Real-rooted atlas

A059473 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(18\,x^{2} + 24\,x^{3} + 2\,n - n x - 30\,n x^{2} - 36\,n x^{3} + 5\,n^{2} x + 12\,n^{2} x^{2} + 12\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(20\,x^{2} + 66\,x^{3} + 48\,x^{4} - 2\,n x - 16\,n x^{2} - 36\,n x^{3} - 24\,n x^{4}\right)\,P_{n-1}'(x) + \left(2\,x^{2} + 14\,x^{3} + 24\,x^{4} + 12\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
24181840
385612080264
4161525087123561744
532384175238083960158411520
6649285376159442559221144704876096
71282176152965766412608016019210976031360502656

All 15 rows

Technical details
Node id
n-a4c3f229438a576f · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ab8a6f908e9a31d4

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,     4;
  2:   4,    18,     18;
  3:   8,    56,    120,     80;
  4:  16,   152,    508,    712,     356;
  5:  32,   384,   1752,   3808,    3960,    1584;
  6:  64,   928,   5376,  15944,   25592,   21144,    7048;
  7: 128,  2176,  15296,  57664,  126080,  160192,  109760,   31360;
  8: 256,  4992,  41280, 189120,  525456,  907776,  953760,  558144,  139536;
  9: 512, 11264, 107136, 578304, 1953312, 4284864, 6111168, 5471232, 2793888, 620864;
%F Formula
The row polynomials satisfy n*P(n,x) = (2*n+(5*n^2-n)*x+(12*n^2-30*n+18)*x^2+(12*n^2-36*n+24)*x^3)*P(n-1,x) - (2*n*x+(16*n-20)*x^2+(36*n-66)*x^3+(24*n-48)*x^4)*d/dx P(n-1,x) + (2*x^2+14*x^3+24*x^4+12*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 - n)*x + (12*n^2 - 30*n + 18)*x^2 + (12*n^2 - 36*n + 24)*x^3)*P[n - 1, x] - (2*n*x + (16*n - 20)*x^2 + (36*n - 66)*x^3 + (24*n - 48)*x^4)*D[P[n - 1, x], x] + (2*x^2 + 14*x^3 + 24*x^4 + 12*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 - n)*x + (12*n^2 - 30*n + 18)*x^2 + (12*n^2 - 36*n + 24)*x^3)*v[i - 1] - (2*n*x + (16*n - 20)*x^2 + (36*n - 66)*x^3 + (24*n - 48)*x^4)*deriv(v[i - 1], x) + (2*x^2 + 14*x^3 + 24*x^4 + 12*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

Candidate equalities ?

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