Real-rooted atlas

A059473 → 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

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(62 + 414\,x + 1140\,x^{2} + 1660\,x^{3} + 1350\,x^{4} + 582\,x^{5} + 104\,x^{6} - 85\,n - 576\,n x - 1612\,n x^{2} - 2386\,n x^{3} - 1971\,n x^{4} - 862\,n x^{5} - 156\,n x^{6} + 27\,n^{2} + 180\,n^{2} x + 502\,n^{2} x^{2} + 748\,n^{2} x^{3} + 627\,n^{2} x^{4} + 280\,n^{2} x^{5} + 52\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(78 + 648\,x + 2278\,x^{2} + 4400\,x^{3} + 5050\,x^{4} + 3448\,x^{5} + 1298\,x^{6} + 208\,x^{7} - 36\,n - 300\,n x - 1062\,n x^{2} - 2072\,n x^{3} - 2408\,n x^{4} - 1668\,n x^{5} - 638\,n x^{6} - 104\,n x^{7}\right)\,P_{n-1}'(x) + \left(12 + 120\,x + 518\,x^{2} + 1262\,x^{3} + 1900\,x^{4} + 1812\,x^{5} + 1070\,x^{6} + 358\,x^{7} + 52\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
2185440112
3803605362641240
4356213647804728174413744
515841188035488527763907211520152320
6704863432237032470712523920309888760961688128
7313603292801477312367104054567044851648238912050265618709120

All 15 rows

Technical details
Node id
n-6abbd94f049bfefa · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-8042f30bf8ddfc41

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,       6;
  2:     18,      54,       40;
  3:     80,     360,      536,       264;
  4:    356,    2136,     4780,      4728,      1744;
  5:   1584,   11880,    35488,     52776,     39072,     11520;
  6:   7048,   63432,   237032,    470712,    523920,    309888,     76096;
  7:  31360,  329280,  1477312,   3671040,   5456704,   4851648,   2389120,    502656;
  8: 139536, 1674432,  8767776,  26165376,  48673296,  57792960,  42773568,  18041472,   3320320;
  9: 620864, 8381664, 50173440, 174791232, 390534336, 580339296, 573566976, 363553920, 134102528, 21932544;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (27*n^2-85*n+62+(180*n^2-576*n+414)*x+(502*n^2-1612*n+1140)*x^2+(748*n^2-2386*n+1660)*x^3+(627*n^2-1971*n+1350)*x^4+(280*n^2-862*n+582)*x^5+(52*n^2-156*n+104)*x^6)*P(n-1,x) - (36*n-78+(300*n-648)*x+(1062*n-2278)*x^2+(2072*n-4400)*x^3+(2408*n-5050)*x^4+(1668*n-3448)*x^5+(638*n-1298)*x^6+(104*n-208)*x^7)*d/dx P(n-1,x) + (12+120*x+518*x^2+1262*x^3+1900*x^4+1812*x^5+1070*x^6+358*x^7+52*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[((27*n^2 - 85*n + 62 + (180*n^2 - 576*n + 414)*x + (502*n^2 - 1612*n + 1140)*x^2 + (748*n^2 - 2386*n + 1660)*x^3 + (627*n^2 - 1971*n + 1350)*x^4 + (280*n^2 - 862*n + 582)*x^5 + (52*n^2 - 156*n + 104)*x^6)*P[n - 1, x] - (36*n - 78 + (300*n - 648)*x + (1062*n - 2278)*x^2 + (2072*n - 4400)*x^3 + (2408*n - 5050)*x^4 + (1668*n - 3448)*x^5 + (638*n - 1298)*x^6 + (104*n - 208)*x^7)*D[P[n - 1, x], x] + (12 + 120*x + 518*x^2 + 1262*x^3 + 1900*x^4 + 1812*x^5 + 1070*x^6 + 358*x^7 + 52*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + 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] = ((27*n^2 - 85*n + 62 + (180*n^2 - 576*n + 414)*x + (502*n^2 - 1612*n + 1140)*x^2 + (748*n^2 - 2386*n + 1660)*x^3 + (627*n^2 - 1971*n + 1350)*x^4 + (280*n^2 - 862*n + 582)*x^5 + (52*n^2 - 156*n + 104)*x^6)*v[i - 1] - (36*n - 78 + (300*n - 648)*x + (1062*n - 2278)*x^2 + (2072*n - 4400)*x^3 + (2408*n - 5050)*x^4 + (1668*n - 3448)*x^5 + (638*n - 1298)*x^6 + (104*n - 208)*x^7)*deriv(v[i - 1], x) + (12 + 120*x + 518*x^2 + 1262*x^3 + 1900*x^4 + 1812*x^5 + 1070*x^6 + 358*x^7 + 52*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + 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