Real-rooted atlas

A219512 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-2\,n + 8\,n x - 12\,n x^{2} - 6\,n^{2} x + 21\,n^{2} x^{2} - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-n x + 4\,n x^{2} - 3\,n x^{3} - 3\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-2 + 2\,x + 3\,n - 8\,n x + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2123042
333282315630
47815725490378010920
517169005261410674051975218400
636026514376056158598021798008108104979520
774193942225987316955244467808754752783014189175127807680

All 15 rows

Technical details
Node id
n-ff843986b6896ff2 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-20a4ddb391280576

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:    3;
  2:   12,      30;
  3:   33,     282,      315;
  4:   78,    1572,     5490,       3780;
  5:  171,    6900,    52614,     106740,      51975;
  6:  360,   26514,   376056,    1585980,    2179800,      810810;
  7:  741,   93942,  2259873,   16955244,   46780875,    47527830,    14189175;
  8: 1506,  315912, 12139578,  148052664,  704566350,  1399298760,  1111597830,   275675400;
  9: 3039, 1026048, 60405012, 1128729816, 8483912190, 28395642960, 43175602260, 27889029000, 5892561675;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(2*n+(6*n^2-8*n)*x+(9*n^3-21*n^2+12*n)*x^2)*P(n-2,x) - (n*x+(3*n^2-4*n)*x^2-(3*n^2-3*n)*x^3)*d/dx P(n-2,x) + (3*n-2+(6*n^2-8*n+2)*x)*P(n-1,x) + ((n-1)*x-(n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + (6*n^2 - 8*n)*x + (9*n^3 - 21*n^2 + 12*n)*x^2)*P[n - 2, x] - (n*x + (3*n^2 - 4*n)*x^2 - (3*n^2 - 3*n)*x^3)*D[P[n - 2, x], x] + (3*n - 2 + (6*n^2 - 8*n + 2)*x)*P[n - 1, x] + ((n - 1)*x - (n - 1)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + (6*n^2 - 8*n)*x + (9*n^3 - 21*n^2 + 12*n)*x^2)*v[i - 2] - (n*x + (3*n^2 - 4*n)*x^2 - (3*n^2 - 3*n)*x^3)*deriv(v[i - 2], x) + (3*n - 2 + (6*n^2 - 8*n + 2)*x)*v[i - 1] + ((n - 1)*x - (n - 1)*x^2)*deriv(v[i - 1], x))/(n - 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.