Real-rooted atlas

A008292 → γ → D+1

nonnegative coefficients

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

Recurrence

\(\left(-7 + 8\,x + 2\,n\right)\,P_{n}(x) = \left(10 - 121\,x + 518\,x^{2} - 680\,x^{3} + 192\,x^{4} - 4\,n + 62\,n x - 342\,n x^{2} + 360\,n x^{3} - 64\,n x^{4} - 8\,n^{2} x + 88\,n^{2} x^{2} - 48\,n^{2} x^{3} - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(5\,x - 63\,x^{2} + 248\,x^{3} - 336\,x^{4} + 128\,x^{5} - 2\,n x + 32\,n x^{2} - 120\,n x^{3} + 96\,n x^{4} - 4\,n^{2} x^{2} + 16\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-19 + 143\,x - 172\,x^{2} + 32\,x^{3} + 6\,n - 62\,n x + 40\,n x^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(-7\,x + 36\,x^{2} - 32\,x^{3} + 2\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
111
211
3325
49817
523541693
653324136513
7115155415362723477
8241638414976396825569

All 15 rows

Technical details
Node id
n-a402f9522bdbeaf3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7c7aeadd45ef900d

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:    3,     2;
   4:    9,     8;
   5:   23,    54,     16;
   6:   53,   324,    136;
   7:  115,  1554,   1536,    272;
   8:  241,  6384,  14976,   3968;
   9:  495, 23726, 114528,  66048,   7936;
  10: 1005, 82556, 731208, 937728, 176896;
%F Formula
The row polynomials satisfy (2*n-7+8*x)*P(n,x) = -(4*n-10+(8*n^2-62*n+121)*x+(8*n^3-88*n^2+342*n-518)*x^2+(48*n^2-360*n+680)*x^3+(64*n-192)*x^4)*P(n-2,x) - ((2*n-5)*x+(4*n^2-32*n+63)*x^2-(16*n^2-120*n+248)*x^3-(96*n-336)*x^4-128*x^5)*d/dx P(n-2,x) + (6*n-19+(8*n^2-62*n+143)*x+(40*n-172)*x^2+32*x^3)*P(n-1,x) + ((2*n-7)*x-(8*n-36)*x^2-32*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[(-(4*n - 10 + (8*n^2 - 62*n + 121)*x + (8*n^3 - 88*n^2 + 342*n - 518)*x^2 + (48*n^2 - 360*n + 680)*x^3 + (64*n - 192)*x^4)*P[n - 2, x] - ((2*n - 5)*x + (4*n^2 - 32*n + 63)*x^2 - (16*n^2 - 120*n + 248)*x^3 - (96*n - 336)*x^4 - 128*x^5)*D[P[n - 2, x], x] + (6*n - 19 + (8*n^2 - 62*n + 143)*x + (40*n - 172)*x^2 + 32*x^3)*P[n - 1, x] + ((2*n - 7)*x - (8*n - 36)*x^2 - 32*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] = (-(4*n - 10 + (8*n^2 - 62*n + 121)*x + (8*n^3 - 88*n^2 + 342*n - 518)*x^2 + (48*n^2 - 360*n + 680)*x^3 + (64*n - 192)*x^4)*v[i - 2] - ((2*n - 5)*x + (4*n^2 - 32*n + 63)*x^2 - (16*n^2 - 120*n + 248)*x^3 - (96*n - 336)*x^4 - 128*x^5)*deriv(v[i - 2], x) + (6*n - 19 + (8*n^2 - 62*n + 143)*x + (40*n - 172)*x^2 + 32*x^3)*v[i - 1] + ((2*n - 7)*x - (8*n - 36)*x^2 - 32*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.