Real-rooted atlas

A249251 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 - 2\,x + n + 2\,n x\right)\,P_{n}(x) = \left(16 + 24\,x + 8\,x^{2} - 5\,n - 4\,n x\right)\,P_{n-1}(x) + \left(13 + 29\,x + 20\,x^{2} + 4\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
133
2181230
38711436237
4378738468961680
5154840143825158424011211
66102197282509215672480057671970
7234219070214430612061255800135361344449721

All 15 rows

Technical details
Node id
n-31ecf072bbd1fb56 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-e2ee8f4a02068769

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:      3;
  2:     18,      12;
  3:     87,     114,      36;
  4:    378,     738,     468,      96;
  5:   1548,    4014,    3825,    1584,     240;
  6:   6102,   19728,   25092,   15672,    4800,     576;
  7:  23421,   90702,  144306,  120612,   55800,   13536,   1344;
  8:  88146,  397722,  759708,  795780,  493320,  180864,  36288,  3072;
  9: 326769, 1683294, 3752649, 4726248, 3675975, 1807056, 548016, 93696, 6912;
%F Formula
The row polynomials satisfy (n-1+(2*n-2)*x)*P(n,x) = -(5*n-16+(4*n-24)*x-8*x^2)*P(n-1,x) + (13+29*x+20*x^2+4*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(5*n - 16 + (4*n - 24)*x - 8*x^2)*P[n - 1, x] + (13 + 29*x + 20*x^2 + 4*x^3)*D[P[n - 1, x], x])/(n - 1 + (2*n - 2)*x)]];
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] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(5*n - 16 + (4*n - 24)*x - 8*x^2)*v[i - 1] + (13 + 29*x + 20*x^2 + 4*x^3)*deriv(v[i - 1], x))/(n - 1 + (2*n - 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.