Real-rooted atlas

row-factor cofactor of A101842 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,x + n\right)\,P_{n}(x) = \left(2 - 9\,x + 14\,x^{2} - 9\,x^{3} + 2\,x^{4} - 2\,n + 7\,n x - 12\,n x^{2} + 10\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x + 5\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(x - 5\,x^{2} + 9\,x^{3} - 7\,x^{4} + 2\,x^{5} - n x + 4\,n x^{2} - 6\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-5 + 12\,x - 9\,x^{2} + 2\,x^{3} + 3\,n - 7\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 4\,x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
256112
3123314160
427158144301360
55866112085306212520
61212502843971801791126120160
7248883351150780953708457752541181440

All 15 rows

Technical details
Node id
n-aa11a1baf2a96f75 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d762c6db5469cb8f

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:   12,    33,      14,       1;
  4:   27,   158,     144,      30,       1;
  5:   58,   661,    1208,     530,      62,       1;
  6:  121,  2502,    8439,    7180,    1791,     126,      1;
  7:  248,  8833,   51150,   78095,   37084,    5775,    254,     1;
  8:  503, 29718,  279310,  712994,  597360,  175882,  18122,   510,    1;
  9: 1014, 96693, 1413416, 5696608, 7862124, 4041506, 790072, 55944, 1022, 1;
%F Formula
The row polynomials satisfy (n-2+2*x)*P(n,x) = -(2*n-2+(2*n^2-7*n+9)*x+(n^3-5*n^2+12*n-14)*x^2+(3*n^2-10*n+9)*x^3+(2*n-2)*x^4)*P(n-2,x) - ((n-1)*x+(n^2-4*n+5)*x^2-(n^2-6*n+9)*x^3-(3*n-7)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n-5+(2*n^2-7*n+12)*x+(5*n-9)*x^2+2*x^3)*P(n-1,x) + ((n-2)*x-(n-4)*x^2-2*x^3)*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 - 2 + (2*n^2 - 7*n + 9)*x + (n^3 - 5*n^2 + 12*n - 14)*x^2 + (3*n^2 - 10*n + 9)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] - ((n - 1)*x + (n^2 - 4*n + 5)*x^2 - (n^2 - 6*n + 9)*x^3 - (3*n - 7)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n - 5 + (2*n^2 - 7*n + 12)*x + (5*n - 9)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 2)*x - (n - 4)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 2 + 2*x)]];
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 - 2 + (2*n^2 - 7*n + 9)*x + (n^3 - 5*n^2 + 12*n - 14)*x^2 + (3*n^2 - 10*n + 9)*x^3 + (2*n - 2)*x^4)*v[i - 2] - ((n - 1)*x + (n^2 - 4*n + 5)*x^2 - (n^2 - 6*n + 9)*x^3 - (3*n - 7)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n - 5 + (2*n^2 - 7*n + 12)*x + (5*n - 9)*x^2 + 2*x^3)*v[i - 1] + ((n - 2)*x - (n - 4)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 2 + 2*x)); 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

Same fingerprint ?

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.