Real-rooted atlas

A116414 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-52\,n - 10\,n x + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-130 - 64\,x - 10\,x^{2} - 130\,n - 77\,n x - 10\,n x^{2} + n x^{3} - 13\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3} + 5\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-169 - 195\,x - 63\,x^{2} - 5\,x^{3} + 13\,n x^{2} + 10\,n x^{3} + n x^{4} + 13\,n^{2} x^{2} + 10\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1527
22220345
395138454282
44098204238051737
5176045043210952125610557
67573235802153788001800180763477
732585119534133425698601962530422458378324

All 15 rows

Technical details
Node id
n-e2eb2d94d1c8ba7b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5bb9b91dbe4d6d19

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:      5,       2;
  2:     22,      20,       3;
  3:     95,     138,      45,       4;
  4:    409,     820,     423,      80,       5;
  5:   1760,    4504,    3210,     952,     125,      6;
  6:   7573,   23580,   21537,    8800,    1800,    180,     7;
  7:  32585,  119534,  133425,   69860,   19625,   3042,   245,    8;
  8: 140206,  592100,  781815,  500800,  180050,  38220,  4753,  320,   9;
  9: 603275, 2882310, 4396950, 3336840, 1467375, 398034, 67620, 7008, 405, 10;
%F Formula
The row polynomials satisfy -(52*n+10*n*x-2*n^3*x^2)*P(n,x) = -(130*n+130+(13*n^2+77*n+64)*x-(5*n^3+5*n^2-10*n-10)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (169+195*x-(13*n^2+13*n-63)*x^2-(10*n^2+10*n-5)*x^3-(n^2+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[(-(130*n + 130 + (13*n^2 + 77*n + 64)*x - (5*n^3 + 5*n^2 - 10*n - 10)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (169 + 195*x - (13*n^2 + 13*n - 63)*x^2 - (10*n^2 + 10*n - 5)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-52*n - 10*n*x + 2*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] = (-(130*n + 130 + (13*n^2 + 77*n + 64)*x - (5*n^3 + 5*n^2 - 10*n - 10)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (169 + 195*x - (13*n^2 + 13*n - 63)*x^2 - (10*n^2 + 10*n - 5)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-52*n - 10*n*x + 2*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