Real-rooted atlas

A008292 → γ (scale 1/4) → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + 4\,x + n\right)\,P_{n}(x) = \left(-12\,x - 32\,x^{2} - 28\,x^{3} + 24\,x^{4} - 3\,n + 8\,n x + 33\,n x^{2} + 22\,n x^{3} - 8\,n x^{4} + 4\,n^{2} + 6\,n^{2} x - 4\,n^{2} x^{2} - 6\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-8\,x^{2} - 16\,x^{3} + 8\,x^{4} + 16\,x^{5} - 2\,n x + 4\,n x^{2} + 18\,n x^{3} + 12\,n x^{4} + 2\,n^{2} x + 4\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(2 - 6\,x - 28\,x^{2} + 8\,x^{3} - 4\,n + 10\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 6\,x^{2} - 8\,x^{3} - 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3Pn(1)
111
222
38210
4401656
524015216408
6168015042723456
71344015792451227234016
8120960177408721927936378496

All 15 rows

Technical details
Node id
n-e65f1cf2dc36553a · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-e639584d673dfbeb

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:        2;
   3:        8,        2;
   4:       40,       16;
   5:      240,      152,       16;
   6:     1680,     1504,      272;
   7:    13440,    15792,     4512,     272;
   8:   120960,   177408,    72192,    7936;
   9:  1209600,  2136960,  1152000,  200704,   7936;
  10: 13305600, 27578880, 18720000, 4671488, 353792;
%F Formula
The row polynomials satisfy (n-1+4*x)*P(n,x) = -(n^3-4*n^2+3*n+(2*n^3-6*n^2-8*n+12)*x+(n^3+4*n^2-33*n+32)*x^2+(6*n^2-22*n+28)*x^3+(8*n-24)*x^4)*P(n-2,x) + ((2*n^2-2*n)*x+(4*n^2+4*n-8)*x^2+(2*n^2+18*n-16)*x^3+(12*n+8)*x^4+16*x^5)*d/dx P(n-2,x) + (2*n^2-4*n+2+(2*n^2-6)*x+(10*n-28)*x^2+8*x^3)*P(n-1,x) - ((2*n-2)*x+(2*n+6)*x^2+8*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 + 3*n + (2*n^3 - 6*n^2 - 8*n + 12)*x + (n^3 + 4*n^2 - 33*n + 32)*x^2 + (6*n^2 - 22*n + 28)*x^3 + (8*n - 24)*x^4)*P[n - 2, x] + ((2*n^2 - 2*n)*x + (4*n^2 + 4*n - 8)*x^2 + (2*n^2 + 18*n - 16)*x^3 + (12*n + 8)*x^4 + 16*x^5)*D[P[n - 2, x], x] + (2*n^2 - 4*n + 2 + (2*n^2 - 6)*x + (10*n - 28)*x^2 + 8*x^3)*P[n - 1, x] - ((2*n - 2)*x + (2*n + 6)*x^2 + 8*x^3)*D[P[n - 1, x], x])/(n - 1 + 4*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] = 2; for(n = 3, N, my(i = n); v[i] = (-(n^3 - 4*n^2 + 3*n + (2*n^3 - 6*n^2 - 8*n + 12)*x + (n^3 + 4*n^2 - 33*n + 32)*x^2 + (6*n^2 - 22*n + 28)*x^3 + (8*n - 24)*x^4)*v[i - 2] + ((2*n^2 - 2*n)*x + (4*n^2 + 4*n - 8)*x^2 + (2*n^2 + 18*n - 16)*x^3 + (12*n + 8)*x^4 + 16*x^5)*deriv(v[i - 2], x) + (2*n^2 - 4*n + 2 + (2*n^2 - 6)*x + (10*n - 28)*x^2 + 8*x^3)*v[i - 1] - ((2*n - 2)*x + (2*n + 6)*x^2 + 8*x^3)*deriv(v[i - 1], x))/(n - 1 + 4*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.