Real-rooted atlas

A114655 → 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

\(\left(1 + 2\,x + x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(3 + 10\,x + 11\,x^{2} + 4\,x^{3} + 3\,n + 10\,n x + 11\,n x^{2} + 4\,n x^{3}\right)\,P_{n-1}(x) + \left(2 + 10\,x + 18\,x^{2} + 14\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
122
26814
3226040122
4903725042241190
5394218444804032134412434
61806125603456047040316808448136094
7855871604247320451440459360247104549121540298
8415864067001690920387464052852804292288192192036608017879414

All 15 rows

Technical details
Node id
n-c797a96ea6c06aee · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9a4b425d2a954ae8

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:       2;
   2:       6,        8;
   3:      22,       60,       40;
   4:      90,      372,      504,       224;
   5:     394,     2184,     4480,      4032,      1344;
   6:    1806,    12560,    34560,     47040,     31680,      8448;
   7:    8558,    71604,   247320,    451440,    459360,    247104,     54912;
   8:   41586,   406700,  1690920,   3874640,   5285280,   4292288,   1921920,    366080;
   9:  206098,  2306832, 11216128,  30942912,  52980928,  57657600,  38950912,  14936064,   2489344;
  10: 1037718, 13081752, 72828288, 235015872, 484481088, 661699584, 598800384, 346245120, 116093952, 17199104;
%F Formula
The row polynomials satisfy (n+1+(2*n+2)*x+(n+1)*x^2)*P(n,x) = (3*n+3+(10*n+10)*x+(11*n+11)*x^2+(4*n+4)*x^3)*P(n-1,x) + (2+10*x+18*x^2+14*x^3+4*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + 3 + (10*n + 10)*x + (11*n + 11)*x^2 + (4*n + 4)*x^3)*P[n - 1, x] + (2 + 10*x + 18*x^2 + 14*x^3 + 4*x^4)*D[P[n - 1, x], x])/(n + 1 + (2*n + 2)*x + (n + 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((3*n + 3 + (10*n + 10)*x + (11*n + 11)*x^2 + (4*n + 4)*x^3)*v[i - 1] + (2 + 10*x + 18*x^2 + 14*x^3 + 4*x^4)*deriv(v[i - 1], x))/(n + 1 + (2*n + 2)*x + (n + 1)*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

Same fingerprint ?

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.