)*x^2-64*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(6*n - 15 + (16*n^2 - 124*n + 234)*x + (32*n^3 - 352*n^2 + 1288*n - 1696)*x^2 + (192*n^2 - 1440*n + 2656)*x^3 + (256*n - 768)*x^4)*P[n - 2, x] - ((4*n - 10)*x + (16*n^2 - 112*n + 196)*x^2 - (64*n^2 - 480*n + 928)*x^3 - (384*n - 1344)*x^4 - 512*x^5)*D[P[n - 2, x], x] + (8*n - 26 + (16*n^2 - 124*n + 270)*x + (80*n - 344)*x^2 + 64*x^3)*P[n - 1, x] + ((4*n - 14)*x - (16*n - 72)*x^2 - 64*x^3)*D[P[n - 1, x], x])/(2*n - 7 + 8*x)]];
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; for(n = 3, N, my(i = n); v[i] = (-(6*n - 15 + (16*n^2 - 124*n + 234)*x + (32*n^3 - 352*n^2 + 1288*n - 1696)*x^2 + (192*n^2 - 1440*n + 2656)*x^3 + (256*n - 768)*x^4)*v[i - 2] - ((4*n - 10)*x + (16*n^2 - 112*n + 196)*x^2 - (64*n^2 - 480*n + 928)*x^3 - (384*n - 1344)*x^4 - 512*x^5)*deriv(v[i - 2], x) + (8*n - 26 + (16*n^2 - 124*n + 270)*x + (80*n - 344)*x^2 + 64*x^3)*v[i - 1] + ((4*n - 14)*x - (16*n - 72)*x^2 - 64*x^3)*deriv(v[i - 1], x))/(2*n - 7 + 8*x)); 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

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.