Real-rooted atlas

A108838 → x ↦ x+1 → basis A013610

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(3 + 7\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(3 + 18\,x + 27\,x^{2} - n - 6\,n x - 9\,n x^{2} - 2\,n^{2} - 12\,n^{2} x - 18\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(6 + 24\,n + 24\,n x + 12\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 7 + 6\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
17613
228481894
312131224354730
45501896226810801625956
52591111901813513500445548650357
6125366508813338013608071280174961458437318
76192137564893177012118688788503470046633943743877774

All 15 rows

Technical details
Node id
n-d57ac55d3b3604dc · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-3abc2859e25348b7

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:       2;
  1:       7,        6;
  2:      28,       48,       18;
  3:     121,      312,      243,       54;
  4:     550,     1896,     2268,     1080,      162;
  5:    2591,    11190,    18135,    13500,     4455,      486;
  6:   12536,    65088,   133380,   136080,    71280,    17496,     1458;
  7:   61921,   375648,   931770,  1211868,   878850,   347004,    66339,    4374;
  8:  310954,  2158608,  6288912,  9961056,  9276120,  5116608,  1592136,  244944,  13122;
  9: 1582791, 12373974, 41423562, 77429520, 88295508, 62950608, 27586818, 6980904, 885735, 39366;
%F Formula
The row polynomials satisfy (2*n^2+7*n+3)*P(n,x) = -(2*n^2+n-3+(12*n^2+6*n-18)*x+(18*n^2+9*n-27)*x^2)*P(n-2,x) + (12*n^2+24*n+6+(12*n^2+24*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 7 + 6*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 7 + 6*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 + n - 3 + (12*n^2 + 6*n - 18)*x + (18*n^2 + 9*n - 27)*x^2)*P[n - 2, x] + (12*n^2 + 24*n + 6 + (12*n^2 + 24*n)*x)*P[n - 1, x])/(2*n^2 + 7*n + 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 7 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^2 + n - 3 + (12*n^2 + 6*n - 18)*x + (18*n^2 + 9*n - 27)*x^2)*v[i - 2] + (12*n^2 + 24*n + 6 + (12*n^2 + 24*n)*x)*v[i - 1])/(2*n^2 + 7*n + 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

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. [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