Real-rooted atlas

A111125 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(30 + 12\,x - 6\,x^{2} - 20\,n - 8\,n x + 13\,n x^{2} - 9\,n^{2} x^{2} + 2\,n^{3} x^{2}\right)\,P_{n}(x) = \left(20 + 18\,x + 4\,x^{2} - 30\,n - 17\,n x + 4\,n x^{2} + 2\,n x^{3} - 5\,n^{2} x - 11\,n^{2} x^{2} - 3\,n^{2} x^{3} + 3\,n^{3} x^{2} + n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 50\,x - 29\,x^{2} - 4\,x^{3} - 10\,n x^{2} - 12\,n x^{3} - 2\,n x^{4} + 5\,n^{2} x^{2} + 6\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2426
31114328
42962304125
576230180525543
61997808553928062312
752125083555226472511479694
813647786135721114049251206154840155
9357123578488404925227975941418622009164701
109349700901683152017361405656060616415272025210670058
1124476205390561240780376645890341082118090267203807310112707392
126407959503218234902887712277257517377087357842124004122951503741210875545
131677611707864580177810314080112814758199132411658814513763588306095067764441343467067
14439204486408218143640357902324396445036397458211796348846400266805957640086999871252014172965804
1511498511376279055925265121248136165349665153750582101886015492931201759754746327608882611206001098560215685616194
Technical details
Node id
n-277de05a692d65ac · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f47845d34095612f

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:    1;
   2:    4,     2;
   3:   11,    14,      3;
   4:   29,    62,     30,      4;
   5:   76,   230,    180,     52,      5;
   6:  199,   780,    855,    392,     80,     6;
   7:  521,  2508,   3555,   2264,    725,   114,     7;
   8: 1364,  7786,  13572,  11140,   4925,  1206,   154,    8;
   9: 3571, 23578,  48840,  49252,  27975,  9414,  1862,  200,   9;
  10: 9349, 70090, 168315, 201736, 140565, 60606, 16415, 2720, 252, 10;
%F Formula
The row polynomials satisfy -(20*n-30+(8*n-12)*x-(2*n^3-9*n^2+13*n-6)*x^2)*P(n,x) = -(30*n-20+(5*n^2+17*n-18)*x-(3*n^3-11*n^2+4*n+4)*x^2-(n^3-3*n^2+2*n)*x^3)*P(n-1,x) - (25+50*x-(5*n^2-10*n-29)*x^2-(6*n^2-12*n-4)*x^3-(n^2-2*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(30*n - 20 + (5*n^2 + 17*n - 18)*x - (3*n^3 - 11*n^2 + 4*n + 4)*x^2 - (n^3 - 3*n^2 + 2*n)*x^3)*P[n - 1, x] - (25 + 50*x - (5*n^2 - 10*n - 29)*x^2 - (6*n^2 - 12*n - 4)*x^3 - (n^2 - 2*n)*x^4)*D[P[n - 1, x], x])/(-20*n + 30 - (8*n - 12)*x + (2*n^3 - 9*n^2 + 13*n - 6)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(30*n - 20 + (5*n^2 + 17*n - 18)*x - (3*n^3 - 11*n^2 + 4*n + 4)*x^2 - (n^3 - 3*n^2 + 2*n)*x^3)*v[i - 1] - (25 + 50*x - (5*n^2 - 10*n - 29)*x^2 - (6*n^2 - 12*n - 4)*x^3 - (n^2 - 2*n)*x^4)*deriv(v[i - 1], x))/(-20*n + 30 - (8*n - 12)*x + (2*n^3 - 9*n^2 + 13*n - 6)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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