Real-rooted atlas

A112007 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 2\,x + n\right)\,P_{n}(x) = \left(3\,x^{2} - 7\,x^{3} + 2\,x^{4} + n - 7\,n x + 4\,n x^{2} + 8\,n x^{3} - 2\,n x^{4} + 5\,n^{2} x - 3\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} + 7\,x^{3} - 7\,x^{4} + 2\,x^{5} + n x - 4\,n x^{3} + 3\,n x^{4} - n^{2} x + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-4 + 10\,x - 9\,x^{2} + 2\,x^{3} - n - 3\,n x + 5\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x + 4\,x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1314
21410125
382102251210
456411004845612205
544281250887561908119127720
6390241502641545005456068102461405405
73814561911744271736414272202955502291450116756750

All 15 rows

Technical details
Node id
n-61a1938677a824c3 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-853f0e1aa93cb97b

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,         1;
  2:       14,        10,         1;
  3:       82,       102,        25,         1;
  4:      564,      1100,       484,        56,         1;
  5:     4428,     12508,      8756,      1908,       119,        1;
  6:    39024,    150264,    154500,     54560,      6810,      246,       1;
  7:   381456,   1911744,   2717364,   1427220,    295550,    22914,     501,      1;
  8:  4096800,  25786512,  48346488,  35502064,  11078000,  1466060,   74288,   1012,    1;
  9: 47967840, 368712864, 878901624, 860334440, 379182384, 76721960, 6858096, 235056, 2035, 1;
%F Formula
The row polynomials satisfy (n-2+2*x)*P(n,x) = -(n^3-n+(2*n^3-5*n^2+7*n)*x+(n^3-4*n-3)*x^2+(3*n^2-8*n+7)*x^3+(2*n-2)*x^4)*P(n-2,x) - ((n^2-n)*x+2*x^2-(n^2-4*n+7)*x^3-(3*n-7)*x^4-2*x^5)*d/dx P(n-2,x) + (2*n^2-n-4+(2*n^2-3*n+10)*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) = 3 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n + (2*n^3 - 5*n^2 + 7*n)*x + (n^3 - 4*n - 3)*x^2 + (3*n^2 - 8*n + 7)*x^3 + (2*n - 2)*x^4)*P[n - 2, x] - ((n^2 - n)*x + 2*x^2 - (n^2 - 4*n + 7)*x^3 - (3*n - 7)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (2*n^2 - n - 4 + (2*n^2 - 3*n + 10)*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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - n + (2*n^3 - 5*n^2 + 7*n)*x + (n^3 - 4*n - 3)*x^2 + (3*n^2 - 8*n + 7)*x^3 + (2*n - 2)*x^4)*v[i - 2] - ((n^2 - n)*x + 2*x^2 - (n^2 - 4*n + 7)*x^3 - (3*n - 7)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (2*n^2 - n - 4 + (2*n^2 - 3*n + 10)*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.