Real-rooted atlas

A081577 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(9\,x^{2} + 6\,x^{3} + 2\,n - 2\,n x - 15\,n x^{2} - 9\,n x^{3} + 4\,n^{2} x + 6\,n^{2} x^{2} + 3\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(19\,x^{2} + 33\,x^{3} + 12\,x^{4} - 2\,n x - 14\,n x^{2} - 18\,n x^{3} - 6\,n x^{4}\right)\,P_{n-1}'(x) + \left(2\,x^{2} + 13\,x^{3} + 12\,x^{4} + 3\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2412622
38404816112
41611223217644580
53228886410886001202992
664704278450564544196832815448
712816648192197122528017600627289679744
82563840226566835211577611404864704195842448411664
95128704599042181124634886001924784642288646019266882125120
1010241945615308865433616819202727168284505619015687858561827201827210970464
1120484300838092818718725664768111083521447219212607488724723226356485491204992056632576
12409694208927744515686417996800415825926543360070993920530058242671232086721281636608136384292353088
13819220480022200321377894454556672145561600269795328352960512327373824213640192958059522808524848439043726081509207808
14163844423685234688358973441591541764825374721033037824159041126417755115521436039168831753216335895552897489921425177610179847790949760
Technical details
Node id
n-df560e0f56fd533e · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ea902a17f4c48b3d

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:   4,   12,     6;
  3:   8,   40,    48,     16;
  4:  16,  112,   232,    176,     44;
  5:  32,  288,   864,   1088,    600,    120;
  6:  64,  704,  2784,   5056,   4544,   1968,    328;
  7: 128, 1664,  8192,  19712,  25280,  17600,   6272,    896;
  8: 256, 3840, 22656,  68352, 115776, 114048,  64704,  19584,  2448;
  9: 512, 8704, 59904, 218112, 463488, 600192, 478464, 228864, 60192, 6688;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n+(4*n^2-2*n)*x+(6*n^2-15*n+9)*x^2+(3*n^2-9*n+6)*x^3)*P(n-1,x) - (2*n*x+(14*n-19)*x^2+(18*n-33)*x^3+(6*n-12)*x^4)*d/dx P(n-1,x) + (2*x^2+13*x^3+12*x^4+3*x^5)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + (4*n^2 - 2*n)*x + (6*n^2 - 15*n + 9)*x^2 + (3*n^2 - 9*n + 6)*x^3)*P[n - 1, x] - (2*n*x + (14*n - 19)*x^2 + (18*n - 33)*x^3 + (6*n - 12)*x^4)*D[P[n - 1, x], x] + (2*x^2 + 13*x^3 + 12*x^4 + 3*x^5)*D[P[n - 1, x], {x, 2}])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + (4*n^2 - 2*n)*x + (6*n^2 - 15*n + 9)*x^2 + (3*n^2 - 9*n + 6)*x^3)*v[i - 1] - (2*n*x + (14*n - 19)*x^2 + (18*n - 33)*x^3 + (6*n - 12)*x^4)*deriv(v[i - 1], x) + (2*x^2 + 13*x^3 + 12*x^4 + 3*x^5)*deriv(deriv(v[i - 1], x), x))/(n)); 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

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