Real-rooted atlas

A156366 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 8\,x + 15\,x^{2} + 10\,x^{3} + 2\,x^{4} + 2\,n x + 6\,n x^{2} + 6\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}(x) + \left(x + x^{2} - 3\,x^{3} - 5\,x^{4} - 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 14\,x + 22\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01142237
1136168160365
2182834212814564501
31176341217344297681590466605
41366125341117283600964624802026721149877
5174843104626048335321677664007946368295168022687565
6115141420423202832265878409855180817670707215003545648361216503589781
71304845490815389280189091848104874460829279293444265493504309254016088040550412420052205

All 15 rows

Technical details
Node id
n-5a275fa244db7a09 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a6397fa9532fc789

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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,    14,      22;
  1: 1,    36,     168,       160;
  2: 1,    82,     834,      2128,       1456;
  3: 1,   176,    3412,     17344,      29768,       15904;
  4: 1,   366,   12534,    111728,     360096,      462480,       202672;
  5: 1,   748,   43104,    626048,    3353216,     7766400,      7946368,       2951680;
  6: 1,  1514,  142042,   3202832,   26587840,    98551808,    176707072,     150035456,      48361216;
  7: 1,  3048,  454908,  15389280,  189091848,  1048744608,   2927929344,    4265493504,    3092540160,     880405504;
  8: 1,  6118, 1428750,  70704048, 1244224272,  9880792272,  40215082224,   89360774400,  109382840064,   69161591296,   17630351872;
  9: 1, 12260, 4426984, 314358816, 7728766368, 85159095744, 484171320000, 1537565597952, 2825299383168, 2977906000384, 1668889100288, 385148108800;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(2*n+8)*x+(6*n+15)*x^2+(6*n+10)*x^3+(2*n+2)*x^4)*P(n-1,x) + (x+x^2-3*x^3-5*x^4-2*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 14*x + 22*x^2.
%t Mathematica
P[0, x_] := 1 + 14*x + 22*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (2*n + 8)*x + (6*n + 15)*x^2 + (6*n + 10)*x^3 + (2*n + 2)*x^4)*P[n - 1, x] + (x + x^2 - 3*x^3 - 5*x^4 - 2*x^5)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 14*x + 22*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (2*n + 8)*x + (6*n + 15)*x^2 + (6*n + 10)*x^3 + (2*n + 2)*x^4)*v[i - 1] + (x + x^2 - 3*x^3 - 5*x^4 - 2*x^5)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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

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