Real-rooted atlas

A081577 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(3\,n + 9\,n x + 9\,n x^{2} + 3\,n x^{3}\right)\,P_{n}(x) = \left(59 + 447\,x + 1350\,x^{2} + 2110\,x^{3} + 1815\,x^{4} + 819\,x^{5} + 152\,x^{6} - 78\,n - 594\,n x - 1833\,n x^{2} - 2943\,n x^{3} - 2601\,n x^{4} - 1203\,n x^{5} - 228\,n x^{6} + 25\,n^{2} + 180\,n^{2} x + 546\,n^{2} x^{2} + 884\,n^{2} x^{3} + 801\,n^{2} x^{4} + 384\,n^{2} x^{5} + 76\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(46 + 487\,x + 2049\,x^{2} + 4550\,x^{3} + 5840\,x^{4} + 4371\,x^{5} + 1777\,x^{6} + 304\,x^{7} - 20\,n - 212\,n x - 903\,n x^{2} - 2044\,n x^{3} - 2686\,n x^{4} - 2064\,n x^{5} - 863\,n x^{6} - 152\,n x^{7}\right)\,P_{n-1}'(x) + \left(4 + 56\,x + 319\,x^{2} + 971\,x^{3} + 1750\,x^{4} + 1934\,x^{5} + 1291\,x^{6} + 479\,x^{7} + 76\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1257
26303369
316120282205623
4444401578239012895741
5120150072481686018834808552647
63284920299049404016107014235050737483349
7896156801148644558001056592142944010447463183654436383

All 15 rows

Technical details
Node id
n-d51db9bb42c74cd5 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-6b3356d6162feba7

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:    2,      5;
  2:    6,     30,      33;
  3:   16,    120,     282,     205;
  4:   44,    440,    1578,    2390,     1289;
  5:  120,   1500,    7248,   16860,    18834,     8085;
  6:  328,   4920,   29904,   94040,   161070,   142350,     50737;
  7:  896,  15680,  114864,  455800,  1056592,  1429440,   1044746,   318365;
  8: 2448,  48960,  419760, 2012400,  5893956, 10789560,  12050226,  7506630,  1997721;
  9: 6688, 150480, 1478016, 8309280, 29440008, 68120100, 102884880, 97784100, 53070210, 12535525;
%F Formula
The row polynomials satisfy (3*n+9*n*x+9*n*x^2+3*n*x^3)*P(n,x) = (25*n^2-78*n+59+(180*n^2-594*n+447)*x+(546*n^2-1833*n+1350)*x^2+(884*n^2-2943*n+2110)*x^3+(801*n^2-2601*n+1815)*x^4+(384*n^2-1203*n+819)*x^5+(76*n^2-228*n+152)*x^6)*P(n-1,x) - (20*n-46+(212*n-487)*x+(903*n-2049)*x^2+(2044*n-4550)*x^3+(2686*n-5840)*x^4+(2064*n-4371)*x^5+(863*n-1777)*x^6+(152*n-304)*x^7)*d/dx P(n-1,x) + (4+56*x+319*x^2+971*x^3+1750*x^4+1934*x^5+1291*x^6+479*x^7+76*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 - 78*n + 59 + (180*n^2 - 594*n + 447)*x + (546*n^2 - 1833*n + 1350)*x^2 + (884*n^2 - 2943*n + 2110)*x^3 + (801*n^2 - 2601*n + 1815)*x^4 + (384*n^2 - 1203*n + 819)*x^5 + (76*n^2 - 228*n + 152)*x^6)*P[n - 1, x] - (20*n - 46 + (212*n - 487)*x + (903*n - 2049)*x^2 + (2044*n - 4550)*x^3 + (2686*n - 5840)*x^4 + (2064*n - 4371)*x^5 + (863*n - 1777)*x^6 + (152*n - 304)*x^7)*D[P[n - 1, x], x] + (4 + 56*x + 319*x^2 + 971*x^3 + 1750*x^4 + 1934*x^5 + 1291*x^6 + 479*x^7 + 76*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 - 78*n + 59 + (180*n^2 - 594*n + 447)*x + (546*n^2 - 1833*n + 1350)*x^2 + (884*n^2 - 2943*n + 2110)*x^3 + (801*n^2 - 2601*n + 1815)*x^4 + (384*n^2 - 1203*n + 819)*x^5 + (76*n^2 - 228*n + 152)*x^6)*v[i - 1] - (20*n - 46 + (212*n - 487)*x + (903*n - 2049)*x^2 + (2044*n - 4550)*x^3 + (2686*n - 5840)*x^4 + (2064*n - 4371)*x^5 + (863*n - 1777)*x^6 + (152*n - 304)*x^7)*deriv(v[i - 1], x) + (4 + 56*x + 319*x^2 + 971*x^3 + 1750*x^4 + 1934*x^5 + 1291*x^6 + 479*x^7 + 76*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