Real-rooted atlas

A145597 → γ → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(24\,n - 24\,n x + 4\,n^{2} - 22\,n^{2} x + 27\,n^{3} x + 6\,n^{3} x^{2} + 5\,n^{4} x + 13\,n^{4} x^{2} + 8\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(48 - 176\,x + 64\,x^{2} - 32\,x^{3} + 32\,n - 4\,n x - 96\,n x^{2} - 80\,n x^{3} + 4\,n^{2} + 96\,n^{2} x - 52\,n^{2} x^{2} - 40\,n^{2} x^{3} + 37\,n^{3} x + 96\,n^{3} x^{2} + 52\,n^{3} x^{3} + 5\,n^{4} x + 61\,n^{4} x^{2} + 68\,n^{4} x^{3} + 10\,n^{5} x^{2} + 28\,n^{5} x^{3} + n^{6} x^{2} + 4\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(16\,x - 72\,x^{2} + 48\,x^{3} - 64\,x^{4} + 8\,n x - 8\,n x^{2} - 40\,n x^{3} - 224\,n x^{4} + 34\,n^{2} x^{2} - 60\,n^{2} x^{3} - 304\,n^{2} x^{4} + 10\,n^{3} x^{2} + 10\,n^{3} x^{3} - 200\,n^{3} x^{4} + 16\,n^{4} x^{3} - 64\,n^{4} x^{4} + 2\,n^{5} x^{3} - 8\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
133
261218
3106070
41518081276
5214205671008
62884022684483584
73615126804403212384

All 15 rows

Technical details
Node id
n-8f1128324610fe73 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8c1c541ae5fb5ecf

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:  3;
  2:  6,   12;
  3: 10,   60;
  4: 15,  180,    81;
  5: 21,  420,   567;
  6: 28,  840,  2268,   448;
  7: 36, 1512,  6804,  4032;
  8: 45, 2520, 17010, 20160,  2250;
  9: 55, 3960, 37422, 73920, 24750;
%F Formula
The row polynomials satisfy (4*n^2+24*n+(5*n^4+27*n^3-22*n^2-24*n)*x+(n^6+8*n^5+13*n^4+6*n^3)*x^2)*P(n,x) = (4*n^2+32*n+48+(5*n^4+37*n^3+96*n^2-4*n-176)*x+(n^6+10*n^5+61*n^4+96*n^3-52*n^2-96*n+64)*x^2+(4*n^6+28*n^5+68*n^4+52*n^3-40*n^2-80*n-32)*x^3)*P(n-1,x) + ((8*n+16)*x+(10*n^3+34*n^2-8*n-72)*x^2+(2*n^5+16*n^4+10*n^3-60*n^2-40*n+48)*x^3-(8*n^5+64*n^4+200*n^3+304*n^2+224*n+64)*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[((4*n^2 + 32*n + 48 + (5*n^4 + 37*n^3 + 96*n^2 - 4*n - 176)*x + (n^6 + 10*n^5 + 61*n^4 + 96*n^3 - 52*n^2 - 96*n + 64)*x^2 + (4*n^6 + 28*n^5 + 68*n^4 + 52*n^3 - 40*n^2 - 80*n - 32)*x^3)*P[n - 1, x] + ((8*n + 16)*x + (10*n^3 + 34*n^2 - 8*n - 72)*x^2 + (2*n^5 + 16*n^4 + 10*n^3 - 60*n^2 - 40*n + 48)*x^3 - (8*n^5 + 64*n^4 + 200*n^3 + 304*n^2 + 224*n + 64)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 24*n + (5*n^4 + 27*n^3 - 22*n^2 - 24*n)*x + (n^6 + 8*n^5 + 13*n^4 + 6*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] = ((4*n^2 + 32*n + 48 + (5*n^4 + 37*n^3 + 96*n^2 - 4*n - 176)*x + (n^6 + 10*n^5 + 61*n^4 + 96*n^3 - 52*n^2 - 96*n + 64)*x^2 + (4*n^6 + 28*n^5 + 68*n^4 + 52*n^3 - 40*n^2 - 80*n - 32)*x^3)*v[i - 1] + ((8*n + 16)*x + (10*n^3 + 34*n^2 - 8*n - 72)*x^2 + (2*n^5 + 16*n^4 + 10*n^3 - 60*n^2 - 40*n + 48)*x^3 - (8*n^5 + 64*n^4 + 200*n^3 + 304*n^2 + 224*n + 64)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 24*n + (5*n^4 + 27*n^3 - 22*n^2 - 24*n)*x + (n^6 + 8*n^5 + 13*n^4 + 6*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