Real-rooted atlas

A357613 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-2\,n - 6\,n x - 6\,n x^{2} - 2\,n x^{3} + 4\,n^{2} + 12\,n^{2} x + 12\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n}(x) = \left(6 + 33\,x + 69\,x^{2} + 69\,x^{3} + 33\,x^{4} + 6\,x^{5} - 27\,n - 146\,n x - 302\,n x^{2} - 300\,n x^{3} - 143\,n x^{4} - 26\,n x^{5} + 27\,n^{2} + 141\,n^{2} x + 281\,n^{2} x^{2} + 267\,n^{2} x^{3} + 120\,n^{2} x^{4} + 20\,n^{2} x^{5}\right)\,P_{n-1}(x) + \left(3\,x + 18\,x^{2} + 42\,x^{3} + 48\,x^{4} + 27\,x^{5} + 6\,x^{6} - 9\,n x - 55\,n x^{2} - 131\,n x^{3} - 153\,n x^{4} - 88\,n x^{5} - 20\,n x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1358
2155041106
3844206933771574
4495330081908964364924598
53003250258294013662011181536365395768
6185641856407698601694420208708513638303693056488704
711628013566006755868186139803064012030128280163834653800305107794898

All 15 rows

Technical details
Node id
n-68a79d6800b9e1df · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-bad64ddddd969224

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:       3,        5;
  2:      15,       50,        41;
  3:      84,      420,       693,        377;
  4:     495,     3300,      8190,       8964,       3649;
  5:    3003,    25025,     82940,     136620,     111815,      36365;
  6:   18564,   185640,    769860,    1694420,    2087085,    1363830,     369305;
  7:  116280,  1356600,   6755868,   18613980,   30640120,   30128280,   16383465,    3800305;
  8:  735471,  9806280,  57004332,  188675928,  388869390,  511004088,  418054140,  194654216,   39490049;
  9: 4686825, 70302375, 467240400, 1805739936, 4471750206, 7358114610, 8044270920, 5633882280, 2293485003, 413442773;
%F Formula
The row polynomials satisfy (4*n^2-2*n+(12*n^2-6*n)*x+(12*n^2-6*n)*x^2+(4*n^2-2*n)*x^3)*P(n,x) = (27*n^2-27*n+6+(141*n^2-146*n+33)*x+(281*n^2-302*n+69)*x^2+(267*n^2-300*n+69)*x^3+(120*n^2-143*n+33)*x^4+(20*n^2-26*n+6)*x^5)*P(n-1,x) - ((9*n-3)*x+(55*n-18)*x^2+(131*n-42)*x^3+(153*n-48)*x^4+(88*n-27)*x^5+(20*n-6)*x^6)*d/dx 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[((27*n^2 - 27*n + 6 + (141*n^2 - 146*n + 33)*x + (281*n^2 - 302*n + 69)*x^2 + (267*n^2 - 300*n + 69)*x^3 + (120*n^2 - 143*n + 33)*x^4 + (20*n^2 - 26*n + 6)*x^5)*P[n - 1, x] - ((9*n - 3)*x + (55*n - 18)*x^2 + (131*n - 42)*x^3 + (153*n - 48)*x^4 + (88*n - 27)*x^5 + (20*n - 6)*x^6)*D[P[n - 1, x], x])/(4*n^2 - 2*n + (12*n^2 - 6*n)*x + (12*n^2 - 6*n)*x^2 + (4*n^2 - 2*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] = ((27*n^2 - 27*n + 6 + (141*n^2 - 146*n + 33)*x + (281*n^2 - 302*n + 69)*x^2 + (267*n^2 - 300*n + 69)*x^3 + (120*n^2 - 143*n + 33)*x^4 + (20*n^2 - 26*n + 6)*x^5)*v[i - 1] - ((9*n - 3)*x + (55*n - 18)*x^2 + (131*n - 42)*x^3 + (153*n - 48)*x^4 + (88*n - 27)*x^5 + (20*n - 6)*x^6)*deriv(v[i - 1], x))/(4*n^2 - 2*n + (12*n^2 - 6*n)*x + (12*n^2 - 6*n)*x^2 + (4*n^2 - 2*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. [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.