Real-rooted atlas

A085478 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(60 + 96\,x - 48\,x^{2} - 40\,n - 64\,n x + 104\,n x^{2} - 72\,n^{2} x^{2} + 16\,n^{3} x^{2}\right)\,P_{n}(x) = \left(80 + 168\,x + 64\,x^{2} - 60\,n - 116\,n x + 16\,n x^{2} + 32\,n x^{3} - 20\,n^{2} x - 104\,n^{2} x^{2} - 48\,n^{2} x^{3} + 24\,n^{3} x^{2} + 16\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 140\,x - 212\,x^{2} - 64\,x^{3} - 40\,n x^{2} - 96\,n x^{3} - 32\,n x^{4} + 20\,n^{2} x^{2} + 48\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2246
35201237
413769632217
534260504352801230
6898402196236811201926805
723326168604126089200326444836973

All 15 rows

Technical details
Node id
n-d7e4b9ce95d524b0 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-64fd7e0d0c81c245

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:    1;
  2:    2,     4;
  3:    5,    20,     12;
  4:   13,    76,     96,     32;
  5:   34,   260,    504,    352,     80;
  6:   89,   840,   2196,   2368,   1120,    192;
  7:  233,  2616,   8604,  12608,   9200,   3264,    448;
  8:  610,  7940,  31464,  58400,  58000,  31680,   8960,  1024;
  9: 1597, 23644, 109608, 246496, 310800, 230976, 100352, 23552, 2304;
%F Formula
The row polynomials satisfy -(40*n-60+(64*n-96)*x-(16*n^3-72*n^2+104*n-48)*x^2)*P(n,x) = -(60*n-80+(20*n^2+116*n-168)*x-(24*n^3-104*n^2+16*n+64)*x^2-(16*n^3-48*n^2+32*n)*x^3)*P(n-1,x) - (25+140*x-(20*n^2-40*n-212)*x^2-(48*n^2-96*n-64)*x^3-(16*n^2-32*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[(-(60*n - 80 + (20*n^2 + 116*n - 168)*x - (24*n^3 - 104*n^2 + 16*n + 64)*x^2 - (16*n^3 - 48*n^2 + 32*n)*x^3)*P[n - 1, x] - (25 + 140*x - (20*n^2 - 40*n - 212)*x^2 - (48*n^2 - 96*n - 64)*x^3 - (16*n^2 - 32*n)*x^4)*D[P[n - 1, x], x])/(-40*n + 60 - (64*n - 96)*x + (16*n^3 - 72*n^2 + 104*n - 48)*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] = (-(60*n - 80 + (20*n^2 + 116*n - 168)*x - (24*n^3 - 104*n^2 + 16*n + 64)*x^2 - (16*n^3 - 48*n^2 + 32*n)*x^3)*v[i - 1] - (25 + 140*x - (20*n^2 - 40*n - 212)*x^2 - (48*n^2 - 96*n - 64)*x^3 - (16*n^2 - 32*n)*x^4)*deriv(v[i - 1], x))/(-40*n + 60 - (64*n - 96)*x + (16*n^3 - 72*n^2 + 104*n - 48)*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