Real-rooted atlas

A081579 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(15\,x^{2} + 10\,x^{3} + 2\,n - 4\,n x - 25\,n x^{2} - 15\,n x^{3} + 6\,n^{2} x + 10\,n^{2} x^{2} + 5\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(31\,x^{2} + 55\,x^{3} + 20\,x^{4} - 2\,n x - 22\,n x^{2} - 30\,n x^{3} - 10\,n x^{4}\right)\,P_{n-1}'(x) + \left(2\,x^{2} + 21\,x^{3} + 20\,x^{4} + 5\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
2416828
38567224160
41616040032080976
5324161600224012802565824
664102453761152011200499283235008
7128243216256482566976051584188162688209920

All 15 rows

Technical details
Node id
n-6a9cdd4a77ba1516 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-75d085a9c44fa5cf

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,    16,      8;
  3:   8,    56,     72,     24;
  4:  16,   160,    400,    320,      80;
  5:  32,   416,   1600,   2240,    1280,     256;
  6:  64,  1024,   5376,  11520,   11200,    4992,     832;
  7: 128,  2432,  16256,  48256,   69760,   51584,   18816,   2688;
  8: 256,  5632,  45824, 176128,  349696,  378368,  225536,  69632,   8704;
  9: 512, 12800, 122880, 583680, 1502720, 2207232, 1899520, 947200, 253440, 28160;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n+(6*n^2-4*n)*x+(10*n^2-25*n+15)*x^2+(5*n^2-15*n+10)*x^3)*P(n-1,x) - (2*n*x+(22*n-31)*x^2+(30*n-55)*x^3+(10*n-20)*x^4)*d/dx P(n-1,x) + (2*x^2+21*x^3+20*x^4+5*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 + (6*n^2 - 4*n)*x + (10*n^2 - 25*n + 15)*x^2 + (5*n^2 - 15*n + 10)*x^3)*P[n - 1, x] - (2*n*x + (22*n - 31)*x^2 + (30*n - 55)*x^3 + (10*n - 20)*x^4)*D[P[n - 1, x], x] + (2*x^2 + 21*x^3 + 20*x^4 + 5*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 + (6*n^2 - 4*n)*x + (10*n^2 - 25*n + 15)*x^2 + (5*n^2 - 15*n + 10)*x^3)*v[i - 1] - (2*n*x + (22*n - 31)*x^2 + (30*n - 55)*x^3 + (10*n - 20)*x^4)*deriv(v[i - 1], x) + (2*x^2 + 21*x^3 + 20*x^4 + 5*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