Real-rooted atlas

A113214 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(10\,n + 9\,n x + n x^{2} + 5\,n^{2} x + 2\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(50 + 50\,x + 18\,x^{2} + 2\,x^{3} - 20\,n + 27\,n x + 23\,n x^{2} + 4\,n x^{3} - 15\,n^{2} x + n^{2} x^{2} + 2\,n^{2} x^{3} - 2\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(25 + 50\,x + 34\,x^{2} + 10\,x^{3} + x^{4} + 25\,n x + 40\,n x^{2} + 17\,n x^{3} + 2\,n x^{4} + 5\,n^{2} x^{2} + 6\,n^{2} x^{3} + n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2410317
372621458
4115878365188
518120225172556596
6292365706123207871852
74745013291844135553410585672

All 15 rows

Technical details
Node id
n-8102ee907cf8c943 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b4f2292904ff7bd1

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:   3,    2;
  2:   4,   10,    3;
  3:   7,   26,   21,     4;
  4:  11,   58,   78,    36,     5;
  5:  18,  120,  225,   172,    55,     6;
  6:  29,  236,  570,   612,   320,    78,    7;
  7:  47,  450, 1329,  1844,  1355,   534,  105,    8;
  8:  76,  838, 2928,  4988,  4745,  2622,  826,  136,   9;
  9: 123, 1534, 6189, 12508, 14640, 10476, 4613, 1208, 171, 10;
%F Formula
The row polynomials satisfy (10*n+(5*n^2+9*n)*x+(n^3+2*n^2+n)*x^2)*P(n,x) = -(20*n-50+(15*n^2-27*n-50)*x+(2*n^3-n^2-23*n-18)*x^2-(2*n^2+4*n+2)*x^3)*P(n-1,x) + (25+(25*n+50)*x+(5*n^2+40*n+34)*x^2+(6*n^2+17*n+10)*x^3+(n^2+2*n+1)*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[(-(20*n - 50 + (15*n^2 - 27*n - 50)*x + (2*n^3 - n^2 - 23*n - 18)*x^2 - (2*n^2 + 4*n + 2)*x^3)*P[n - 1, x] + (25 + (25*n + 50)*x + (5*n^2 + 40*n + 34)*x^2 + (6*n^2 + 17*n + 10)*x^3 + (n^2 + 2*n + 1)*x^4)*D[P[n - 1, x], x])/(10*n + (5*n^2 + 9*n)*x + (n^3 + 2*n^2 + n)*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] = (-(20*n - 50 + (15*n^2 - 27*n - 50)*x + (2*n^3 - n^2 - 23*n - 18)*x^2 - (2*n^2 + 4*n + 2)*x^3)*v[i - 1] + (25 + (25*n + 50)*x + (5*n^2 + 40*n + 34)*x^2 + (6*n^2 + 17*n + 10)*x^3 + (n^2 + 2*n + 1)*x^4)*deriv(v[i - 1], x))/(10*n + (5*n^2 + 9*n)*x + (n^3 + 2*n^2 + n)*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. [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
  3. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.