Real-rooted atlas

A116412 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(9 + 27\,x + 27\,x^{2} + 9\,x^{3} + 9\,n + 27\,n x + 27\,n x^{2} + 9\,n x^{3}\right)\,P_{n}(x) = \left(293 + 1935\,x + 5199\,x^{2} + 7333\,x^{3} + 5760\,x^{4} + 2400\,x^{5} + 416\,x^{6} - 300\,n - 2133\,n x - 6183\,n x^{2} - 9375\,n x^{3} - 7857\,n x^{4} - 3456\,n x^{5} - 624\,n x^{6} + 79\,n^{2} + 540\,n^{2} x + 1578\,n^{2} x^{2} + 2492\,n^{2} x^{3} + 2223\,n^{2} x^{4} + 1056\,n^{2} x^{5} + 208\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(175 + 1720\,x + 6834\,x^{2} + 14468\,x^{3} + 17807\,x^{4} + 12828\,x^{5} + 5032\,x^{6} + 832\,x^{7} - 65\,n - 644\,n x - 2634\,n x^{2} - 5812\,n x^{3} - 7513\,n x^{4} - 5712\,n x^{5} - 2372\,n x^{6} - 416\,n x^{7}\right)\,P_{n-1}'(x) + \left(13 + 173\,x + 946\,x^{2} + 2798\,x^{3} + 4945\,x^{4} + 5393\,x^{5} + 3568\,x^{6} + 1316\,x^{7} + 208\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14711
2135046109
3432554893041091
4142113833454276200810909
5469473618799366673515613264109091
615491887094416248348362238277872876161090909
7511672985440823146114628704893342756213702457875210909091

All 15 rows

Technical details
Node id
n-8aff585f4784cf2c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-e7ebe070162a5d1b

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,       7;
  2:    13,      50,      46;
  3:    43,     255,     489,      304;
  4:   142,    1138,    3345,     4276,      2008;
  5:   469,    4736,   18799,    36667,     35156,     13264;
  6:  1549,   18870,   94416,   248348,    362238,    277872,     87616;
  7:  5116,   72985,  440823,  1461146,   2870489,   3342756,   2137024,    578752;
  8: 16897,  276266, 1955341,  7824284,  19359985,  30333656,  29393536,  16107968,   3822976;
  9: 55807, 1028745, 8349831, 39162099, 116965170, 230700663, 300515304, 249320016, 119558592, 25252864;
%F Formula
The row polynomials satisfy (9*n+9+(27*n+27)*x+(27*n+27)*x^2+(9*n+9)*x^3)*P(n,x) = (79*n^2-300*n+293+(540*n^2-2133*n+1935)*x+(1578*n^2-6183*n+5199)*x^2+(2492*n^2-9375*n+7333)*x^3+(2223*n^2-7857*n+5760)*x^4+(1056*n^2-3456*n+2400)*x^5+(208*n^2-624*n+416)*x^6)*P(n-1,x) - (65*n-175+(644*n-1720)*x+(2634*n-6834)*x^2+(5812*n-14468)*x^3+(7513*n-17807)*x^4+(5712*n-12828)*x^5+(2372*n-5032)*x^6+(416*n-832)*x^7)*d/dx P(n-1,x) + (13+173*x+946*x^2+2798*x^3+4945*x^4+5393*x^5+3568*x^6+1316*x^7+208*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[((79*n^2 - 300*n + 293 + (540*n^2 - 2133*n + 1935)*x + (1578*n^2 - 6183*n + 5199)*x^2 + (2492*n^2 - 9375*n + 7333)*x^3 + (2223*n^2 - 7857*n + 5760)*x^4 + (1056*n^2 - 3456*n + 2400)*x^5 + (208*n^2 - 624*n + 416)*x^6)*P[n - 1, x] - (65*n - 175 + (644*n - 1720)*x + (2634*n - 6834)*x^2 + (5812*n - 14468)*x^3 + (7513*n - 17807)*x^4 + (5712*n - 12828)*x^5 + (2372*n - 5032)*x^6 + (416*n - 832)*x^7)*D[P[n - 1, x], x] + (13 + 173*x + 946*x^2 + 2798*x^3 + 4945*x^4 + 5393*x^5 + 3568*x^6 + 1316*x^7 + 208*x^8)*D[P[n - 1, x], {x, 2}])/(9*n + 9 + (27*n + 27)*x + (27*n + 27)*x^2 + (9*n + 9)*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] = ((79*n^2 - 300*n + 293 + (540*n^2 - 2133*n + 1935)*x + (1578*n^2 - 6183*n + 5199)*x^2 + (2492*n^2 - 9375*n + 7333)*x^3 + (2223*n^2 - 7857*n + 5760)*x^4 + (1056*n^2 - 3456*n + 2400)*x^5 + (208*n^2 - 624*n + 416)*x^6)*v[i - 1] - (65*n - 175 + (644*n - 1720)*x + (2634*n - 6834)*x^2 + (5812*n - 14468)*x^3 + (7513*n - 17807)*x^4 + (5712*n - 12828)*x^5 + (2372*n - 5032)*x^6 + (416*n - 832)*x^7)*deriv(v[i - 1], x) + (13 + 173*x + 946*x^2 + 2798*x^3 + 4945*x^4 + 5393*x^5 + 3568*x^6 + 1316*x^7 + 208*x^8)*deriv(deriv(v[i - 1], x), x))/(9*n + 9 + (27*n + 27)*x + (27*n + 27)*x^2 + (9*n + 9)*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