Real-rooted atlas

A281000 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + n x^{2} + 2\,n^{2} + 4\,n^{2} x + 2\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-2 - 12\,x - 24\,x^{2} - 20\,x^{3} - 6\,x^{4} + 28\,n x + 83\,n x^{2} + 82\,n x^{3} + 27\,n x^{4} + 8\,n^{2} + 14\,n^{2} x - 8\,n^{2} x^{2} - 26\,n^{2} x^{3} - 12\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-4\,x - 15\,x^{2} - 21\,x^{3} - 13\,x^{4} - 3\,x^{5} + 16\,n x + 60\,n x^{2} + 84\,n x^{3} + 52\,n x^{4} + 12\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2347
310302161
435175266127603
5126924226823048356457
64624620161702607019855579872975
717162230810381823509228113817110641835857013
86435105105621621184684530730702915640147703531057210356323

All 15 rows

Technical details
Node id
n-9c03bbfe0b1292b9 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2444250a0fed39b2

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     3,       4;
   3:    10,      30,       21;
   4:    35,     175,      266,      127;
   5:   126,     924,     2268,     2304,       835;
   6:   462,    4620,    16170,    26070,     19855,      5798;
   7:  1716,   22308,   103818,   235092,    281138,    171106,     41835;
   8:  6435,  105105,   621621,  1846845,   3073070,   2915640,   1477035,    310572;
   9: 24310,  486200,  3539536, 13205192,  28588560,  37499280,  29459640,  12778152,   2356779;
  10: 92378, 2217072, 19399380, 88128612, 237780972, 403456716, 436143024, 292211640, 110797569, 18199284;
%F Formula
The row polynomials satisfy (2*n^2+n+(4*n^2+2*n)*x+(2*n^2+n)*x^2)*P(n,x) = (8*n^2-2+(14*n^2+28*n-12)*x-(8*n^2-83*n+24)*x^2-(26*n^2-82*n+20)*x^3-(12*n^2-27*n+6)*x^4)*P(n-1,x) + ((16*n-4)*x+(60*n-15)*x^2+(84*n-21)*x^3+(52*n-13)*x^4+(12*n-3)*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^2 - 2 + (14*n^2 + 28*n - 12)*x - (8*n^2 - 83*n + 24)*x^2 - (26*n^2 - 82*n + 20)*x^3 - (12*n^2 - 27*n + 6)*x^4)*P[n - 1, x] + ((16*n - 4)*x + (60*n - 15)*x^2 + (84*n - 21)*x^3 + (52*n - 13)*x^4 + (12*n - 3)*x^5)*D[P[n - 1, x], x])/(2*n^2 + n + (4*n^2 + 2*n)*x + (2*n^2 + n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^2 - 2 + (14*n^2 + 28*n - 12)*x - (8*n^2 - 83*n + 24)*x^2 - (26*n^2 - 82*n + 20)*x^3 - (12*n^2 - 27*n + 6)*x^4)*v[i - 1] + ((16*n - 4)*x + (60*n - 15)*x^2 + (84*n - 21)*x^3 + (52*n - 13)*x^4 + (12*n - 3)*x^5)*deriv(v[i - 1], x))/(2*n^2 + n + (4*n^2 + 2*n)*x + (2*n^2 + n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.