Real-rooted atlas

A121574 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(4\,n + 12\,n x + 12\,n x^{2} + 4\,n x^{3}\right)\,P_{n}(x) = \left(106 + 752\,x + 2170\,x^{2} + 3280\,x^{3} + 2750\,x^{4} + 1216\,x^{5} + 222\,x^{6} - 145\,n - 1038\,n x - 3043\,n x^{2} - 4684\,n x^{3} - 3999\,n x^{4} - 1798\,n x^{5} - 333\,n x^{6} + 51\,n^{2} + 342\,n^{2} x + 969\,n^{2} x^{2} + 1476\,n^{2} x^{3} + 1269\,n^{2} x^{4} + 582\,n^{2} x^{5} + 111\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(108 + 996\,x + 3804\,x^{2} + 7860\,x^{3} + 9540\,x^{4} + 6828\,x^{5} + 2676\,x^{6} + 444\,x^{7} - 52\,n - 470\,n x - 1784\,n x^{2} - 3698\,n x^{3} - 4532\,n x^{4} - 3290\,n x^{5} - 1312\,n x^{6} - 222\,n x^{7}\right)\,P_{n-1}'(x) + \left(13 + 150\,x + 730\,x^{2} + 1966\,x^{3} + 3220\,x^{4} + 3298\,x^{5} + 2070\,x^{6} + 730\,x^{7} + 111\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1358
210342872
333169283155640
4109746188820948595696
53603084104521751614516476050688
6118912234519751166921460279657826377451072
7392747171240963678527113747711352416246091461654014080

All 15 rows

Technical details
Node id
n-3f8f01a9d7330c7a · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-96cf975a14bd14a1

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:     3,      5;
  2:    10,     34,      28;
  3:    33,    169,     283,      155;
  4:   109,    746,    1888,     2094,      859;
  5:   360,   3084,   10452,    17516,    14516,     4760;
  6:  1189,  12234,   51975,   116692,   146027,    96578,    26377;
  7:  3927,  47171,  240963,   678527,  1137477,  1135241,   624609,   146165;
  8: 12970, 178138, 1063222,  3601690,  7573786, 10123870,  8400802,  3956782,   809956;
  9: 42837, 662143, 4521784, 17905072, 45304282, 75961270, 84399760, 59924680, 24672265, 4488275;
%F Formula
The row polynomials satisfy (4*n+12*n*x+12*n*x^2+4*n*x^3)*P(n,x) = (51*n^2-145*n+106+(342*n^2-1038*n+752)*x+(969*n^2-3043*n+2170)*x^2+(1476*n^2-4684*n+3280)*x^3+(1269*n^2-3999*n+2750)*x^4+(582*n^2-1798*n+1216)*x^5+(111*n^2-333*n+222)*x^6)*P(n-1,x) - (52*n-108+(470*n-996)*x+(1784*n-3804)*x^2+(3698*n-7860)*x^3+(4532*n-9540)*x^4+(3290*n-6828)*x^5+(1312*n-2676)*x^6+(222*n-444)*x^7)*d/dx P(n-1,x) + (13+150*x+730*x^2+1966*x^3+3220*x^4+3298*x^5+2070*x^6+730*x^7+111*x^8)*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[((51*n^2 - 145*n + 106 + (342*n^2 - 1038*n + 752)*x + (969*n^2 - 3043*n + 2170)*x^2 + (1476*n^2 - 4684*n + 3280)*x^3 + (1269*n^2 - 3999*n + 2750)*x^4 + (582*n^2 - 1798*n + 1216)*x^5 + (111*n^2 - 333*n + 222)*x^6)*P[n - 1, x] - (52*n - 108 + (470*n - 996)*x + (1784*n - 3804)*x^2 + (3698*n - 7860)*x^3 + (4532*n - 9540)*x^4 + (3290*n - 6828)*x^5 + (1312*n - 2676)*x^6 + (222*n - 444)*x^7)*D[P[n - 1, x], x] + (13 + 150*x + 730*x^2 + 1966*x^3 + 3220*x^4 + 3298*x^5 + 2070*x^6 + 730*x^7 + 111*x^8)*D[P[n - 1, x], {x, 2}])/(4*n + 12*n*x + 12*n*x^2 + 4*n*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] = ((51*n^2 - 145*n + 106 + (342*n^2 - 1038*n + 752)*x + (969*n^2 - 3043*n + 2170)*x^2 + (1476*n^2 - 4684*n + 3280)*x^3 + (1269*n^2 - 3999*n + 2750)*x^4 + (582*n^2 - 1798*n + 1216)*x^5 + (111*n^2 - 333*n + 222)*x^6)*v[i - 1] - (52*n - 108 + (470*n - 996)*x + (1784*n - 3804)*x^2 + (3698*n - 7860)*x^3 + (4532*n - 9540)*x^4 + (3290*n - 6828)*x^5 + (1312*n - 2676)*x^6 + (222*n - 444)*x^7)*deriv(v[i - 1], x) + (13 + 150*x + 730*x^2 + 1966*x^3 + 3220*x^4 + 3298*x^5 + 2070*x^6 + 730*x^7 + 111*x^8)*deriv(deriv(v[i - 1], x), x))/(4*n + 12*n*x + 12*n*x^2 + 4*n*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