Real-rooted atlas

A144436 → D

nonnegative coefficients

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

Recurrence

\(\left(3 + n\right)\,P_{n}(x) = \left(-64\,x^{2} - 16\,n x^{2}\right)\,P_{n-4}(x) + \left(-32\,x + 48\,x^{2} - 8\,n x - 16\,n x^{2} - 8\,n^{2} x^{2}\right)\,P_{n-3}(x) + \left(-16\,x^{2} + 16\,x^{3} - 4\,n x^{2} + 4\,n x^{3}\right)\,P_{n-3}'(x) + \left(-8 + 40\,x - 2\,n + 4\,n x + 6\,n x^{2} - 2\,n^{2} x - n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 4\,x^{2} - n x - 2\,n x^{2} + 3\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(10 - 6\,x + 3\,n + 4\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 8 + 2\,x\)
\(P_{3}(x) = 23 + 46\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
28210
32346372
4543401624560
51171636245446854680
6244651021672130201220642672
7499233941453291937725848529947424480

All 15 rows

Technical details
Node id
n-43d2f559facec194 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-9259c5061b0f9acf

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:    8,      2;
  3:   23,     46,       3;
  4:   54,    340,     162,        4;
  5:  117,   1636,    2454,      468,        5;
  6:  244,   6510,   21672,    13020,     1220,       6;
  7:  499,  23394,  145329,   193772,    58485,    2994,      7;
  8: 1010,  79120,  830922,  2053304,  1384870,  237360,   7070,     8;
  9: 2033, 257512, 4294812, 17688984, 22111230, 8589624, 901292, 16264, 9;
%F Formula
The row polynomials satisfy (n+3)*P(n,x) = -(16*n+64)*x^2*P(n-4,x) - ((8*n+32)*x+(8*n^2+16*n-48)*x^2)*P(n-3,x) - ((4*n+16)*x^2-(4*n+16)*x^3)*d/dx P(n-3,x) - (2*n+8+(2*n^2-4*n-40)*x+(n^3+n^2-6*n)*x^2)*P(n-2,x) - ((n+4)*x+(n^2+2*n-4)*x^2-(n^2+3*n)*x^3)*d/dx P(n-2,x) + (3*n+10+(2*n^2+4*n-6)*x)*P(n-1,x) + ((n+3)*x-(n+3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 8 + 2*x, P(3,x) = 23 + 46*x + 3*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 8 + 2*x;
P[3, x_] := 23 + 46*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(16*n + 64)*x^2*P[n - 4, x] - ((8*n + 32)*x + (8*n^2 + 16*n - 48)*x^2)*P[n - 3, x] - ((4*n + 16)*x^2 - (4*n + 16)*x^3)*D[P[n - 3, x], x] - (2*n + 8 + (2*n^2 - 4*n - 40)*x + (n^3 + n^2 - 6*n)*x^2)*P[n - 2, x] - ((n + 4)*x + (n^2 + 2*n - 4)*x^2 - (n^2 + 3*n)*x^3)*D[P[n - 2, x], x] + (3*n + 10 + (2*n^2 + 4*n - 6)*x)*P[n - 1, x] + ((n + 3)*x - (n + 3)*x^2)*D[P[n - 1, x], x])/(n + 3)]];
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; v[3] = 8 + 2*x; v[4] = 23 + 46*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(16*n + 64)*x^2*v[i - 4] - ((8*n + 32)*x + (8*n^2 + 16*n - 48)*x^2)*v[i - 3] - ((4*n + 16)*x^2 - (4*n + 16)*x^3)*deriv(v[i - 3], x) - (2*n + 8 + (2*n^2 - 4*n - 40)*x + (n^3 + n^2 - 6*n)*x^2)*v[i - 2] - ((n + 4)*x + (n^2 + 2*n - 4)*x^2 - (n^2 + 3*n)*x^3)*deriv(v[i - 2], x) + (3*n + 10 + (2*n^2 + 4*n - 6)*x)*v[i - 1] + ((n + 3)*x - (n + 3)*x^2)*deriv(v[i - 1], x))/(n + 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.