Real-rooted atlas

A104684 → basis A013610 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-12 + 27\,x - 27\,x^{2} + 12\,n - 63\,n x + 81\,n x^{2} + 36\,n^{2} x - 81\,n^{2} x^{2} + 27\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-132 + 81\,x + 513\,x^{2} + 162\,x^{3} + 84\,n - 513\,n x - 783\,n x^{2} + 162\,n x^{3} + 288\,n^{2} x - 27\,n^{2} x^{2} - 567\,n^{2} x^{3} + 189\,n^{3} x^{2} + 243\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-8 + 144\,x^{2} + 243\,x^{3} + 81\,x^{4} - 48\,n x - 198\,n x^{2} - 135\,n x^{3} + 81\,n x^{4} - 36\,n^{2} x^{2} - 162\,n^{2} x^{3} - 162\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2369
313482788
463342405108918
53212304421225924059834
61683150303726036720141751458106326
78989960483009154104002612256998451031152664
84863960509422923814003020358627516227543214891749612497148

All 15 rows

Technical details
Node id
n-b213ae83c910518e · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9f5cd98eb75e902b

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,        6;
   3:      13,       48,        27;
   4:      63,      342,       405,       108;
   5:     321,     2304,      4212,      2592,       405;
   6:    1683,    15030,     37260,     36720,     14175,      1458;
   7:    8989,    96048,    300915,    410400,    261225,     69984,      5103;
   8:   48639,   605094,   2292381,   4003020,   3586275,   1622754,    321489,    17496;
   9:  265729,  3772416,  16768080,  35731584,  41447700,  26780544,   9144576,  1399680,   59049;
  10: 1462563, 23331942, 119038896, 299630016, 427529340, 361169928, 178319232, 47869056, 5845851, 196830;
%F Formula
The row polynomials satisfy (12*n-12+(36*n^2-63*n+27)*x+(27*n^3-81*n^2+81*n-27)*x^2)*P(n,x) = (84*n-132+(288*n^2-513*n+81)*x+(189*n^3-27*n^2-783*n+513)*x^2+(243*n^3-567*n^2+162*n+162)*x^3)*P(n-1,x) - (8+48*n*x+(36*n^2+198*n-144)*x^2+(162*n^2+135*n-243)*x^3+(162*n^2-81*n-81)*x^4)*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[((84*n - 132 + (288*n^2 - 513*n + 81)*x + (189*n^3 - 27*n^2 - 783*n + 513)*x^2 + (243*n^3 - 567*n^2 + 162*n + 162)*x^3)*P[n - 1, x] - (8 + 48*n*x + (36*n^2 + 198*n - 144)*x^2 + (162*n^2 + 135*n - 243)*x^3 + (162*n^2 - 81*n - 81)*x^4)*D[P[n - 1, x], x])/(12*n - 12 + (36*n^2 - 63*n + 27)*x + (27*n^3 - 81*n^2 + 81*n - 27)*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] = ((84*n - 132 + (288*n^2 - 513*n + 81)*x + (189*n^3 - 27*n^2 - 783*n + 513)*x^2 + (243*n^3 - 567*n^2 + 162*n + 162)*x^3)*v[i - 1] - (8 + 48*n*x + (36*n^2 + 198*n - 144)*x^2 + (162*n^2 + 135*n - 243)*x^3 + (162*n^2 - 81*n - 81)*x^4)*deriv(v[i - 1], x))/(12*n - 12 + (36*n^2 - 63*n + 27)*x + (27*n^3 - 81*n^2 + 81*n - 27)*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