Real-rooted atlas

A081580 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(-12 + x^{2} + 12\,n - 6\,n x - n x^{2} + 6\,n^{2} x\right)\,P_{n}(x) = \left(-204 - 240\,x - 43\,x^{2} - 2\,x^{3} + 300\,n + 318\,n x + 29\,n x^{2} - 72\,n^{2} - 66\,n^{2} x\right)\,P_{n-1}(x) + \left(-144 - 312\,x - 198\,x^{2} - 31\,x^{3} - x^{4} + 144\,n + 288\,n x + 150\,n x^{2} + 6\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
29211
34232377
4202248694523
5855148076512053225
63567793059551740185619383
71430838944387301712033202647112693

All 15 rows

Technical details
Node id
n-30eb8d3a18f0fa5e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-d4f8bf288351b326

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      0;
  1:      1;
  2:      9,      2;
  3:     42,     32,       3;
  4:    202,    248,      69,      4;
  5:    855,   1480,     765,    120,      5;
  6:   3567,   7930,    5955,   1740,    185,     6;
  7:  14308,  38944,   38730,  17120,   3320,   264,    7;
  8:  56484, 181824,  225126, 134280,  39840,  5652,  357,   8;
  9: 219069, 814416, 1208718, 912528, 371130, 80352, 8883, 464, 9;
%F Formula
The row polynomials satisfy (12*n-12+(6*n^2-6*n)*x-(n-1)*x^2)*P(n,x) = -(72*n^2-300*n+204+(66*n^2-318*n+240)*x-(29*n-43)*x^2+2*x^3)*P(n-1,x) + (144*n-144+(288*n-312)*x+(150*n-198)*x^2+(6*n-31)*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(72*n^2 - 300*n + 204 + (66*n^2 - 318*n + 240)*x - (29*n - 43)*x^2 + 2*x^3)*P[n - 1, x] + (144*n - 144 + (288*n - 312)*x + (150*n - 198)*x^2 + (6*n - 31)*x^3 - x^4)*D[P[n - 1, x], x])/(12*n - 12 + (6*n^2 - 6*n)*x - (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(72*n^2 - 300*n + 204 + (66*n^2 - 318*n + 240)*x - (29*n - 43)*x^2 + 2*x^3)*v[i - 1] + (144*n - 144 + (288*n - 312)*x + (150*n - 198)*x^2 + (6*n - 31)*x^3 - x^4)*deriv(v[i - 1], x))/(12*n - 12 + (6*n^2 - 6*n)*x - (n - 1)*x^2)); 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.