Real-rooted atlas

A059473 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(3\,n + 9\,n x + 9\,n x^{2} + 3\,n x^{3}\right)\,P_{n}(x) = \left(62 + 486\,x + 1500\,x^{2} + 2380\,x^{3} + 2070\,x^{4} + 942\,x^{5} + 176\,x^{6} - 75\,n - 600\,n x - 1932\,n x^{2} - 3210\,n x^{3} - 2913\,n x^{4} - 1374\,n x^{5} - 264\,n x^{6} + 25\,n^{2} + 180\,n^{2} x + 558\,n^{2} x^{2} + 932\,n^{2} x^{3} + 873\,n^{2} x^{4} + 432\,n^{2} x^{5} + 88\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(46 + 496\,x + 2142\,x^{2} + 4880\,x^{3} + 6410\,x^{4} + 4896\,x^{5} + 2026\,x^{6} + 352\,x^{7} - 20\,n - 212\,n x - 918\,n x^{2} - 2128\,n x^{3} - 2872\,n x^{4} - 2268\,n x^{5} - 974\,n x^{6} - 176\,n x^{7}\right)\,P_{n-1}'(x) + \left(4 + 56\,x + 322\,x^{2} + 998\,x^{3} + 1840\,x^{4} + 2084\,x^{5} + 1426\,x^{6} + 542\,x^{7} + 88\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141014
21890108216
380600146411603304
4356356013116210801246450576
515841980097632237240283968133920774144
67048105720653208212708038489283668640143891211849536
73136054880040759681664960040387904581692804605299215460480181376384

All 15 rows

Technical details
Node id
n-ec67ab7cdd04664e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-cd5e6dee4b7eabae

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,       10;
  2:     18,       90,       108;
  3:     80,      600,      1464,      1160;
  4:    356,     3560,     13116,     21080,      12464;
  5:   1584,    19800,     97632,    237240,     283968,     133920;
  6:   7048,   105720,    653208,   2127080,    3848928,    3668640,     1438912;
  7:  31360,   548800,   4075968,  16649600,   40387904,   58169280,    46052992,    15460480;
  8: 139536,  2790720,  24211872, 118990080,  362241936,  699399360,   836260416,   566110080,  166116096;
  9: 620864, 13969440, 138645504, 796528320, 2918748096, 7073431200, 11335937280, 11583849600, 6848593920, 1784844800;
%F Formula
The row polynomials satisfy (3*n+9*n*x+9*n*x^2+3*n*x^3)*P(n,x) = (25*n^2-75*n+62+(180*n^2-600*n+486)*x+(558*n^2-1932*n+1500)*x^2+(932*n^2-3210*n+2380)*x^3+(873*n^2-2913*n+2070)*x^4+(432*n^2-1374*n+942)*x^5+(88*n^2-264*n+176)*x^6)*P(n-1,x) - (20*n-46+(212*n-496)*x+(918*n-2142)*x^2+(2128*n-4880)*x^3+(2872*n-6410)*x^4+(2268*n-4896)*x^5+(974*n-2026)*x^6+(176*n-352)*x^7)*d/dx P(n-1,x) + (4+56*x+322*x^2+998*x^3+1840*x^4+2084*x^5+1426*x^6+542*x^7+88*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[((25*n^2 - 75*n + 62 + (180*n^2 - 600*n + 486)*x + (558*n^2 - 1932*n + 1500)*x^2 + (932*n^2 - 3210*n + 2380)*x^3 + (873*n^2 - 2913*n + 2070)*x^4 + (432*n^2 - 1374*n + 942)*x^5 + (88*n^2 - 264*n + 176)*x^6)*P[n - 1, x] - (20*n - 46 + (212*n - 496)*x + (918*n - 2142)*x^2 + (2128*n - 4880)*x^3 + (2872*n - 6410)*x^4 + (2268*n - 4896)*x^5 + (974*n - 2026)*x^6 + (176*n - 352)*x^7)*D[P[n - 1, x], x] + (4 + 56*x + 322*x^2 + 998*x^3 + 1840*x^4 + 2084*x^5 + 1426*x^6 + 542*x^7 + 88*x^8)*D[P[n - 1, x], {x, 2}])/(3*n + 9*n*x + 9*n*x^2 + 3*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] = ((25*n^2 - 75*n + 62 + (180*n^2 - 600*n + 486)*x + (558*n^2 - 1932*n + 1500)*x^2 + (932*n^2 - 3210*n + 2380)*x^3 + (873*n^2 - 2913*n + 2070)*x^4 + (432*n^2 - 1374*n + 942)*x^5 + (88*n^2 - 264*n + 176)*x^6)*v[i - 1] - (20*n - 46 + (212*n - 496)*x + (918*n - 2142)*x^2 + (2128*n - 4880)*x^3 + (2872*n - 6410)*x^4 + (2268*n - 4896)*x^5 + (974*n - 2026)*x^6 + (176*n - 352)*x^7)*deriv(v[i - 1], x) + (4 + 56*x + 322*x^2 + 998*x^3 + 1840*x^4 + 2084*x^5 + 1426*x^6 + 542*x^7 + 88*x^8)*deriv(deriv(v[i - 1], x), x))/(3*n + 9*n*x + 9*n*x^2 + 3*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