Real-rooted atlas

A123160 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(5 + 5\,x - 7\,n - 7\,n x + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n}(x) = \left(-9 - 27\,x - 27\,x^{2} - 9\,x^{3} + 9\,n + 27\,n x + 27\,n x^{2} + 9\,n x^{3} - 2\,n^{2} - 6\,n^{2} x - 6\,n^{2} x^{2} - 2\,n^{2} x^{3}\right)\,P_{n-2}(x) + \left(14 + 58\,x + 44\,x^{2} - 16\,n - 64\,n x - 48\,n x^{2} + 4\,n^{2} + 16\,n^{2} x + 12\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2123
316815
4112393890
5120114252192579
6130260960160510023858
71425102760751510098533626262
8156903665026005560286292328814181380
9172148414168741302264084033963891121571841265955
101902304275521842127444081863120282931223915618641468906706
11111034204992041286021112566935880146961601944328514625630478000863058530
121132489585470853380535180822091916610750801121599051314502208907198326572086448716876
1311566798139700165231012413808624457682157582245145648758336453408768788385405606281483213443206387790
14118292042196483030885267876181605077766725209921996043049418391259060631994285784532416327074759783076479422992276180
151210121943341525312307544323783816813001896440832679929350117675724066330394444384330637220037799798127197390706664666890936165364807308
Technical details
Node id
n-f9db47211e4f0c6d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9f75e8bcfcc04e6d

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: 1,  2;
   3: 1,  6,    8;
   4: 1, 12,   39,    38;
   5: 1, 20,  114,   252,    192;
   6: 1, 30,  260,   960,   1605,   1002;
   7: 1, 42,  510,  2760,   7515,  10098,    5336;
   8: 1, 56,  903,  6650,  26005,  56028,   62923,   28814;
   9: 1, 72, 1484, 14168,  74130, 226408,  403396,  389112,  157184;
  10: 1, 90, 2304, 27552, 184212, 744408, 1863120, 2829312, 2391561, 864146;
%F Formula
The row polynomials satisfy (2*n^2-7*n+5+(2*n^2-7*n+5)*x)*P(n,x) = -(2*n^2-9*n+9+(6*n^2-27*n+27)*x+(6*n^2-27*n+27)*x^2+(2*n^2-9*n+9)*x^3)*P(n-2,x) + (4*n^2-16*n+14+(16*n^2-64*n+58)*x+(12*n^2-48*n+44)*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 9*n + 9 + (6*n^2 - 27*n + 27)*x + (6*n^2 - 27*n + 27)*x^2 + (2*n^2 - 9*n + 9)*x^3)*P[n - 2, x] + (4*n^2 - 16*n + 14 + (16*n^2 - 64*n + 58)*x + (12*n^2 - 48*n + 44)*x^2)*P[n - 1, x])/(2*n^2 - 7*n + 5 + (2*n^2 - 7*n + 5)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 9*n + 9 + (6*n^2 - 27*n + 27)*x + (6*n^2 - 27*n + 27)*x^2 + (2*n^2 - 9*n + 9)*x^3)*v[i - 2] + (4*n^2 - 16*n + 14 + (16*n^2 - 64*n + 58)*x + (12*n^2 - 48*n + 44)*x^2)*v[i - 1])/(2*n^2 - 7*n + 5 + (2*n^2 - 7*n + 5)*x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.