Real-rooted atlas

A104684 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x - x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-19 - 82\,x - 131\,x^{2} - 92\,x^{3} - 24\,x^{4} + 11\,n + 46\,n x + 71\,n x^{2} + 48\,n x^{3} + 12\,n x^{4}\right)\,P_{n-1}(x) + \left(-4 - 26\,x - 66\,x^{2} - 82\,x^{3} - 50\,x^{4} - 12\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2347
313342269
463246318136763
53211668323427728868881
61683109202823036340232905944106407
7898969942226020388240373830191292406361298949
8486394413221711416367668047254903633252154719628148816065483

All 15 rows

Technical details
Node id
n-f7742e88082e85c3 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c2c66460b34de1f0

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:      13,       34,       22;
   4:      63,      246,      318,       136;
   5:     321,     1668,     3234,      2772,       886;
   6:    1683,    10920,    28230,     36340,     23290,      5944;
   7:    8989,    69942,   226020,    388240,    373830,    191292,     40636;
   8:   48639,   441322,  1711416,   3676680,   4725490,   3633252,   1547196,    281488;
   9:  265729,  2754568, 12462604,  32141368,  51678970,  53043592,  33938884,  12375592,  1968934;
  10: 1462563, 17051724, 88170372, 265373976, 512335278, 657939744, 561998892, 307885896, 98159778, 13875544;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x+(n-1)*x^2)*P(n,x) = (11*n-19+(46*n-82)*x+(71*n-131)*x^2+(48*n-92)*x^3+(12*n-24)*x^4)*P(n-1,x) - (4+26*x+66*x^2+82*x^3+50*x^4+12*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[((11*n - 19 + (46*n - 82)*x + (71*n - 131)*x^2 + (48*n - 92)*x^3 + (12*n - 24)*x^4)*P[n - 1, x] - (4 + 26*x + 66*x^2 + 82*x^3 + 50*x^4 + 12*x^5)*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] = 1; for(n = 2, N, my(i = n); v[i] = ((11*n - 19 + (46*n - 82)*x + (71*n - 131)*x^2 + (48*n - 92)*x^3 + (12*n - 24)*x^4)*v[i - 1] - (4 + 26*x + 66*x^2 + 82*x^3 + 50*x^4 + 12*x^5)*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

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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023