Real-rooted atlas

A008288 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 3\,x - 3\,x^{2} - x^{3} + n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(116 + 777\,x + 2153\,x^{2} + 3161\,x^{3} + 2595\,x^{4} + 1130\,x^{5} + 204\,x^{6} - 93\,n - 624\,n x - 1737\,n x^{2} - 2567\,n x^{3} - 2124\,n x^{4} - 933\,n x^{5} - 170\,n x^{6} + 18\,n^{2} + 120\,n^{2} x + 334\,n^{2} x^{2} + 496\,n^{2} x^{3} + 414\,n^{2} x^{4} + 184\,n^{2} x^{5} + 34\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(76 + 631\,x + 2218\,x^{2} + 4286\,x^{3} + 4924\,x^{4} + 3367\,x^{5} + 1270\,x^{6} + 204\,x^{7} - 24\,n - 200\,n x - 707\,n x^{2} - 1376\,n x^{3} - 1594\,n x^{4} - 1100\,n x^{5} - 419\,n x^{6} - 68\,n x^{7}\right)\,P_{n-1}'(x) + \left(8 + 80\,x + 345\,x^{2} + 839\,x^{3} + 1260\,x^{4} + 1198\,x^{5} + 705\,x^{6} + 235\,x^{7} + 34\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2235
35151131
412548039185
5291743883811391111
6705251564231316984956665
71691521567111211123937263176339991
8408428419184474907018461920301966279239945

All 15 rows

Technical details
Node id
n-6b10b31dc92d1722 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2a11099686bc538f

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    2,     3;
   3:    5,    15,     11;
   4:   12,    54,     80,     39;
   5:   29,   174,    388,    381,     139;
   6:   70,   525,   1564,   2313,    1698,     495;
   7:  169,  1521,   5671,  11211,   12393,    7263,    1763;
   8:  408,  4284,  19184,  47490,   70184,   61920,   30196,    6279;
   9:  985, 11820,  61792, 183798,  340206,  401256,  294490,  122961,  22363;
  10: 2378, 32103, 191896, 666582, 1482828, 2190582, 2149072, 1350099, 492838, 79647;
%F Formula
The row polynomials satisfy (n-1+(3*n-3)*x+(3*n-3)*x^2+(n-1)*x^3)*P(n,x) = (18*n^2-93*n+116+(120*n^2-624*n+777)*x+(334*n^2-1737*n+2153)*x^2+(496*n^2-2567*n+3161)*x^3+(414*n^2-2124*n+2595)*x^4+(184*n^2-933*n+1130)*x^5+(34*n^2-170*n+204)*x^6)*P(n-1,x) - (24*n-76+(200*n-631)*x+(707*n-2218)*x^2+(1376*n-4286)*x^3+(1594*n-4924)*x^4+(1100*n-3367)*x^5+(419*n-1270)*x^6+(68*n-204)*x^7)*d/dx P(n-1,x) + (8+80*x+345*x^2+839*x^3+1260*x^4+1198*x^5+705*x^6+235*x^7+34*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((18*n^2 - 93*n + 116 + (120*n^2 - 624*n + 777)*x + (334*n^2 - 1737*n + 2153)*x^2 + (496*n^2 - 2567*n + 3161)*x^3 + (414*n^2 - 2124*n + 2595)*x^4 + (184*n^2 - 933*n + 1130)*x^5 + (34*n^2 - 170*n + 204)*x^6)*P[n - 1, x] - (24*n - 76 + (200*n - 631)*x + (707*n - 2218)*x^2 + (1376*n - 4286)*x^3 + (1594*n - 4924)*x^4 + (1100*n - 3367)*x^5 + (419*n - 1270)*x^6 + (68*n - 204)*x^7)*D[P[n - 1, x], x] + (8 + 80*x + 345*x^2 + 839*x^3 + 1260*x^4 + 1198*x^5 + 705*x^6 + 235*x^7 + 34*x^8)*D[P[n - 1, x], {x, 2}])/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((18*n^2 - 93*n + 116 + (120*n^2 - 624*n + 777)*x + (334*n^2 - 1737*n + 2153)*x^2 + (496*n^2 - 2567*n + 3161)*x^3 + (414*n^2 - 2124*n + 2595)*x^4 + (184*n^2 - 933*n + 1130)*x^5 + (34*n^2 - 170*n + 204)*x^6)*v[i - 1] - (24*n - 76 + (200*n - 631)*x + (707*n - 2218)*x^2 + (1376*n - 4286)*x^3 + (1594*n - 4924)*x^4 + (1100*n - 3367)*x^5 + (419*n - 1270)*x^6 + (68*n - 204)*x^7)*deriv(v[i - 1], x) + (8 + 80*x + 345*x^2 + 839*x^3 + 1260*x^4 + 1198*x^5 + 705*x^6 + 235*x^7 + 34*x^8)*deriv(deriv(v[i - 1], x), x))/(n - 1 + (3*n - 3)*x + (3*n - 3)*x^2 + (n - 1)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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