Real-rooted atlas

A121574 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-11 - 2\,x + x^{2} + 14\,n - n x - n x^{2} - 3\,n^{2} + 3\,n^{2} x\right)\,P_{n}(x) = \left(-27 - 38\,x - 17\,x^{2} - 2\,x^{3} + 79\,n + 76\,n x + 13\,n x^{2} - 24\,n^{2} - 12\,n^{2} x\right)\,P_{n-1}(x) + \left(-26 - 59\,x - 45\,x^{2} - 13\,x^{3} - x^{4} + 39\,n + 69\,n x + 33\,n x^{2} + 3\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2729
33522360
4155152454356
56428583997651980
6255043202757824115610572
79841202021643767561475162754880

All 15 rows

Technical details
Node id
n-2935343de73ad24f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9b575975b98d4265

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:      0;
  1:      1;
  2:      7,      2;
  3:     35,     22,      3;
  4:    155,    152,     45,      4;
  5:    642,    858,    399,     76,      5;
  6:   2550,   4320,   2757,    824,    115,     6;
  7:   9841,  20202,  16437,   6756,   1475,   162,    7;
  8:  37189,  89708,  88851,  46684,  14015,  2400,  217,   8;
  9: 138305, 383386, 447855, 287192, 110320, 25950, 3647, 280, 9;
%F Formula
The row polynomials satisfy -(3*n^2-14*n+11-(3*n^2-n-2)*x+(n-1)*x^2)*P(n,x) = -(24*n^2-79*n+27+(12*n^2-76*n+38)*x-(13*n-17)*x^2+2*x^3)*P(n-1,x) + (39*n-26+(69*n-59)*x+(33*n-45)*x^2+(3*n-13)*x^3-x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n^2 - 79*n + 27 + (12*n^2 - 76*n + 38)*x - (13*n - 17)*x^2 + 2*x^3)*P[n - 1, x] + (39*n - 26 + (69*n - 59)*x + (33*n - 45)*x^2 + (3*n - 13)*x^3 - x^4)*D[P[n - 1, x], x])/(-3*n^2 + 14*n - 11 + (3*n^2 - n - 2)*x - (n - 1)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(24*n^2 - 79*n + 27 + (12*n^2 - 76*n + 38)*x - (13*n - 17)*x^2 + 2*x^3)*v[i - 1] + (39*n - 26 + (69*n - 59)*x + (33*n - 45)*x^2 + (3*n - 13)*x^3 - x^4)*deriv(v[i - 1], x))/(-3*n^2 + 14*n - 11 + (3*n^2 - n - 2)*x - (n - 1)*x^2)); 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.