Real-rooted atlas

A090452 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 6\,x + 6\,x^{2} + 2\,x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(2 + 22\,x + 86\,x^{2} + 162\,x^{3} + 160\,x^{4} + 80\,x^{5} + 16\,x^{6} + n + 2\,n x - 14\,n x^{2} - 60\,n x^{3} - 91\,n x^{4} - 62\,n x^{5} - 16\,n x^{6}\right)\,P_{n-1}(x) + \left(2\,x + 16\,x^{2} + 52\,x^{3} + 88\,x^{4} + 82\,x^{5} + 40\,x^{6} + 8\,x^{7}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115612
2111437244171
31181264498708763602700
412627215385221110001413610176315245522
51355003970197556508514483021604020752011624028896802872
61458318655577952639258520551970640325410037533602877456131904027385614631759
71561288168421430948471403636220115680652753499048967380642174966035153638489248149313922661504273366252

All 15 rows

Technical details
Node id
n-bcea29165d888af2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-dd60ceb7c2348550

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,  5,    6;
  2: 1, 11,   43,    72,     44;
  3: 1, 18,  126,   449,    870,     876,      360;
  4: 1, 26,  272,  1538,   5221,   11000,    14136,     10176,      3152;
  5: 1, 35,  500,  3970,  19755,   65085,   144830,    216040,    207520,     116240,      28896;
  6: 1, 45,  831,  8655,  57795,  263925,   852055,   1970640,   3254100,    3753360,    2877456,     1319040,      273856;
  7: 1, 56, 1288, 16842, 143094,  847140,  3636220,  11568065,  27534990,   48967380,   64217496,    60351536,    38489248,    14931392,     2661504;
  8: 1, 68, 1896, 30184, 314804, 2314872, 12535936,  51344900, 161540425,  393162576,  739919600,  1067897152,  1160882016,   920294144,   502402816,   168929280,   26380544;
  9: 1, 81, 2682, 50808, 633852, 5618844, 37062816, 187354620, 739682901, 2307005909, 5715438102, 11248322256, 17490220608, 21230393184, 19702232640, 13504056576, 6443778816, 1911889152, 265655808;
%F Formula
The row polynomials satisfy (n+2+(3*n+6)*x+(3*n+6)*x^2+(n+2)*x^3)*P(n,x) = (n+2+(2*n+22)*x-(14*n-86)*x^2-(60*n-162)*x^3-(91*n-160)*x^4-(62*n-80)*x^5-(16*n-16)*x^6)*P(n-1,x) + (2*x+16*x^2+52*x^3+88*x^4+82*x^5+40*x^6+8*x^7)*d/dx 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 + 2 + (2*n + 22)*x - (14*n - 86)*x^2 - (60*n - 162)*x^3 - (91*n - 160)*x^4 - (62*n - 80)*x^5 - (16*n - 16)*x^6)*P[n - 1, x] + (2*x + 16*x^2 + 52*x^3 + 88*x^4 + 82*x^5 + 40*x^6 + 8*x^7)*D[P[n - 1, x], x])/(n + 2 + (3*n + 6)*x + (3*n + 6)*x^2 + (n + 2)*x^3)]];
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 + 2 + (2*n + 22)*x - (14*n - 86)*x^2 - (60*n - 162)*x^3 - (91*n - 160)*x^4 - (62*n - 80)*x^5 - (16*n - 16)*x^6)*v[i - 1] + (2*x + 16*x^2 + 52*x^3 + 88*x^4 + 82*x^5 + 40*x^6 + 8*x^7)*deriv(v[i - 1], x))/(n + 2 + (3*n + 6)*x + (3*n + 6)*x^2 + (n + 2)*x^3)); 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