Real-rooted atlas

A008459 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(2\,n^{2} - 3\,n^{2} x + 6\,n^{3} x + 2\,n^{4} x^{2}\right)\,P_{n}(x) = \left(4\,x^{2} - 4\,n - 26\,n x - 44\,n x^{2} - 16\,n x^{3} + 8\,n^{2} + 24\,n^{2} x + 4\,n^{2} x^{2} - 4\,n^{2} x^{3} + 24\,n^{3} x + 56\,n^{3} x^{2} + 24\,n^{3} x^{3} + 8\,n^{4} x^{2} + 12\,n^{4} x^{3}\right)\,P_{n-1}(x) + \left(x^{2} + 2\,x^{3} - 4\,n x - 20\,n x^{2} - 32\,n x^{3} - 16\,n x^{4} - 12\,n^{2} x^{2} - 36\,n^{2} x^{3} - 24\,n^{2} x^{4} - 4\,n^{3} x^{3} - 8\,n^{3} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
26483690
320240432128820
4701120324025604007390
52525040201602688012000115265484
692422176113400215040168000483843136571060
73432960965987521478400168000087091217561681924911400

All 15 rows

Technical details
Node id
n-2ec52acd342aaa2b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-84fb8546f9af188a

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:     2,       8;
  2:     6,      48,       36;
  3:    20,     240,      432,      128;
  4:    70,    1120,     3240,     2560,       400;
  5:   252,    5040,    20160,    26880,     12000,      1152;
  6:   924,   22176,   113400,   215040,    168000,     48384,     3136;
  7:  3432,   96096,   598752,  1478400,   1680000,    870912,   175616,     8192;
  8: 12870,  411840,  3027024,  9225216,  13860000,  10644480,  3951360,   589824,   20736;
  9: 48620, 1750320, 14826240, 53813760, 100900800, 103783680, 57953280, 16220160, 1866240, 51200;
%F Formula
The row polynomials satisfy (2*n^2+(6*n^3-3*n^2)*x+2*n^4*x^2)*P(n,x) = (8*n^2-4*n+(24*n^3+24*n^2-26*n)*x+(8*n^4+56*n^3+4*n^2-44*n+4)*x^2+(12*n^4+24*n^3-4*n^2-16*n)*x^3)*P(n-1,x) - (4*n*x+(12*n^2+20*n-1)*x^2+(4*n^3+36*n^2+32*n-2)*x^3+(8*n^3+24*n^2+16*n)*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[((8*n^2 - 4*n + (24*n^3 + 24*n^2 - 26*n)*x + (8*n^4 + 56*n^3 + 4*n^2 - 44*n + 4)*x^2 + (12*n^4 + 24*n^3 - 4*n^2 - 16*n)*x^3)*P[n - 1, x] - (4*n*x + (12*n^2 + 20*n - 1)*x^2 + (4*n^3 + 36*n^2 + 32*n - 2)*x^3 + (8*n^3 + 24*n^2 + 16*n)*x^4)*D[P[n - 1, x], x])/(2*n^2 + (6*n^3 - 3*n^2)*x + 2*n^4*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] = ((8*n^2 - 4*n + (24*n^3 + 24*n^2 - 26*n)*x + (8*n^4 + 56*n^3 + 4*n^2 - 44*n + 4)*x^2 + (12*n^4 + 24*n^3 - 4*n^2 - 16*n)*x^3)*v[i - 1] - (4*n*x + (12*n^2 + 20*n - 1)*x^2 + (4*n^3 + 36*n^2 + 32*n - 2)*x^3 + (8*n^3 + 24*n^2 + 16*n)*x^4)*deriv(v[i - 1], x))/(2*n^2 + (6*n^3 - 3*n^2)*x + 2*n^4*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