Real-rooted atlas

A116414 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-104\,n - 40\,n x + 16\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-260 - 256\,x - 80\,x^{2} - 260\,n - 308\,n x - 80\,n x^{2} + 16\,n x^{3} - 52\,n^{2} x + 40\,n^{2} x^{2} + 32\,n^{2} x^{3} + 40\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-169 - 390\,x - 252\,x^{2} - 40\,x^{3} + 52\,n x^{2} + 80\,n x^{3} + 16\,n x^{4} + 52\,n^{2} x^{2} + 80\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1549
222401274
39527618032583
440916401692640804461
517609008128407616200019233416
67573471608614870400288005760448246289
732585239068533700558880314000973441568010241792281

All 15 rows

Technical details
Node id
n-59ec8492b3db58d9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-a660609bf9979ab5

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:      5,       4;
  2:     22,      40,       12;
  3:     95,     276,      180,       32;
  4:    409,    1640,     1692,      640,       80;
  5:   1760,    9008,    12840,     7616,     2000,      192;
  6:   7573,   47160,    86148,    70400,    28800,     5760,     448;
  7:  32585,  239068,   533700,   558880,   314000,    97344,   15680,   1024;
  8: 140206, 1184200,  3127260,  4006400,  2880800,  1223040,  304192,  40960,   2304;
  9: 603275, 5764620, 17587800, 26694720, 23478000, 12737088, 4327680, 897024, 103680, 5120;
%F Formula
The row polynomials satisfy -(104*n+40*n*x-16*n^3*x^2)*P(n,x) = -(260*n+260+(52*n^2+308*n+256)*x-(40*n^3+40*n^2-80*n-80)*x^2-(16*n^3+32*n^2+16*n)*x^3)*P(n-1,x) - (169+390*x-(52*n^2+52*n-252)*x^2-(80*n^2+80*n-40)*x^3-(16*n^2+16*n)*x^4)*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[(-(260*n + 260 + (52*n^2 + 308*n + 256)*x - (40*n^3 + 40*n^2 - 80*n - 80)*x^2 - (16*n^3 + 32*n^2 + 16*n)*x^3)*P[n - 1, x] - (169 + 390*x - (52*n^2 + 52*n - 252)*x^2 - (80*n^2 + 80*n - 40)*x^3 - (16*n^2 + 16*n)*x^4)*D[P[n - 1, x], x])/(-104*n - 40*n*x + 16*n^3*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] = (-(260*n + 260 + (52*n^2 + 308*n + 256)*x - (40*n^3 + 40*n^2 - 80*n - 80)*x^2 - (16*n^3 + 32*n^2 + 16*n)*x^3)*v[i - 1] - (169 + 390*x - (52*n^2 + 52*n - 252)*x^2 - (80*n^2 + 80*n - 40)*x^3 - (16*n^2 + 16*n)*x^4)*deriv(v[i - 1], x))/(-104*n - 40*n*x + 16*n^3*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