Real-rooted atlas

A082985 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-100 - 60\,x + 4\,x^{2} + 12\,x^{3} + 200\,n - 130\,n x - 68\,n x^{2} - 10\,n x^{3} + 500\,n^{2} x - 180\,n^{2} x^{2} - 116\,n^{2} x^{3} - 4\,n^{2} x^{4} + 600\,n^{3} x^{2} + 56\,n^{3} x^{3} - 8\,n^{3} x^{4} + 240\,n^{4} x^{3} + 16\,n^{4} x^{4} + 32\,n^{5} x^{4}\right)\,P_{n}(x) = \left(-400 - 1090\,x - 704\,x^{2} + 86\,x^{3} + 120\,x^{4} - 4\,x^{5} + 800\,n + 830\,n x - 1532\,n x^{2} - 2246\,n x^{3} - 748\,n x^{4} - 24\,n x^{5} + 2700\,n^{2} x + 4380\,n^{2} x^{2} + 1696\,n^{2} x^{3} - 196\,n^{2} x^{4} - 108\,n^{2} x^{5} + 3000\,n^{3} x^{2} + 4344\,n^{3} x^{3} + 1760\,n^{3} x^{4} + 168\,n^{3} x^{5} + 1120\,n^{4} x^{3} + 1264\,n^{4} x^{4} + 304\,n^{4} x^{5} + 128\,n^{5} x^{4} + 96\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(-125 - 450\,x - 510\,x^{2} - 144\,x^{3} + 67\,x^{4} + 18\,x^{5} - 8\,x^{6} - 750\,n x - 2750\,n x^{2} - 3810\,n x^{3} - 2442\,n x^{4} - 696\,n x^{5} - 64\,n x^{6} - 1200\,n^{2} x^{2} - 3500\,n^{2} x^{3} - 3712\,n^{2} x^{4} - 1684\,n^{2} x^{5} - 272\,n^{2} x^{6} - 600\,n^{3} x^{3} - 1400\,n^{3} x^{4} - 1056\,n^{3} x^{5} - 256\,n^{3} x^{6} - 80\,n^{4} x^{4} - 144\,n^{4} x^{5} - 64\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141216
2116045116
329224315112680
476756153910562253652
5199242064356512275039618712
65217488245703286420800608463792964
7136422620882901468801241255518812005960451432

All 15 rows

Technical details
Node id
n-6edb74122424e6aa · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-f01b6e496fe8ddd7

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:    4,     12;
  2:   11,     60,      45;
  3:   29,    224,     315,     112;
  4:   76,    756,    1539,    1056,     225;
  5:  199,   2420,    6435,    6512,    2750,     396;
  6:  521,   7488,   24570,   32864,   20800,    6084,     637;
  7: 1364,  22620,   88290,  146880,  124125,   55188,   12005,    960;
  8: 3571,  67116,  303705,  604384,  639625,  383112,  128282,  21760,  1377;
  9: 9349, 196384, 1010781, 2341712, 2979200, 2248308, 1022238, 269952, 36936, 1900;
%F Formula
The row polynomials satisfy (200*n-100+(500*n^2-130*n-60)*x+(600*n^3-180*n^2-68*n+4)*x^2+(240*n^4+56*n^3-116*n^2-10*n+12)*x^3+(32*n^5+16*n^4-8*n^3-4*n^2)*x^4)*P(n,x) = (800*n-400+(2700*n^2+830*n-1090)*x+(3000*n^3+4380*n^2-1532*n-704)*x^2+(1120*n^4+4344*n^3+1696*n^2-2246*n+86)*x^3+(128*n^5+1264*n^4+1760*n^3-196*n^2-748*n+120)*x^4+(96*n^5+304*n^4+168*n^3-108*n^2-24*n-4)*x^5)*P(n-1,x) - (125+(750*n+450)*x+(1200*n^2+2750*n+510)*x^2+(600*n^3+3500*n^2+3810*n+144)*x^3+(80*n^4+1400*n^3+3712*n^2+2442*n-67)*x^4+(144*n^4+1056*n^3+1684*n^2+696*n-18)*x^5+(64*n^4+256*n^3+272*n^2+64*n+8)*x^6)*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[((800*n - 400 + (2700*n^2 + 830*n - 1090)*x + (3000*n^3 + 4380*n^2 - 1532*n - 704)*x^2 + (1120*n^4 + 4344*n^3 + 1696*n^2 - 2246*n + 86)*x^3 + (128*n^5 + 1264*n^4 + 1760*n^3 - 196*n^2 - 748*n + 120)*x^4 + (96*n^5 + 304*n^4 + 168*n^3 - 108*n^2 - 24*n - 4)*x^5)*P[n - 1, x] - (125 + (750*n + 450)*x + (1200*n^2 + 2750*n + 510)*x^2 + (600*n^3 + 3500*n^2 + 3810*n + 144)*x^3 + (80*n^4 + 1400*n^3 + 3712*n^2 + 2442*n - 67)*x^4 + (144*n^4 + 1056*n^3 + 1684*n^2 + 696*n - 18)*x^5 + (64*n^4 + 256*n^3 + 272*n^2 + 64*n + 8)*x^6)*D[P[n - 1, x], x])/(200*n - 100 + (500*n^2 - 130*n - 60)*x + (600*n^3 - 180*n^2 - 68*n + 4)*x^2 + (240*n^4 + 56*n^3 - 116*n^2 - 10*n + 12)*x^3 + (32*n^5 + 16*n^4 - 8*n^3 - 4*n^2)*x^4)]];
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] = ((800*n - 400 + (2700*n^2 + 830*n - 1090)*x + (3000*n^3 + 4380*n^2 - 1532*n - 704)*x^2 + (1120*n^4 + 4344*n^3 + 1696*n^2 - 2246*n + 86)*x^3 + (128*n^5 + 1264*n^4 + 1760*n^3 - 196*n^2 - 748*n + 120)*x^4 + (96*n^5 + 304*n^4 + 168*n^3 - 108*n^2 - 24*n - 4)*x^5)*v[i - 1] - (125 + (750*n + 450)*x + (1200*n^2 + 2750*n + 510)*x^2 + (600*n^3 + 3500*n^2 + 3810*n + 144)*x^3 + (80*n^4 + 1400*n^3 + 3712*n^2 + 2442*n - 67)*x^4 + (144*n^4 + 1056*n^3 + 1684*n^2 + 696*n - 18)*x^5 + (64*n^4 + 256*n^3 + 272*n^2 + 64*n + 8)*x^6)*deriv(v[i - 1], x))/(200*n - 100 + (500*n^2 - 130*n - 60)*x + (600*n^3 - 180*n^2 - 68*n + 4)*x^2 + (240*n^4 + 56*n^3 - 116*n^2 - 10*n + 12)*x^3 + (32*n^5 + 16*n^4 - 8*n^3 - 4*n^2)*x^4)); 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