Real-rooted atlas

A108838 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(9 + 18\,n + 3\,n x + 11\,n^{2} + 7\,n^{2} x + 2\,n^{3} + 5\,n^{3} x + n^{4} x\right)\,P_{n}(x) = \left(18 + 42\,n + 36\,n x + 3\,n x^{2} + 32\,n^{2} + 54\,n^{2} x + 14\,n^{2} x^{2} + 8\,n^{3} + 26\,n^{3} x + 12\,n^{3} x^{2} + 4\,n^{4} x + 3\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-9\,x - 12\,x^{2} - 3\,x^{3} - 12\,n x - 22\,n x^{2} - 10\,n x^{3} - 4\,n^{2} x - 12\,n^{2} x^{2} - 8\,n^{2} x^{3} - 2\,n^{3} x^{2} - 2\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1549
21424644
342112638225
4132480432128101182
5429198024751200225126321
61430800812870880027503601434232
7486232032630635605625025554453916187137

All 15 rows

Technical details
Node id
n-398a4845fac4d9f1 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f0acff1b7583ce22

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:     2;
  1:     5,      4;
  2:    14,     24,       6;
  3:    42,    112,      63,       8;
  4:   132,    480,     432,     128,      10;
  5:   429,   1980,    2475,    1200,     225,     12;
  6:  1430,   8008,   12870,    8800,    2750,    360,     14;
  7:  4862,  32032,   63063,   56056,   25025,   5544,    539,    16;
  8: 16796, 127296,  297024,  326144,  191100,  61152,  10192,   768,   18;
  9: 58786, 503880, 1360476, 1782144, 1299480, 550368, 133770, 17472, 1053, 20;
%F Formula
The row polynomials satisfy (2*n^3+11*n^2+18*n+9+(n^4+5*n^3+7*n^2+3*n)*x)*P(n,x) = (8*n^3+32*n^2+42*n+18+(4*n^4+26*n^3+54*n^2+36*n)*x+(3*n^4+12*n^3+14*n^2+3*n)*x^2)*P(n-1,x) - ((4*n^2+12*n+9)*x+(2*n^3+12*n^2+22*n+12)*x^2+(2*n^3+8*n^2+10*n+3)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.
%t Mathematica
P[0, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((8*n^3 + 32*n^2 + 42*n + 18 + (4*n^4 + 26*n^3 + 54*n^2 + 36*n)*x + (3*n^4 + 12*n^3 + 14*n^2 + 3*n)*x^2)*P[n - 1, x] - ((4*n^2 + 12*n + 9)*x + (2*n^3 + 12*n^2 + 22*n + 12)*x^2 + (2*n^3 + 8*n^2 + 10*n + 3)*x^3)*D[P[n - 1, x], x])/(2*n^3 + 11*n^2 + 18*n + 9 + (n^4 + 5*n^3 + 7*n^2 + 3*n)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = ((8*n^3 + 32*n^2 + 42*n + 18 + (4*n^4 + 26*n^3 + 54*n^2 + 36*n)*x + (3*n^4 + 12*n^3 + 14*n^2 + 3*n)*x^2)*v[i - 1] - ((4*n^2 + 12*n + 9)*x + (2*n^3 + 12*n^2 + 22*n + 12)*x^2 + (2*n^3 + 8*n^2 + 10*n + 3)*x^3)*deriv(v[i - 1], x))/(2*n^3 + 11*n^2 + 18*n + 9 + (n^4 + 5*n^3 + 7*n^2 + 3*n)*x)); 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

Same fingerprint ?

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