Real-rooted atlas

A144698 → D+1

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + n\right)\,P_{n}(x) = \left(-4 - 6\,x - 4\,x^{2} - 6\,x^{3} - 4\,x^{4} - 2\,n - 5\,n x - 9\,n x^{2} - 8\,n x^{3} - 2\,n x^{4} - 2\,n^{2} x - 4\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x - 2\,x^{2} + 2\,x^{4} + 2\,x^{5} - n x - 2\,n x^{2} + 3\,n x^{4} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(4 + 9\,x + 6\,x^{2} + 2\,x^{3} + 3\,n + 5\,n x + 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(x + x^{2} - 2\,x^{3} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 5 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1549
214451675
33325830564660
4721133299418452566300
51514330215002801010505102465520
631015249129000303490231090576454096740880
762951010690591267098035767151756986308705163849072000

All 15 rows

Technical details
Node id
n-510f2912b6542c00 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-68412ed67a0586d4

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:    5,      4;
  2:   14,     45,       16;
  3:   33,    258,      305,        64;
  4:   72,   1133,     2994,      1845,       256;
  5:  151,   4330,    21500,     28010,     10505,      1024;
  6:  310,  15249,   129000,    303490,    231090,     57645,      4096;
  7:  629,  51010,   690591,   2670980,   3576715,   1756986,    308705,    16384;
  8: 1268, 164973,  3423610,  20457419,  44052720,  37335907,  12623522,  1625445,   65536;
  9: 2547, 521538, 16082114, 142092790, 462940716, 621616478, 357424102, 87066666, 8454905, 262144;
%F Formula
The row polynomials satisfy (n+1+2*x)*P(n,x) = -(2*n+4+(2*n^2+5*n+6)*x+(n^3+4*n^2+9*n+4)*x^2+(3*n^2+8*n+6)*x^3+(2*n+4)*x^4)*P(n-2,x) - ((n+2)*x+(n^2+2*n+2)*x^2-n^2*x^3-(3*n+2)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n+4+(2*n^2+5*n+9)*x+(5*n+6)*x^2+2*x^3)*P(n-1,x) + ((n+1)*x-(n-1)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 4 + (2*n^2 + 5*n + 6)*x + (n^3 + 4*n^2 + 9*n + 4)*x^2 + (3*n^2 + 8*n + 6)*x^3 + (2*n + 4)*x^4)*P[n - 2, x] - ((n + 2)*x + (n^2 + 2*n + 2)*x^2 - n^2*x^3 - (3*n + 2)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n + 4 + (2*n^2 + 5*n + 9)*x + (5*n + 6)*x^2 + 2*x^3)*P[n - 1, x] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n + 1 + 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] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 4 + (2*n^2 + 5*n + 6)*x + (n^3 + 4*n^2 + 9*n + 4)*x^2 + (3*n^2 + 8*n + 6)*x^3 + (2*n + 4)*x^4)*v[i - 2] - ((n + 2)*x + (n^2 + 2*n + 2)*x^2 - n^2*x^3 - (3*n + 2)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n + 4 + (2*n^2 + 5*n + 9)*x + (5*n + 6)*x^2 + 2*x^3)*v[i - 1] + ((n + 1)*x - (n - 1)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n + 1 + 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.