Real-rooted atlas

A056241 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-2 + 4\,x - 18\,x^{2} + 14\,x^{3} - 4\,x^{4} + 3\,n - 17\,n x + 32\,n x^{2} - 22\,n x^{3} + 4\,n x^{4} - 2\,n^{2} + 18\,n^{2} x - 18\,n^{2} x^{2} + 8\,n^{2} x^{3} + n^{3} - 5\,n^{3} x + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-2 - 8\,x - 30\,x^{2} + 8\,x^{3} + 14\,x^{4} - 12\,x^{5} + n - 3\,n x + 31\,n x^{2} - 6\,n x^{3} - 38\,n x^{4} + 12\,n x^{5} + 16\,n^{2} x - 2\,n^{2} x^{2} - 14\,n^{2} x^{3} + 24\,n^{2} x^{4} + n^{3} - 5\,n^{3} x + n^{3} x^{2} + 12\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-4\,x - 8\,x^{2} - 4\,x^{3} - 8\,x^{4} - 4\,x^{5} - 8\,x^{6} - 2\,n x - 2\,n x^{2} - 18\,n x^{3} - 16\,n x^{4} - 16\,n x^{5} + 2\,n^{2} x - 6\,n^{2} x^{2} - 6\,n^{2} x^{3} - 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
245110
37189135
4114849141123
516105180105201427
6222015135011952711460
7293501232178511623293514923

All 15 rows

Technical details
Node id
n-c14d9c29051f3e6c · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3e4d9cd930a6e230

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,   1;
  2:  4,   5,    1;
  3:  7,  18,    9,     1;
  4: 11,  48,   49,    14,     1;
  5: 16, 105,  180,   105,    20,     1;
  6: 22, 201,  513,   501,   195,    27,    1;
  7: 29, 350, 1232,  1785,  1162,   329,   35,   1;
  8: 37, 568, 2618,  5212,  5027,  2380,  518,  44,  1;
  9: 46, 873, 5076, 13182, 17442, 12231, 4452, 774, 54, 1;
%F Formula
The row polynomials satisfy (n^3-2*n^2+3*n-2-(5*n^3-18*n^2+17*n-4)*x+(4*n^3-18*n^2+32*n-18)*x^2+(8*n^2-22*n+14)*x^3+(4*n-4)*x^4)*P(n,x) = (n^3+n-2-(5*n^3-16*n^2+3*n+8)*x+(n^3-2*n^2+31*n-30)*x^2+(12*n^3-14*n^2-6*n+8)*x^3+(24*n^2-38*n+14)*x^4+(12*n-12)*x^5)*P(n-1,x) + ((2*n^2-2*n-4)*x-(6*n^2+2*n+8)*x^2-(6*n^2+18*n+4)*x^3-(8*n^2+16*n+8)*x^4-(16*n+4)*x^5-8*x^6)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + n - 2 - (5*n^3 - 16*n^2 + 3*n + 8)*x + (n^3 - 2*n^2 + 31*n - 30)*x^2 + (12*n^3 - 14*n^2 - 6*n + 8)*x^3 + (24*n^2 - 38*n + 14)*x^4 + (12*n - 12)*x^5)*P[n - 1, x] + ((2*n^2 - 2*n - 4)*x - (6*n^2 + 2*n + 8)*x^2 - (6*n^2 + 18*n + 4)*x^3 - (8*n^2 + 16*n + 8)*x^4 - (16*n + 4)*x^5 - 8*x^6)*D[P[n - 1, x], x])/(n^3 - 2*n^2 + 3*n - 2 - (5*n^3 - 18*n^2 + 17*n - 4)*x + (4*n^3 - 18*n^2 + 32*n - 18)*x^2 + (8*n^2 - 22*n + 14)*x^3 + (4*n - 4)*x^4)]];
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 + x; for(n = 2, N, my(i = n + 1); v[i] = ((n^3 + n - 2 - (5*n^3 - 16*n^2 + 3*n + 8)*x + (n^3 - 2*n^2 + 31*n - 30)*x^2 + (12*n^3 - 14*n^2 - 6*n + 8)*x^3 + (24*n^2 - 38*n + 14)*x^4 + (12*n - 12)*x^5)*v[i - 1] + ((2*n^2 - 2*n - 4)*x - (6*n^2 + 2*n + 8)*x^2 - (6*n^2 + 18*n + 4)*x^3 - (8*n^2 + 16*n + 8)*x^4 - (16*n + 4)*x^5 - 8*x^6)*deriv(v[i - 1], x))/(n^3 - 2*n^2 + 3*n - 2 - (5*n^3 - 18*n^2 + 17*n - 4)*x + (4*n^3 - 18*n^2 + 32*n - 18)*x^2 + (8*n^2 - 22*n + 14)*x^3 + (4*n - 4)*x^4)); 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.