Real-rooted atlas

A081577 → x ↦ x+1 → reverse

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(18\,x^{2} + 24\,x^{3} + n - 2\,n x - 30\,n x^{2} - 36\,n x^{3} + 4\,n^{2} x + 12\,n^{2} x^{2} + 12\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(19\,x^{2} + 66\,x^{3} + 48\,x^{4} - n x - 14\,n x^{2} - 36\,n x^{3} - 24\,n x^{4}\right)\,P_{n-1}'(x) + \left(x^{2} + 13\,x^{3} + 24\,x^{4} + 12\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
216613
3110241651
4114588844205
5118108272300120819
612217463211369843283277
7126256123231604400313689613107

All 15 rows

Technical details
Node id
n-564b9e756ea9e1c7 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-3c392e037ce8dc6c

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: 1,  2;
  2: 1,  6,   6;
  3: 1, 10,  24,   16;
  4: 1, 14,  58,   88,    44;
  5: 1, 18, 108,  272,   300,   120;
  6: 1, 22, 174,  632,  1136,   984,   328;
  7: 1, 26, 256, 1232,  3160,  4400,  3136,   896;
  8: 1, 30, 354, 2136,  7236, 14256, 16176,  9792,  2448;
  9: 1, 34, 468, 3408, 14484, 37512, 59808, 57216, 30096, 6688;
%F Formula
The row polynomials satisfy n*P(n,x) = (n+(4*n^2-2*n)*x+(12*n^2-30*n+18)*x^2+(12*n^2-36*n+24)*x^3)*P(n-1,x) - (n*x+(14*n-19)*x^2+(36*n-66)*x^3+(24*n-48)*x^4)*d/dx P(n-1,x) + (x^2+13*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[((n + (4*n^2 - 2*n)*x + (12*n^2 - 30*n + 18)*x^2 + (12*n^2 - 36*n + 24)*x^3)*P[n - 1, x] - (n*x + (14*n - 19)*x^2 + (36*n - 66)*x^3 + (24*n - 48)*x^4)*D[P[n - 1, x], x] + (x^2 + 13*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] = ((n + (4*n^2 - 2*n)*x + (12*n^2 - 30*n + 18)*x^2 + (12*n^2 - 36*n + 24)*x^3)*v[i - 1] - (n*x + (14*n - 19)*x^2 + (36*n - 66)*x^3 + (24*n - 48)*x^4)*deriv(v[i - 1], x) + (x^2 + 13*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