Real-rooted atlas

A372873 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-60\,x + 120\,x^{2} - 32\,x^{3} - 2\,n + 148\,n x - 184\,n x^{2} + 32\,n x^{3} + 4\,n^{2} - 120\,n^{2} x + 64\,n^{2} x^{2} - 2\,n^{3} + 32\,n^{3} x\right)\,P_{n}(x) = \left(6\,x - 84\,x^{2} + 544\,x^{3} - 192\,x^{4} + 3\,n - 62\,n x + 516\,n x^{2} - 928\,n x^{3} + 192\,n x^{4} - 4\,n^{2} + 84\,n^{2} x - 624\,n^{2} x^{2} + 384\,n^{2} x^{3} + n^{3} - 28\,n^{3} x + 192\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(4\,x^{2} - 72\,x^{3} + 256\,x^{4} - 128\,x^{5} + 2\,n x - 48\,n x^{2} + 224\,n x^{3} - 256\,n x^{4} - 2\,n^{2} x + 40\,n^{2} x^{2} - 128\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
219414
301230850
402738816179
5003033024032632
60032411260624642192
7000561414431215681287478

All 15 rows

Technical details
Node id
n-4e27c7d8f83a8b6d · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-36781b99b303c661

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,  2;
  2: 1,  9,  4;
  3: 0, 12, 30,   8;
  4: 0,  2, 73,  88,   16;
  5: 0,  0, 30, 330,  240,   32;
  6: 0,  0,  3, 241, 1260,  624,    64;
  7: 0,  0,  0,  56, 1414, 4312,  1568,   128;
  8: 0,  0,  0,   4,  561, 6832, 13664,  3840,  256;
  9: 0,  0,  0,   0,   90, 4050, 28896, 40896, 9216, 512;
%F Formula
The row polynomials satisfy -(2*n^3-4*n^2+2*n-(32*n^3-120*n^2+148*n-60)*x-(64*n^2-184*n+120)*x^2-(32*n-32)*x^3)*P(n,x) = (n^3-4*n^2+3*n-(28*n^3-84*n^2+62*n-6)*x+(192*n^3-624*n^2+516*n-84)*x^2+(384*n^2-928*n+544)*x^3+(192*n-192)*x^4)*P(n-1,x) - ((2*n^2-2*n)*x-(40*n^2-48*n+4)*x^2+(128*n^2-224*n+72)*x^3+(256*n-256)*x^4+128*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 4*n^2 + 3*n - (28*n^3 - 84*n^2 + 62*n - 6)*x + (192*n^3 - 624*n^2 + 516*n - 84)*x^2 + (384*n^2 - 928*n + 544)*x^3 + (192*n - 192)*x^4)*P[n - 1, x] - ((2*n^2 - 2*n)*x - (40*n^2 - 48*n + 4)*x^2 + (128*n^2 - 224*n + 72)*x^3 + (256*n - 256)*x^4 + 128*x^5)*D[P[n - 1, x], x])/(-2*n^3 + 4*n^2 - 2*n + (32*n^3 - 120*n^2 + 148*n - 60)*x + (64*n^2 - 184*n + 120)*x^2 + (32*n - 32)*x^3)]];
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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 - 4*n^2 + 3*n - (28*n^3 - 84*n^2 + 62*n - 6)*x + (192*n^3 - 624*n^2 + 516*n - 84)*x^2 + (384*n^2 - 928*n + 544)*x^3 + (192*n - 192)*x^4)*v[i - 1] - ((2*n^2 - 2*n)*x - (40*n^2 - 48*n + 4)*x^2 + (128*n^2 - 224*n + 72)*x^3 + (256*n - 256)*x^4 + 128*x^5)*deriv(v[i - 1], x))/(-2*n^3 + 4*n^2 - 2*n + (32*n^3 - 120*n^2 + 148*n - 60)*x + (64*n^2 - 184*n + 120)*x^2 + (32*n - 32)*x^3)); 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.