Real-rooted atlas

A078812 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-400 - 120\,x + 656\,x^{2} + 480\,x^{3} - 64\,x^{4} + 400\,n + 120\,n x - 2256\,n x^{2} - 1440\,n x^{3} + 320\,n x^{4} + 2400\,n^{2} x^{2} + 1440\,n^{2} x^{3} - 640\,n^{2} x^{4} - 800\,n^{3} x^{2} - 480\,n^{3} x^{3} + 640\,n^{3} x^{4} - 320\,n^{4} x^{4} + 64\,n^{5} x^{4}\right)\,P_{n}(x) = \left(600\,n + 380\,n x - 264\,n x^{2} - 304\,n x^{3} - 96\,n x^{4} + 700\,n^{2} x + 2280\,n^{2} x^{2} + 2000\,n^{2} x^{3} + 480\,n^{2} x^{4} - 1200\,n^{3} x^{2} - 1440\,n^{3} x^{3} - 288\,n^{3} x^{4} + 64\,n^{3} x^{5} - 160\,n^{4} x^{3} - 384\,n^{4} x^{4} - 128\,n^{4} x^{5} + 96\,n^{5} x^{4} + 64\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(125 + 450\,x + 440\,x^{2} - 48\,x^{3} - 304\,x^{4} - 96\,x^{5} + 700\,n x^{2} + 2280\,n x^{3} + 2000\,n x^{4} + 480\,n x^{5} - 700\,n^{2} x^{2} - 2280\,n^{2} x^{3} - 1920\,n^{2} x^{4} - 288\,n^{2} x^{5} + 64\,n^{2} x^{6} - 160\,n^{3} x^{4} - 384\,n^{3} x^{5} - 128\,n^{3} x^{6} + 80\,n^{4} x^{4} + 192\,n^{4} x^{5} + 64\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
23811
38483692
421200324128673
555720183615364004547
614424008424110086000115229128
73777632342366144052000207363136179557
8987235121286282950403420002108166585681921075031

All 15 rows

Technical details
Node id
n-f9a60fb52c66bdee · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-e8c880dcea1ddd79

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,      8;
   3:    8,     48,      36;
   4:   21,    200,     324,     128;
   5:   55,    720,    1836,    1536,     400;
   6:  144,   2400,    8424,   11008,    6000,     1152;
   7:  377,   7632,   34236,   61440,   52000,    20736,    3136;
   8:  987,  23512,  128628,  295040,  342000,   210816,   65856,    8192;
   9: 2584,  70800,  457452, 1281024, 1896000,  1596672,  768320,  196608,  20736;
  10: 6765, 209560, 1562328, 5174528, 9349200, 10039680, 6585600, 2588672, 559872, 51200;
%F Formula
The row polynomials satisfy (400*n-400+(120*n-120)*x-(800*n^3-2400*n^2+2256*n-656)*x^2-(480*n^3-1440*n^2+1440*n-480)*x^3+(64*n^5-320*n^4+640*n^3-640*n^2+320*n-64)*x^4)*P(n,x) = (600*n+(700*n^2+380*n)*x-(1200*n^3-2280*n^2+264*n)*x^2-(160*n^4+1440*n^3-2000*n^2+304*n)*x^3+(96*n^5-384*n^4-288*n^3+480*n^2-96*n)*x^4+(64*n^5-128*n^4+64*n^3)*x^5)*P(n-1,x) + (125+450*x-(700*n^2-700*n-440)*x^2-(2280*n^2-2280*n+48)*x^3+(80*n^4-160*n^3-1920*n^2+2000*n-304)*x^4+(192*n^4-384*n^3-288*n^2+480*n-96)*x^5+(64*n^4-128*n^3+64*n^2)*x^6)*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[((600*n + (700*n^2 + 380*n)*x - (1200*n^3 - 2280*n^2 + 264*n)*x^2 - (160*n^4 + 1440*n^3 - 2000*n^2 + 304*n)*x^3 + (96*n^5 - 384*n^4 - 288*n^3 + 480*n^2 - 96*n)*x^4 + (64*n^5 - 128*n^4 + 64*n^3)*x^5)*P[n - 1, x] + (125 + 450*x - (700*n^2 - 700*n - 440)*x^2 - (2280*n^2 - 2280*n + 48)*x^3 + (80*n^4 - 160*n^3 - 1920*n^2 + 2000*n - 304)*x^4 + (192*n^4 - 384*n^3 - 288*n^2 + 480*n - 96)*x^5 + (64*n^4 - 128*n^3 + 64*n^2)*x^6)*D[P[n - 1, x], x])/(400*n - 400 + (120*n - 120)*x - (800*n^3 - 2400*n^2 + 2256*n - 656)*x^2 - (480*n^3 - 1440*n^2 + 1440*n - 480)*x^3 + (64*n^5 - 320*n^4 + 640*n^3 - 640*n^2 + 320*n - 64)*x^4)]];
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] = ((600*n + (700*n^2 + 380*n)*x - (1200*n^3 - 2280*n^2 + 264*n)*x^2 - (160*n^4 + 1440*n^3 - 2000*n^2 + 304*n)*x^3 + (96*n^5 - 384*n^4 - 288*n^3 + 480*n^2 - 96*n)*x^4 + (64*n^5 - 128*n^4 + 64*n^3)*x^5)*v[i - 1] + (125 + 450*x - (700*n^2 - 700*n - 440)*x^2 - (2280*n^2 - 2280*n + 48)*x^3 + (80*n^4 - 160*n^3 - 1920*n^2 + 2000*n - 304)*x^4 + (192*n^4 - 384*n^3 - 288*n^2 + 480*n - 96)*x^5 + (64*n^4 - 128*n^3 + 64*n^2)*x^6)*deriv(v[i - 1], x))/(400*n - 400 + (120*n - 120)*x - (800*n^3 - 2400*n^2 + 2256*n - 656)*x^2 - (480*n^3 - 1440*n^2 + 1440*n - 480)*x^3 + (64*n^5 - 320*n^4 + 640*n^3 - 640*n^2 + 320*n - 64)*x^4)); 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