Real-rooted atlas

A102413 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 13\,x + 11\,x^{2} - 3\,x^{3} - x^{4} - 7\,n + 20\,n x - 16\,n x^{2} + 2\,n x^{3} + n x^{4} + 7\,n^{2} - 9\,n^{2} x + 5\,n^{2} x^{2} + n^{2} x^{3} - 2\,n^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(-2 - 17\,x - 21\,x^{2} + 13\,x^{3} + 3\,x^{4} - n + 6\,n x + 36\,n x^{2} - 10\,n x^{3} - 3\,n x^{4} + 5\,n^{2} + 17\,n^{2} x - 15\,n^{2} x^{2} - 3\,n^{2} x^{3} - 2\,n^{3} - 6\,n^{3} x\right)\,P_{n-1}(x) + \left(-x - 8\,x^{2} - 7\,x^{3} + 35\,x^{4} + 12\,x^{5} + x^{6} - n x + 36\,n x^{3} + 12\,n x^{4} + n x^{5} + 2\,n^{2} x + 12\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
256112
37189135
494040121102
5117012070151287
613108276268108181792
7151545327705041542112151

All 15 rows

Technical details
Node id
n-2d28f1dc96f16019 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-8759dd72caa62f4f

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,   1;
  2:  5,   6,    1;
  3:  7,  18,    9,    1;
  4:  9,  40,   40,   12,    1;
  5: 11,  70,  120,   70,   15,    1;
  6: 13, 108,  276,  268,  108,   18,    1;
  7: 15, 154,  532,  770,  504,  154,   21,   1;
  8: 17, 208,  912, 1808, 1744,  848,  208,  24,  1;
  9: 19, 270, 1440, 3678, 4860, 3438, 1320, 270, 27, 1;
%F Formula
The row polynomials satisfy -(2*n^3-7*n^2+7*n-2-(2*n^3-9*n^2+20*n-13)*x-(5*n^2-16*n+11)*x^2-(n^2+2*n-3)*x^3-(n-1)*x^4)*P(n,x) = -(2*n^3-5*n^2+n+2+(6*n^3-17*n^2-6*n+17)*x+(15*n^2-36*n+21)*x^2+(3*n^2+10*n-13)*x^3+(3*n-3)*x^4)*P(n-1,x) + ((2*n^2-n-1)*x+(12*n^2-8)*x^2+(2*n^2+36*n-7)*x^3+(12*n+35)*x^4+(n+12)*x^5+x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 5*n^2 + n + 2 + (6*n^3 - 17*n^2 - 6*n + 17)*x + (15*n^2 - 36*n + 21)*x^2 + (3*n^2 + 10*n - 13)*x^3 + (3*n - 3)*x^4)*P[n - 1, x] + ((2*n^2 - n - 1)*x + (12*n^2 - 8)*x^2 + (2*n^2 + 36*n - 7)*x^3 + (12*n + 35)*x^4 + (n + 12)*x^5 + x^6)*D[P[n - 1, x], x])/(-2*n^3 + 7*n^2 - 7*n + 2 + (2*n^3 - 9*n^2 + 20*n - 13)*x + (5*n^2 - 16*n + 11)*x^2 + (n^2 + 2*n - 3)*x^3 + (n - 1)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^3 - 5*n^2 + n + 2 + (6*n^3 - 17*n^2 - 6*n + 17)*x + (15*n^2 - 36*n + 21)*x^2 + (3*n^2 + 10*n - 13)*x^3 + (3*n - 3)*x^4)*v[i - 1] + ((2*n^2 - n - 1)*x + (12*n^2 - 8)*x^2 + (2*n^2 + 36*n - 7)*x^3 + (12*n + 35)*x^4 + (n + 12)*x^5 + x^6)*deriv(v[i - 1], x))/(-2*n^3 + 7*n^2 - 7*n + 2 + (2*n^3 - 9*n^2 + 20*n - 13)*x + (5*n^2 - 16*n + 11)*x^2 + (n^2 + 2*n - 3)*x^3 + (n - 1)*x^4)); 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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.