Real-rooted atlas

A100862 → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + 2\,x^{2}\right)\,P_{n}(x) = \left(24 + 132\,x + 282\,x^{2} + 296\,x^{3} + 154\,x^{4} + 32\,x^{5} - 29\,n - 169\,n x - 381\,n x^{2} - 419\,n x^{3} - 226\,n x^{4} - 48\,n x^{5} + 9\,n^{2} + 51\,n^{2} x + 115\,n^{2} x^{2} + 129\,n^{2} x^{3} + 72\,n^{2} x^{4} + 16\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(26 + 192\,x + 574\,x^{2} + 894\,x^{3} + 768\,x^{4} + 346\,x^{5} + 64\,x^{6} - 12\,n - 88\,n x - 264\,n x^{2} - 416\,n x^{3} - 364\,n x^{4} - 168\,n x^{5} - 32\,n x^{6}\right)\,P_{n-1}'(x) + \left(4 + 36\,x + 136\,x^{2} + 280\,x^{3} + 340\,x^{4} + 244\,x^{5} + 96\,x^{6} + 16\,x^{7}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1235
25151131
314639345215
4432585735582011633
5142106531604635336096313325
64994491166803271535730206014899115615
718501942586667212940311115270270129255262531057775

All 15 rows

Technical details
Node id
n-458b13fcf32dab26 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-63cd4a1dc80588bb

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:     2,      3;
  2:     5,     15,      11;
  3:    14,     63,      93,      45;
  4:    43,    258,     573,     558,      201;
  5:   142,   1065,    3160,    4635,     3360,      963;
  6:   499,   4491,   16680,   32715,    35730,    20601,     4899;
  7:  1850,  19425,   86667,  212940,   311115,   270270,   129255,    26253;
  8:  7193,  86316,  449666, 1328040,  2431695,  2826432,  2036454,   831564,  147345;
  9: 29186, 394011, 2347416, 8099406, 17833662, 25983909, 25050564, 15408954, 5487534, 862083;
%F Formula
The row polynomials satisfy (2+4*x+2*x^2)*P(n,x) = (9*n^2-29*n+24+(51*n^2-169*n+132)*x+(115*n^2-381*n+282)*x^2+(129*n^2-419*n+296)*x^3+(72*n^2-226*n+154)*x^4+(16*n^2-48*n+32)*x^5)*P(n-1,x) - (12*n-26+(88*n-192)*x+(264*n-574)*x^2+(416*n-894)*x^3+(364*n-768)*x^4+(168*n-346)*x^5+(32*n-64)*x^6)*d/dx P(n-1,x) + (4+36*x+136*x^2+280*x^3+340*x^4+244*x^5+96*x^6+16*x^7)*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[((9*n^2 - 29*n + 24 + (51*n^2 - 169*n + 132)*x + (115*n^2 - 381*n + 282)*x^2 + (129*n^2 - 419*n + 296)*x^3 + (72*n^2 - 226*n + 154)*x^4 + (16*n^2 - 48*n + 32)*x^5)*P[n - 1, x] - (12*n - 26 + (88*n - 192)*x + (264*n - 574)*x^2 + (416*n - 894)*x^3 + (364*n - 768)*x^4 + (168*n - 346)*x^5 + (32*n - 64)*x^6)*D[P[n - 1, x], x] + (4 + 36*x + 136*x^2 + 280*x^3 + 340*x^4 + 244*x^5 + 96*x^6 + 16*x^7)*D[P[n - 1, x], {x, 2}])/(2 + 4*x + 2*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; for(n = 1, N, my(i = n + 1); v[i] = ((9*n^2 - 29*n + 24 + (51*n^2 - 169*n + 132)*x + (115*n^2 - 381*n + 282)*x^2 + (129*n^2 - 419*n + 296)*x^3 + (72*n^2 - 226*n + 154)*x^4 + (16*n^2 - 48*n + 32)*x^5)*v[i - 1] - (12*n - 26 + (88*n - 192)*x + (264*n - 574)*x^2 + (416*n - 894)*x^3 + (364*n - 768)*x^4 + (168*n - 346)*x^5 + (32*n - 64)*x^6)*deriv(v[i - 1], x) + (4 + 36*x + 136*x^2 + 280*x^3 + 340*x^4 + 244*x^5 + 96*x^6 + 16*x^7)*deriv(deriv(v[i - 1], x), x))/(2 + 4*x + 2*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