Real-rooted atlas

A114687 → θ+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n + 2\,n x + n^{2} - 2\,n^{2} x - 12\,n^{3} x + 4\,n^{3} x^{2} - 8\,n^{4} x + 12\,n^{4} x^{2} + 12\,n^{5} x^{2} + 4\,n^{6} x^{2}\right)\,P_{n}(x) = \left(n + 10\,n x - 12\,n x^{2} + 8\,n x^{3} + n^{2} + 22\,n^{2} x - 48\,n^{2} x^{2} + 48\,n^{2} x^{3} - 12\,n^{3} x - 172\,n^{3} x^{2} + 120\,n^{3} x^{3} - 8\,n^{4} x - 72\,n^{4} x^{2} + 192\,n^{4} x^{3} + 12\,n^{5} x^{2} + 120\,n^{5} x^{3} + 4\,n^{6} x^{2} + 24\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(x - 6\,x^{2} + 12\,x^{3} - 8\,x^{4} + 2\,n x - 24\,n x^{2} + 64\,n x^{3} - 48\,n x^{4} - 40\,n^{2} x^{2} + 140\,n^{2} x^{3} - 120\,n^{2} x^{4} - 16\,n^{3} x^{2} + 112\,n^{3} x^{3} - 160\,n^{3} x^{4} + 44\,n^{4} x^{3} - 88\,n^{4} x^{4} + 8\,n^{5} x^{3} - 16\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111617
2148108157
31966485121257
411602160512020009441
5124054002560030000691268153
61336113408960021000014515221952478381
71448211682508809800001354752614656655363287441

All 15 rows

Technical details
Node id
n-291849ed6f23fd33 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-ce7cd47e90ee931d

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: 1,  16;
  2: 1,  48,   108;
  3: 1,  96,   648,     512;
  4: 1, 160,  2160,    5120,     2000;
  5: 1, 240,  5400,   25600,    30000,     6912;
  6: 1, 336, 11340,   89600,   210000,   145152,    21952;
  7: 1, 448, 21168,  250880,   980000,  1354752,   614656,    65536;
  8: 1, 576, 36288,  602112,  3528000,  8128512,  7375872,  2359296,  186624;
  9: 1, 720, 58320, 1290240, 10584000, 36578304, 55319040, 35389440, 8398080, 512000;
%F Formula
The row polynomials satisfy (n^2+n-(8*n^4+12*n^3+2*n^2-2*n)*x+(4*n^6+12*n^5+12*n^4+4*n^3)*x^2)*P(n,x) = (n^2+n-(8*n^4+12*n^3-22*n^2-10*n)*x+(4*n^6+12*n^5-72*n^4-172*n^3-48*n^2-12*n)*x^2+(24*n^6+120*n^5+192*n^4+120*n^3+48*n^2+8*n)*x^3)*P(n-1,x) + ((2*n+1)*x-(16*n^3+40*n^2+24*n+6)*x^2+(8*n^5+44*n^4+112*n^3+140*n^2+64*n+12)*x^3-(16*n^5+88*n^4+160*n^3+120*n^2+48*n+8)*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[((n^2 + n - (8*n^4 + 12*n^3 - 22*n^2 - 10*n)*x + (4*n^6 + 12*n^5 - 72*n^4 - 172*n^3 - 48*n^2 - 12*n)*x^2 + (24*n^6 + 120*n^5 + 192*n^4 + 120*n^3 + 48*n^2 + 8*n)*x^3)*P[n - 1, x] + ((2*n + 1)*x - (16*n^3 + 40*n^2 + 24*n + 6)*x^2 + (8*n^5 + 44*n^4 + 112*n^3 + 140*n^2 + 64*n + 12)*x^3 - (16*n^5 + 88*n^4 + 160*n^3 + 120*n^2 + 48*n + 8)*x^4)*D[P[n - 1, x], x])/(n^2 + n - (8*n^4 + 12*n^3 + 2*n^2 - 2*n)*x + (4*n^6 + 12*n^5 + 12*n^4 + 4*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] = ((n^2 + n - (8*n^4 + 12*n^3 - 22*n^2 - 10*n)*x + (4*n^6 + 12*n^5 - 72*n^4 - 172*n^3 - 48*n^2 - 12*n)*x^2 + (24*n^6 + 120*n^5 + 192*n^4 + 120*n^3 + 48*n^2 + 8*n)*x^3)*v[i - 1] + ((2*n + 1)*x - (16*n^3 + 40*n^2 + 24*n + 6)*x^2 + (8*n^5 + 44*n^4 + 112*n^3 + 140*n^2 + 64*n + 12)*x^3 - (16*n^5 + 88*n^4 + 160*n^3 + 120*n^2 + 48*n + 8)*x^4)*deriv(v[i - 1], x))/(n^2 + n - (8*n^4 + 12*n^3 + 2*n^2 - 2*n)*x + (4*n^6 + 12*n^5 + 12*n^4 + 4*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