Real-rooted atlas

A008292 → γ (scale 1/2) (shift +1) → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(1 + 4\,x + n\right)\,P_{n}(x) = \left(-2 - 10\,x - 12\,x^{2} + 8\,x^{3} + 16\,x^{4} + 5\,n + 21\,n x + 26\,n x^{2} + 4\,n x^{3} - 8\,n x^{4} + n^{2} - 2\,n^{2} x - 9\,n^{2} x^{2} - 6\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(2\,x + 14\,x^{2} + 36\,x^{3} + 40\,x^{4} + 16\,x^{5} + 2\,n + 13\,n x + 31\,n x^{2} + 32\,n x^{3} + 12\,n x^{4} + n^{2} + 4\,n^{2} x + 5\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-3 - 15\,x - 12\,x^{2} + 8\,x^{3} + 7\,n x + 10\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-1 - 7\,x - 14\,x^{2} - 8\,x^{3} - n - 3\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
2314
39413
43527466
515516234351
67911083384342292
745297656374449616425

All 15 rows

Technical details
Node id
n-a702c544a0c5fe42 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2d7d778b892c56a7

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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:      1;
  2:      3,      1;
  3:      9,      4;
  4:     35,     27,      4;
  5:    155,    162,     34;
  6:    791,   1083,    384,     34;
  7:   4529,   7656,   3744,    496;
  8:  28839,  58391,  37560,   8256,   496;
  9: 201939, 477086, 381810, 117216, 11056;
%F Formula
The row polynomials satisfy (n+1+4*x)*P(n,x) = -(n^3-n^2-5*n+2+(2*n^3+2*n^2-21*n+10)*x+(n^3+9*n^2-26*n+12)*x^2+(6*n^2-4*n-8)*x^3+(8*n-16)*x^4)*P(n-2,x) + (n^2+2*n+(4*n^2+13*n+2)*x+(5*n^2+31*n+14)*x^2+(2*n^2+32*n+36)*x^3+(12*n+40)*x^4+16*x^5)*d/dx P(n-2,x) + (2*n^2-3+(2*n^2+7*n-15)*x+(10*n-12)*x^2+8*x^3)*P(n-1,x) - (n+1+(3*n+7)*x+(2*n+14)*x^2+8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2 - 5*n + 2 + (2*n^3 + 2*n^2 - 21*n + 10)*x + (n^3 + 9*n^2 - 26*n + 12)*x^2 + (6*n^2 - 4*n - 8)*x^3 + (8*n - 16)*x^4)*P[n - 2, x] + (n^2 + 2*n + (4*n^2 + 13*n + 2)*x + (5*n^2 + 31*n + 14)*x^2 + (2*n^2 + 32*n + 36)*x^3 + (12*n + 40)*x^4 + 16*x^5)*D[P[n - 2, x], x] + (2*n^2 - 3 + (2*n^2 + 7*n - 15)*x + (10*n - 12)*x^2 + 8*x^3)*P[n - 1, x] - (n + 1 + (3*n + 7)*x + (2*n + 14)*x^2 + 8*x^3)*D[P[n - 1, x], x])/(n + 1 + 4*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - n^2 - 5*n + 2 + (2*n^3 + 2*n^2 - 21*n + 10)*x + (n^3 + 9*n^2 - 26*n + 12)*x^2 + (6*n^2 - 4*n - 8)*x^3 + (8*n - 16)*x^4)*v[i - 2] + (n^2 + 2*n + (4*n^2 + 13*n + 2)*x + (5*n^2 + 31*n + 14)*x^2 + (2*n^2 + 32*n + 36)*x^3 + (12*n + 40)*x^4 + 16*x^5)*deriv(v[i - 2], x) + (2*n^2 - 3 + (2*n^2 + 7*n - 15)*x + (10*n - 12)*x^2 + 8*x^3)*v[i - 1] - (n + 1 + (3*n + 7)*x + (2*n + 14)*x^2 + 8*x^3)*deriv(v[i - 1], x))/(n + 1 + 4*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.