Real-rooted atlas

A156365 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 6\,x + 10\,x^{2} + 6\,x^{3} + x^{4} + n x + 3\,n x^{2} + 3\,n x^{3} + n x^{4}\right)\,P_{n-1}(x) + \left(x + 2\,x^{2} - 2\,x^{4} - x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 10\,x + 13\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01101324
11259175192
21564268965411920
31119167467349829468323040
4124659794080410803912019847293322560
5150120157218257931347181595116289115458355160960
6110126548810776046940090210822043235592824287692708726192897280
7120352076045037868469470222082222344861561006134633563956618171022475631857945600

All 15 rows

Technical details
Node id
n-4cbdf6d8927688ef · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-31922ed48acb43c6

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,   10,      13;
  1: 1,   25,      91,       75;
  2: 1,   56,     426,      896,        541;
  3: 1,  119,    1674,     6734,       9829,        4683;
  4: 1,  246,    5979,    40804,     108039,      120198,       47293;
  5: 1,  501,   20157,   218257,     931347,     1815951,     1628911,       545835;
  6: 1, 1012,   65488,  1077604,    6940090,    21082204,    32355928,     24287692,      7087261;
  7: 1, 2035,  207604,  5037868,   46947022,   208222234,   486156100,    613463356,    395661817,    102247563;
  8: 1, 4082,  647241, 22659000,  296642754,  1842924876,  6136494618,  11574468984,  12380984925,   6997344146,   1622632573;
  9: 1, 8177, 1994799, 99086871, 1782958698, 15072518058, 68693342238, 181584806286, 286442939589, 265649652437, 133576402051, 28091567595;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+(n+6)*x+(3*n+10)*x^2+(3*n+6)*x^3+(n+1)*x^4)*P(n-1,x) + (x+2*x^2-2*x^4-x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 10*x + 13*x^2.
%t Mathematica
P[0, x_] := 1 + 10*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + (n + 6)*x + (3*n + 10)*x^2 + (3*n + 6)*x^3 + (n + 1)*x^4)*P[n - 1, x] + (x + 2*x^2 - 2*x^4 - 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 + 10*x + 13*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (n + 6)*x + (3*n + 10)*x^2 + (3*n + 6)*x^3 + (n + 1)*x^4)*v[i - 1] + (x + 2*x^2 - 2*x^4 - 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

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