Real-rooted atlas

A144699 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,x + n\right)\,P_{n}(x) = \left(-6 - 13\,x - 18\,x^{2} - 17\,x^{3} - 6\,x^{4} - 2\,n - 9\,n x - 20\,n x^{2} - 14\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x - 7\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-3\,x - 5\,x^{2} + x^{3} + 5\,x^{4} + 2\,x^{5} - n x - 4\,n x^{2} + 2\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(7 + 16\,x + 11\,x^{2} + 2\,x^{3} + 3\,n + 9\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 5\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
16511
2176625108
3403815461251092
48716585364402662511760
518262733786460730279063125136080
637321906223143641380604827186186156251693440
775672797117623454983559045296553709112108667812522619520

All 15 rows

Technical details
Node id
n-778c27541fff7d28 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-c0dc793f2abea0c9

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:    6,      5;
  2:   17,     66,       25;
  3:   40,    381,      546,       125;
  4:   87,   1658,     5364,      4026,        625;
  5:  182,   6273,    37864,     60730,      27906,       3125;
  6:  373,  21906,   223143,    641380,     604827,     186186,      15625;
  7:  756,  72797,  1176234,   5498355,    9045296,    5537091,    1210866,     78125;
  8: 1523, 234258,  5757850,  41154470,  107595684,  112675150,   47782574,   7733946,   390625;
  9: 3058, 737793, 26773440, 280391664, 1096392780, 1796198082, 1283661792, 394865640, 48747426, 1953125;
%F Formula
The row polynomials satisfy (n+2+2*x)*P(n,x) = -(2*n+6+(2*n^2+9*n+13)*x+(n^3+7*n^2+20*n+18)*x^2+(3*n^2+14*n+17)*x^3+(2*n+6)*x^4)*P(n-2,x) - ((n+3)*x+(n^2+4*n+5)*x^2-(n^2+2*n+1)*x^3-(3*n+5)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n+7+(2*n^2+9*n+16)*x+(5*n+11)*x^2+2*x^3)*P(n-1,x) + ((n+2)*x-n*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 5*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 5*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 6 + (2*n^2 + 9*n + 13)*x + (n^3 + 7*n^2 + 20*n + 18)*x^2 + (3*n^2 + 14*n + 17)*x^3 + (2*n + 6)*x^4)*P[n - 2, x] - ((n + 3)*x + (n^2 + 4*n + 5)*x^2 - (n^2 + 2*n + 1)*x^3 - (3*n + 5)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n + 7 + (2*n^2 + 9*n + 16)*x + (5*n + 11)*x^2 + 2*x^3)*P[n - 1, x] + ((n + 2)*x - n*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] = 6 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 6 + (2*n^2 + 9*n + 13)*x + (n^3 + 7*n^2 + 20*n + 18)*x^2 + (3*n^2 + 14*n + 17)*x^3 + (2*n + 6)*x^4)*v[i - 2] - ((n + 3)*x + (n^2 + 4*n + 5)*x^2 - (n^2 + 2*n + 1)*x^3 - (3*n + 5)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n + 7 + (2*n^2 + 9*n + 16)*x + (5*n + 11)*x^2 + 2*x^3)*v[i - 1] + ((n + 2)*x - n*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

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.