Real-rooted atlas

A081578 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(12\,x^{2} + 8\,x^{3} + 2\,n - 3\,n x - 20\,n x^{2} - 12\,n x^{3} + 5\,n^{2} x + 8\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(25\,x^{2} + 44\,x^{3} + 16\,x^{4} - 2\,n x - 18\,n x^{2} - 24\,n x^{3} - 8\,n x^{4}\right)\,P_{n-1}'(x) + \left(2\,x^{2} + 17\,x^{3} + 16\,x^{4} + 4\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2414725
38486020136
41613631224461769
532352120816169101824300
664864398479287452328254724121
7128204811904321284408031776114801640135184

All 15 rows

Technical details
Node id
n-cc4e10049d395146 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d3dc80702d83492e

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,    14,     7;
  3:   8,    48,    60,     20;
  4:  16,   136,   312,    244,     61;
  5:  32,   352,  1208,   1616,    910,     182;
  6:  64,   864,  3984,   7928,   7452,    3282,     547;
  7: 128,  2048, 11904,  32128,  44080,   31776,   11480,   1640;
  8: 256,  4736, 33280, 114560, 211936,  220208,  128560,  39368,   4921;
  9: 512, 10752, 88704, 373248, 881472, 1225152, 1020432, 500064, 132858, 14762;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n+(5*n^2-3*n)*x+(8*n^2-20*n+12)*x^2+(4*n^2-12*n+8)*x^3)*P(n-1,x) - (2*n*x+(18*n-25)*x^2+(24*n-44)*x^3+(8*n-16)*x^4)*d/dx P(n-1,x) + (2*x^2+17*x^3+16*x^4+4*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 + (5*n^2 - 3*n)*x + (8*n^2 - 20*n + 12)*x^2 + (4*n^2 - 12*n + 8)*x^3)*P[n - 1, x] - (2*n*x + (18*n - 25)*x^2 + (24*n - 44)*x^3 + (8*n - 16)*x^4)*D[P[n - 1, x], x] + (2*x^2 + 17*x^3 + 16*x^4 + 4*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 + (5*n^2 - 3*n)*x + (8*n^2 - 20*n + 12)*x^2 + (4*n^2 - 12*n + 8)*x^3)*v[i - 1] - (2*n*x + (18*n - 25)*x^2 + (24*n - 44)*x^3 + (8*n - 16)*x^4)*deriv(v[i - 1], x) + (2*x^2 + 17*x^3 + 16*x^4 + 4*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