Real-rooted atlas

A249252 → reverse → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-2 - 8\,x - 12\,x^{2} - 8\,x^{3} - 2\,x^{4} + 2\,n + 8\,n x + 12\,n x^{2} + 8\,n x^{3} + 2\,n x^{4}\right)\,P_{n}(x) = \left(-4 - 12\,x + 36\,x^{2} + 218\,x^{3} + 412\,x^{4} + 384\,x^{5} + 180\,x^{6} + 34\,x^{7} + 4\,n + 4\,n x - 88\,n x^{2} - 363\,n x^{3} - 632\,n x^{4} - 574\,n x^{5} - 268\,n x^{6} - 51\,n x^{7} + 8\,n^{2} x + 52\,n^{2} x^{2} + 145\,n^{2} x^{3} + 220\,n^{2} x^{4} + 190\,n^{2} x^{5} + 88\,n^{2} x^{6} + 17\,n^{2} x^{7}\right)\,P_{n-1}(x) + \left(4\,x + 70\,x^{2} + 358\,x^{3} + 880\,x^{4} + 1200\,x^{5} + 934\,x^{6} + 390\,x^{7} + 68\,x^{8} - 4\,n x - 44\,n x^{2} - 193\,n x^{3} - 446\,n x^{4} - 594\,n x^{5} - 460\,n x^{6} - 193\,n x^{7} - 34\,n x^{8}\right)\,P_{n-1}'(x) + \left(4\,x^{2} + 40\,x^{3} + 157\,x^{4} + 325\,x^{5} + 390\,x^{6} + 274\,x^{7} + 105\,x^{8} + 17\,x^{9}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1459
28241749
3168012461281
4322245525762171601
5645761968321625327739129
612814086176138881696810728275352049
7256332817856513928596883856443009805296761
8512768048768171520366240487328395560179472349211692001
910241740812774453171213855362348096259275218019367164681243739647049
1020483891232409615595524806784992832013936928131479047988808282693644296155003249
114096860168017924381696156111363811148865126080779882886421070434657104110484121577629313604441
128192188416194355211894784481346561357890562740568323990853124165559363042197441476996244284028856188091788035201
1316384409600463257631395840142320640455967744106292608018268162562316596544214250528014057940966201904801650105642001168510194593769
14327688847361088716880953344406505472145895628838616028167662102528114599261441286608268810680655392636160198425716723926319732567127267358125109649
Technical details
Node id
n-b6acceda4550ad22 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-205a62c6b3f27a2b

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:    4,     5;
  2:    8,    24,     17;
  3:   16,    80,    124,     61;
  4:   32,   224,    552,    576,     217;
  5:   64,   576,   1968,   3216,    2532,     773;
  6:  128,  1408,   6176,  13888,   16968,   10728,    2753;
  7:  256,  3328,  17856,  51392,   85968,   83856,   44300,    9805;
  8:  512,  7680,  48768, 171520,  366240,  487328,  395560,  179472,  34921;
  9: 1024, 17408, 127744, 531712, 1385536, 2348096, 2592752, 1801936, 716468, 124373;
%F Formula
The row polynomials satisfy (2*n-2+(8*n-8)*x+(12*n-12)*x^2+(8*n-8)*x^3+(2*n-2)*x^4)*P(n,x) = (4*n-4+(8*n^2+4*n-12)*x+(52*n^2-88*n+36)*x^2+(145*n^2-363*n+218)*x^3+(220*n^2-632*n+412)*x^4+(190*n^2-574*n+384)*x^5+(88*n^2-268*n+180)*x^6+(17*n^2-51*n+34)*x^7)*P(n-1,x) - ((4*n-4)*x+(44*n-70)*x^2+(193*n-358)*x^3+(446*n-880)*x^4+(594*n-1200)*x^5+(460*n-934)*x^6+(193*n-390)*x^7+(34*n-68)*x^8)*d/dx P(n-1,x) + (4*x^2+40*x^3+157*x^4+325*x^5+390*x^6+274*x^7+105*x^8+17*x^9)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 5*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n - 4 + (8*n^2 + 4*n - 12)*x + (52*n^2 - 88*n + 36)*x^2 + (145*n^2 - 363*n + 218)*x^3 + (220*n^2 - 632*n + 412)*x^4 + (190*n^2 - 574*n + 384)*x^5 + (88*n^2 - 268*n + 180)*x^6 + (17*n^2 - 51*n + 34)*x^7)*P[n - 1, x] - ((4*n - 4)*x + (44*n - 70)*x^2 + (193*n - 358)*x^3 + (446*n - 880)*x^4 + (594*n - 1200)*x^5 + (460*n - 934)*x^6 + (193*n - 390)*x^7 + (34*n - 68)*x^8)*D[P[n - 1, x], x] + (4*x^2 + 40*x^3 + 157*x^4 + 325*x^5 + 390*x^6 + 274*x^7 + 105*x^8 + 17*x^9)*D[P[n - 1, x], {x, 2}])/(2*n - 2 + (8*n - 8)*x + (12*n - 12)*x^2 + (8*n - 8)*x^3 + (2*n - 2)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (8*n^2 + 4*n - 12)*x + (52*n^2 - 88*n + 36)*x^2 + (145*n^2 - 363*n + 218)*x^3 + (220*n^2 - 632*n + 412)*x^4 + (190*n^2 - 574*n + 384)*x^5 + (88*n^2 - 268*n + 180)*x^6 + (17*n^2 - 51*n + 34)*x^7)*v[i - 1] - ((4*n - 4)*x + (44*n - 70)*x^2 + (193*n - 358)*x^3 + (446*n - 880)*x^4 + (594*n - 1200)*x^5 + (460*n - 934)*x^6 + (193*n - 390)*x^7 + (34*n - 68)*x^8)*deriv(v[i - 1], x) + (4*x^2 + 40*x^3 + 157*x^4 + 325*x^5 + 390*x^6 + 274*x^7 + 105*x^8 + 17*x^9)*deriv(deriv(v[i - 1], x), x))/(2*n - 2 + (8*n - 8)*x + (12*n - 12)*x^2 + (8*n - 8)*x^3 + (2*n - 2)*x^4)); 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

Candidate equalities ?

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