Real-rooted atlas

A113413 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(9 + 9\,n\right)\,P_{n}(x) = \left(27 + 12\,x^{2} + 16\,x^{3} + 27\,n + 27\,n x - 36\,n x^{2} - 24\,n x^{3} + 27\,n^{2} x + 24\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-27\,x + 45\,x^{2} + 108\,x^{3} + 32\,x^{4} - 27\,n x - 90\,n x^{2} - 72\,n x^{3} - 16\,n x^{4}\right)\,P_{n-1}'(x) + \left(27\,x^{2} + 81\,x^{3} + 48\,x^{4} + 8\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1336
2918734
327816917194
481324432228411106
5243121522141836699996306
6729437410125115566885204623935954
7218715309430116277551489238415805577204994

All 15 rows

Technical details
Node id
n-8ddd19a092206c5c · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d429c37e7f5118e9

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:     3,      3;
  2:     9,     18,      7;
  3:    27,     81,     69,      17;
  4:    81,    324,    432,     228,      41;
  5:   243,   1215,   2214,    1836,     699,      99;
  6:   729,   4374,  10125,   11556,    6885,    2046,    239;
  7:  2187,  15309,  43011,   62775,   51489,   23841,   5805,    577;
  8:  6561,  52488, 173502,  309096,  324810,  206712,  78120,  16104,  1393;
  9: 19683, 177147, 673596, 1418634, 1823958, 1486998, 770796, 245808, 43923, 3363;
%F Formula
The row polynomials satisfy (9*n+9)*P(n,x) = (27*n+27+(27*n^2+27*n)*x+(24*n^2-36*n+12)*x^2+(8*n^2-24*n+16)*x^3)*P(n-1,x) - ((27*n+27)*x+(90*n-45)*x^2+(72*n-108)*x^3+(16*n-32)*x^4)*d/dx P(n-1,x) + (27*x^2+81*x^3+48*x^4+8*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[((27*n + 27 + (27*n^2 + 27*n)*x + (24*n^2 - 36*n + 12)*x^2 + (8*n^2 - 24*n + 16)*x^3)*P[n - 1, x] - ((27*n + 27)*x + (90*n - 45)*x^2 + (72*n - 108)*x^3 + (16*n - 32)*x^4)*D[P[n - 1, x], x] + (27*x^2 + 81*x^3 + 48*x^4 + 8*x^5)*D[P[n - 1, x], {x, 2}])/(9*n + 9)]];
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] = ((27*n + 27 + (27*n^2 + 27*n)*x + (24*n^2 - 36*n + 12)*x^2 + (8*n^2 - 24*n + 16)*x^3)*v[i - 1] - ((27*n + 27)*x + (90*n - 45)*x^2 + (72*n - 108)*x^3 + (16*n - 32)*x^4)*deriv(v[i - 1], x) + (27*x^2 + 81*x^3 + 48*x^4 + 8*x^5)*deriv(deriv(v[i - 1], x), x))/(9*n + 9)); 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