Real-rooted atlas

A079510 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(x^{2} - n^{2} x^{2}\right)\,P_{n-4}(x) + \left(x^{2} - n x^{2} - 2\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(2\,x + 2\,n x - n x^{2} - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(2\,x + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)
\(P_{2}(x) = 3 + 15\,x + 6\,x^{2}\)
\(P_{3}(x) = 40\,x + 92\,x^{2} + 24\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2315624
30409224156
40304056101201165
500630390645247209780
60031596253892037548504091448
70001008013468041177634646440320943320

All 15 rows

Technical details
Node id
n-bd61c757f258cd6d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-c2ef14fd77a5eac7

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: 2,  2;
  2: 3, 15,   6;
  3: 0, 40,  92,    24;
  4: 0, 30, 405,   610,    120;
  5: 0,  0, 630,  3906,   4524,     720;
  6: 0,  0, 315,  9625,  38920,   37548,     5040;
  7: 0,  0,   0, 10080, 134680,  411776,   346464,    40320;
  8: 0,  0,   0,  3780, 221445, 1866060,  4667292,  3526416,   362880;
  9: 0,  0,   0,     0, 173250, 4266570, 26457200, 56774520, 39296160, 3628800;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2-1)*x^2*P(n-4,x) - (2*n^2+n-1)*x^2*P(n-3,x) + ((2*n+2)*x-(n^2+n)*x^2)*P(n-2,x) + (2*n+2)*x*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 3 + 15*x + 6*x^2, P(3,x) = 40*x + 92*x^2 + 24*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[2, x_] := 3 + 15*x + 6*x^2;
P[3, x_] := 40*x + 92*x^2 + 24*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 1)*x^2*P[n - 4, x] - (2*n^2 + n - 1)*x^2*P[n - 3, x] + ((2*n + 2)*x - (n^2 + n)*x^2)*P[n - 2, x] + (2*n + 2)*x*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 3 + 15*x + 6*x^2; v[4] = 40*x + 92*x^2 + 24*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(n^2 - 1)*x^2*v[i - 4] - (2*n^2 + n - 1)*x^2*v[i - 3] + ((2*n + 2)*x - (n^2 + n)*x^2)*v[i - 2] + (2*n + 2)*x*v[i - 1]); 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.