Real-rooted atlas

A008459 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(-2 - 14\,x - 30\,x^{2} - 26\,x^{3} - 8\,x^{4} + 4\,n + 23\,n x + 42\,n x^{2} + 31\,n x^{3} + 8\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 14\,x^{2} - 30\,x^{3} - 26\,x^{4} - 8\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1257
26303369
320150348245763
4707002490370019218881
5252315015120346503771015525106407
692413860837902604004372203748501279051298949
73432600604379761720950392532051849003664920106792516065483

All 15 rows

Technical details
Node id
n-98a62e1ec8173456 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-be64fe17a53239d2

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:     2,       5;
  2:     6,      30,       33;
  3:    20,     150,      348,      245;
  4:    70,     700,     2490,     3700,      1921;
  5:   252,    3150,    15120,    34650,     37710,     15525;
  6:   924,   13860,    83790,   260400,    437220,    374850,    127905;
  7:  3432,   60060,   437976,  1720950,   3925320,   5184900,   3664920,   1067925;
  8: 12870,  257400,  2198196, 10450440,  30194010,  54192600,  58908780,  35411400,   9004545;
  9: 48620, 1093950, 10707840, 59759700, 209261052, 476125650, 702960720, 648657900, 339138810, 76499525;
%F Formula
The row polynomials satisfy (n+2*n*x+n*x^2)*P(n,x) = (4*n-2+(23*n-14)*x+(42*n-30)*x^2+(31*n-26)*x^3+(8*n-8)*x^4)*P(n-1,x) - (2*x+14*x^2+30*x^3+26*x^4+8*x^5)*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[((4*n - 2 + (23*n - 14)*x + (42*n - 30)*x^2 + (31*n - 26)*x^3 + (8*n - 8)*x^4)*P[n - 1, x] - (2*x + 14*x^2 + 30*x^3 + 26*x^4 + 8*x^5)*D[P[n - 1, x], x])/(n + 2*n*x + n*x^2)]];
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] = ((4*n - 2 + (23*n - 14)*x + (42*n - 30)*x^2 + (31*n - 26)*x^3 + (8*n - 8)*x^4)*v[i - 1] - (2*x + 14*x^2 + 30*x^3 + 26*x^4 + 8*x^5)*deriv(v[i - 1], x))/(n + 2*n*x + n*x^2)); 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. [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